Hn flDemoriam
Xibraq? of
3onas
ffiernarfc
professor of
matbanson
p basics
Carnegie flnstitute of
o
CAMBRIDGE PHYSICAL SERIES
THE
ELECTRON THEORY OF
MATTER
CAMBRIDGE UNIVERSITY PRESS C.
F.
H0tttron:
CLAY, MANAGER FETTER LANE, E.G. 100 PRINCES STREET
lontnm: H. K. LEWIS, 136
GOWER STREET,
WILLIAM WESLEY AND SON, 3SerUtx:
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!Uip?t0:
F.
Toronto: J.
M.
W.C.
ESSEX STREET, STKAN:
ASHER AND CO, A. BROCKHAUS
$efolcrfe: G. P. Bomftftg rmtf (Talntlta:
28
PUTNAM'S SONS
MACMILLAN AND DENT AND SONS,
CO.,
I/n<.
I/rn.
Moo: THE MARUZEN-KAIiUSHIKI-KAISIIA
All rights reserved
THE
ELECTRON THEORY OF
MATTER BY
O. M.A.
W. RICHARDSON (CAMB.), D.Sc. (LoNi).), F.R.S.
WHKATSTONK PROFKSSOR OF PHYSICS AT
KING'S COLLRCK,
Cambridge at
:
the University Press
LONDON
PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS
PREFACE book
based on a course of lectures which I have
is
delivered
graduate students at Princeton University the last few during years. My aim has been to exhibit the extent to which the fundamental facts of physical science may be coordinated by means of the conception of the electron and to
the laws of electrodynamics. In developing the subject I have started from the most elementary beginnings, and I have therefore necessary to include much matter which is to be found in any ordinary text-book of the theory of electricity and magnetism. It is hoped that the lack of conciseness thereby involved may be
found
it
more than atoned appeal.
The
for
by the wider
circle to
which the book may
course of lectures at Princeton on which the book
is
modern founded proved useful as an introduction to the methods mathematical physics in addition to forming a presentation of the of
results of recent physical discovery.
The broad scope of the subject makes it imperative that a good deal of selection should be exercised as to the nature and treatment of the topics considered. In determining these, consideration has been given to importance, interest, and instructiveness, roughly in
the on lor named.
The necessary incompleteness
is
remedied to
references to scientific papers and to other works. These references are intended to supplement the discussion in the
some extent by
text rather than subject. tried,
to
exhibit the historical development of the are not referred to. I have
Thus many important papers
however, to be as accurate as possible in any statements specifically with historical matters.
which deal
For a variety of reasons the book
has, unfortunately, suffered
considerable delay in passing through the press.
I have, however,
some account of the important recent results while correcting the proofs; so that, with some reservations, the book may be regarded as fairly representing the found
it
possible to incorporate
PREFACE
vj state of the subject
to the
up
time of printing. The most serious with the electron theory of metallic
in the part dealing exception is
of Keesom important theoretical papers Leiden the Physical Laboratory, Supplement (Communications from Nos. 30 and 32 (1913)) and Wien (Columbia University Lectures 184 (1913)) as well as the full acand Berlin p. conduction, where the
Sitzungsber. (1913) count of Kamerlingh Onnes' experimental work at low temperatures
No. 139 (1014)) (Comm. Leiden, Supplement No. 34 (1913) and me in time to be dealt with. These papers load one
did not reach to
hope that the
difficulties
which beset the electron theory of may be overcome by the
metallic conduction in its usual form
Planck's theory of radiation. application of the ideas underlying In any event the theories of Chapters xvil and xviu should be valid at sufficiently high temperatures when the results of t/he
quantum theory
Many
coalesce with those of the continuous
other branches of the subject
possibly less aggravated, situation;
of atomic
are in a
similar,
theory.
though
amongst these the questions
i-he spectroscopic emission, X-rays and Al, the magnetic properties of bodies are conspicuous examples.
structure,
present time this field for the experimental
I
am
is
and
indebted to
unquestionably a very fruitful one the theoretical physicist.
)>o|,h
for
the
kindness of the publishers
of
[Jie
following journals and works for permission to reproduce various diagrams, viz. Annalen der Physik, Figs. 25, 30 a, 82 and 42; Journal de Physique, Figs. 35, 48 and 49 Philosophicul Mu<ju~im>, 52 and Lorentz' 56; Figs. Theory of Electrons, Fig. 55; Shirks' Prindpien der Atomdynamik, Figs. 53 and 54. ;
I wish to express
my
thanks to
Dr
C. J. Davisson of
Carnegie Institute, Pittsburg, and to Dr K. T. Compton of Reed College, Oregon, for the assistance I have received from notes they took of Editor of the
my
lectures, as well as to
Cambridge Physical
Mr
T. G. Bedford
Series, for his valuable help
suggestions in reading the proofs.
0.
KING'S COLLEGE,
May, 1914.
W.
RICHARDSON.
and
CONTENTS
.....
CHAP. I.
THE
ORIGIN OK THK ELECTRON THEORY
PAGE 1
ELECTRIC INTENSITY AND POTENTIAL
12
III.
DIELECTRIC MEDIA
39
IV.
TllK ELECTRON THEORY OF DIIOLECTRIC MEDIA
II.
.
.58
.
MAGNETISM
77
VI.
KLKCTROMAGNKTISM
86
VII.
ELECTROMAGNETIC WAVKH
V.
VIII.
115
DISPERSION, ABSORPTION AND SELECTIVE REFLEXION
.
142
.
FUNDAMENTAL EQUATIONS
IX.
TllK
X.
TlIE ACTIVITY OF THE FORCES
182 201
XL
CHARGED SYSTEM
XII.
CHARGE MOVING WITH VARIABLE VELOCITY
XIII.
THE AETHER
XIV.
THE
XV.
RADIATION AND TEMPERATURE
32G
XVI.
TliE THEORY OF MAGNETISM
3(51
XVII.
THE
XVIIL
piuNciPLi']
IN
UNIFORM MOTION
....
217
243 208
OF RELATIVITY
29(>
KINETIC THEORY OF ELECTRONIC CONDUCTION
.
TlIM EQUILIBRIUM THIOORY OF ELECTRONIC CONDUCTORS
.
398
.
441
TYPES OF RADIATION
470
XX.
SPECTROSCOPIC PHUNOMKNA
512
XXL
THE STRUOTUUM OF THK ATOM
XIX.
XXII.
GRAVITATION IN
DUX OF NAMKS
JN'DKX OF M11J.JKCTS
......
DID r >8,S
,
......... .
.
.
.
.
.
.
()(K)
.()()()
CHAPTER
1
THE ORIGIN OF THE ELECTRON THEORY The Electron Theory of atomic theory.
of Matter
It differs
may be looked upon as a form from the form of the atomic theory
with which chemistry is familiar, especially in that it makes the ultimate atoms minute geometrical configurations of electric
A
charge, instead of particles of uncharged matter. large number of different lines of inquiry, often closely interwoven, have led to
the adoption of such a view of the structure of matter. Of these lines of inquiry, however, three may be considered
different
pre-eminently conspicuous.
In the
first place,
although the electron theory has made most it is a logical development
rapid progress in the last two decades, of the views held a century ago by especially of the views to which .
Davy* and
Berzeliusf and
Faraday J was led by
chemical discoveries made somewhat
his electro-
Davy concluded, from phenomena known in his
later.
a general review of the electrochemical day, that the forces between the chemical atoms were of electrical
Shortly afterwards a complete system of chemical structure depending on the same idea was developed by Berzelius and, origin.
original form Berzelius's electrochemical theory was have much in common with insufficiently elastic, its main features The laws of electrolysis on the views modern the most subject.
although in
its
discovered by Faraday led to an important advance by pointing for electricity; for they distinctly to an atomic constitution
showed that each chemical atom invariably transported either a quantity of electricity or an integral multiple of that *
Phil.
Tram.
p. 1 (1807).
| Mtim. Acad. Stockholm (1812)
;
Nicholson'** Journal, voln. xxxiv.
(1818).
$ Kxp.
lies.
R. K. T.
5377, 5*23,
001, 713, 821 arid especially 852,
8(><).
and xxxv.
THE ORIGIN OF THE ELECTRON THEORY
2
which was determined by the chemical This inference from Faraday's electrolytic
a
multiple quantity, atom. valency of the
researches was strongly advocated
The second
line
much
later
of inquiry referred
to
by von Eelmholte*. dealt mainly with
would be impossible adequately to discuss at this stage the complex questions which present themselves in It may be permissible to recall that Maxwell's this conneetip. optical
It
phenomena.
had been found to account
satisfactorily electromagnetic theory behaviour of light, and electromagnetic waves in general,
for the
presented themselves when and dispersion, which depend upon phenomena These were transmission through material media, were considered. found capable of removal by introducing the simple hypothesis in free space;
but that
difficulties
like refraction
that the material media contained particles having appropriate natural frequencies of vibration. Maxwell's
electromagnetic theory
of light naturally suggested that these particles
were electrically charged, and the facts of dielectric polarisation were then found to fall into line, approximately at any rate, with the optical The phenomena. theory of the propagation of light in movingmedia
new
also
made important advances under the influence of the
A
views.
striking confirmation of the correctness
of the
general position was furnished by Zeeman's discovery of the change of the frequency of spectral lines when the emitting source was placed in a strong magnetic field. The magnitude of this change enabled an estimate of the ratio of the to the
charge mass of the particles to be made. The resulting values wens in substantial agreement with those which were obtained at about, the same time by different and more entirely
direct methods In the development of the theory along the lines just indicated f,hr ideas of H. A. Lorentz and J. Larmor f have
had a preponderating h
J
influence.
The
third line of attack was furnished by the experimenKi study of the phenomena accompanying the discharge of cleetrieit v through gases and especially of the properties of the cathode nvs the Roentgen rays and the rays emitted by radioactive sub^n,^' The matter described the ensuing paragraph will serve to
m
'*"*
"Faraday Lecture" '
(1881). ' '
,l.
Trans. A. vol.
OMXCT
.
p.
821 (18M
^ ,
'fBtetrov, Leipsfc"
^
'
Vm f
;
THE ORIGIN OF THE ELECTRON THEORY
3
a general idea of the kind of information which has been supplied by these researches.
The Isolation of
the
Elementary Charge.
The
electron theory may now be said to have developed far the beyond region of hypothesis. Discovery after discovery during the last fifteen years has established indubitably the existence of
a negative electron whose properties are independent of the matter originates. .J. J. Thomson* and Wiechertf, inthat the magnetic deflection of the cathode showed dependently, a in tube at a low pressure proved that they vacuum rays present consisted of negatively charged particles for which the value of e/m, where e is the electric charge and m the mass of one of the 7 About the same particles, was equal to about 1*8 x 10 E.M. units. time LorentzJ showed that Zeeman's discovery of the shift of the spectral lines of an emitting gas, produced by a magnetic
from which
it
pointed to the existence, within the atom, of negatively charged particles which had approximately the same value of e/m.
field,
Now
the value of e/m for the lightest known chemical atom, the of hydrogen, can be obtained very accurately by electrolytic
atom
found to be equal to 9 '5 7 7 x 10 3 E.M. units. would follow that if the charge carried by a cathode
experiments and
Hence
it
is
ray particle or by the particles which emit the spectral lines were identical with the charge carried by a hydrogen atom in electro-
we should
rather expect would be the case from Faraday's electrolytic experiments, then, the mass of these particles must be very much less than that of a hydrogen, atom. lysis, as,
indeed,
This C. T.
was soon put to the test of experiment. had shown that when moist ionised gas is sudden expansion, a cloud forms and the drops of
question
Wilson
R.
1
subjected to
1
water condense on
X"
|
in
preference
to
the
uncharged
Phil.
J. J.
May. V.
vol. XLIV. p.
298 (1897).
Verhandl. der Physik.-okon. (resellach. zu K$mg*berg.i. Pr. (1897). Phil. Mag. V. voi. xun. p. 232 (1897). XittinyxveritL der
||
ions
Thomson 11 utilised this phenomenon in order to the number of ions in a volume of gas containing a measured
molecules. count;
the
Akad. van Wet.
Roy. Hoc. Proc. March 19, 1890.
te
Amsterdam,
vol. v. pp. 181, 2-i2 (189(">).
THE ORIGIN OF
4
ai^j./^
inn,
The amount is as follows Stated briefly the method cloud the in may be the whole of the drops rf water condensed on he of supersaturate produced by calculated from the degree the for formula of Stokes s The :
totalcharge.
|
application
known expansion. rate of
fluid gives the average size of of a sphere in a viscous number n of the these data determine the total
fall
Thus
each drop.
that practically all the assumed, for sufficient .reasons, total charge ne on all The ion. one and only one drops contain out of the chamber them the ions could be determined by sweeping the strength an electrometer before the expansion took place, It
drops
is
into
same as in the condensation Thus the charge on a single ion was obtained by in this way Thomson showed that the
the of the source of ionisation being experiments.
Proceeding
division.
negative
ions liberated in air
from radium each carried the
by Roentgen rays and by the ft rays same charge as the hydrogen ion in
electrolysis.
ions investigated in these experiments are rather cated structures and are not identical with the electron.
The
compliIn the
produced by ultra-violet light falling on a was shown by Thomson that the particles when first, These emitted have the same value ofe/m as the cathode rays. case of the ionisation
metal
it
would not be likely to aggregate together in the presence of gas Wilson* showed that the negative ions molecules, and C. T.
R
from ultra-violet light behaved exactly like those from the other The inference ionising agents in his condensation experiments.
from these experiments therefore is that the particles which form the cathode rays and which are emitted during photoelectric action carry a charge equal to that of the
hydrogen atom in Experiments by Townsendf, on the rate of fall of the clouds produced when the gases evolved from chemical actions in the wet are allowed to bubble through water, occurring way had previously led him to conclude that the ions present in such gases carry the same charge as a hydrogen ion in electrolysis.
electrolysis.
This conclusion has been strengthened by other methods of determining the charge on an electron. One of these depends on the theory of the radiation of electromagnetic energy from hot, bodies.
The theory *
of this
Phil Trans. A.
method vol.
will
be considered in the
cxcn. p. 403 (1899).
t Phil Maq. Feb. 1898.
THE ORIGIN OF THE ELECTRON THEORY
5
One of the recent methods, which is due. to Rutherford f and Geiger, depends upon the properties of radioactive substances. These are found to emit positively charged bodies, called a particles, which carry twice the charge e of an electron and are able to sequel*.
produce a large number of new ions when they pass through a gas. By magnifying this secondary ionisation by means of an auxiliary electric field and also using a very sensitive electrometer, Rutherford
was able to detect the ionisation produced by a single a particle. When a very weak radioactive preparation was used the a particles were emitted at times separated by rather wide and irregular intervals, and as the effect produced by each one separately could be detected, the number emitted by a given amount of the radioactive substance in a given time could be measured. The only other datum which is required to measure e is the quantity of positive electricity which is carried away from the same quantity
of the
preparation by the a rays. obtained by other experiments.
The
This had previously been
drop method has recently been improved by H. A. Wilson]: and R. A. Millikan. The former showed that the charge on the drops could be deduced from the rate of fall under falling
gravitation combined with different electric fields, without makinguse of the degree of supersaturation whilst the latter showed how ;
the drops of water could be replaced by drops of a non- volatile oil. The drops of oil have the great advantage that they do not and by allowing a sufficient number of electrons to evaporate :
combine with them and applying a supporting electric field which just balances the gravitational force, they can be kept under observation for an indefinite length of time. In this way Millikan has shown that the method is capable of yielding results of very groat precision. All the throe methods bust mentioned are quite accurate and exhibit an excellent agreement. It is claimed that the charge e on an electron is known to within 1 per cent. Millikan's|| latest value is = 4r<Sl x 10~ 10 E.s. unit or 1/60 x 10~"JO E.M. unit. *
See chap. xv. t Roy. Soc. Proc. A. Phil.
Mag. VI.
vol. LXXXI. pp. 141,
vol. v. p.
429 (1903).
163 (1908).
THE ORIGIN O*
Application
to the
Atomic Theory.
other important consequences. These experiments have led to of an element is one gram atom Since the charge carried by and the charge from electrolytic experiments, accurately known shown to be been has ion a monovalent electrolytic carried
by
the number of atoms that of a gaseous ion, it follows that equal to to the same degree known atom of any substance is in one
gram
of accuracy as
e.
Since the charge which
by one grain 9*649 x 10 E.M.
carried
is
:{
atom of a monovalent element in electrolysis is one gram atom of any units it follows that the number of atoms in element
is
23 6'02 x 10
to liberate Also, since the charge required C. and 760 mms. is 04327 101. at
.
half a cubic centimetre of unit, it follows that
the
H
2
number
of molecules in one cubic centi-
metre of any gas under standard conditions of temperature and pressure
is
1'60
---.
These values are in agreement with the comparatively inaccurate estimates which had previously been given by methods bused on the kinetic theory of gases and other considerations.
Millikan was also able to observe the changes produced by the combination of single ions with the drops. These experiments, as well as those of Rutherford with the a rays, furnish a very direct. and convincing proof of the atomic theory of matter arid electricity. The consequences of the atomic theory of matter have recently been strikingly verified by experiments in other directions.
Perrin* has shown that the irregular motions of minute suspended in fluids are in accordance with the requirements of the particles kinetic theory of gases.
determination of the
A
study of these motions also leads i<> a number of molecules in one grain 'molecule of
any element and thus to a determination of
by Perrin
The value
e.
<
>btai ne<
1
in agreement with the other recent determinations. An examination of the distribution of velocity and kinetic energy among the electrons emitted by hot bodies, which has been carriell is
out by the writer, partly in collaboration with F. C. Brown f, has shown that the motions of these electrons are in very close accordance with those required by Maxwell's theory, for the molecules
Annals
dp
a him.
o+
Ao
r>Jw,
innrv
i
>->,.,
--
THE ORIGIN OF THE ELECTRON THEORY
/
Electromagnetic Inertia.
About thirty years ago J. J. Thomson* pointed out the extremely important result that an electric charge possessed inertia simply in virtue of the energy of its electromagnetic field, and he succeeded in calculating the magnitude of the
electrical
inertia, or electromagnetic mass, as it is usually called, of a charged This additional mass does not play any important part in sphere. ordinary electrostatic experiments, as it is always small compared
with the mass of the uncharged portion of the conductors and insulators
is
which are
experimented
with.
The
case
is
very
when we
are dealing with charged particles whose mass one only eighteen-hundredth of that of an atom of hydrogen. It
different
was obviously an important experimental problem to determine how much of the inertia of the electron was of the type foreseen by Thomson and how much, if any, was to be attributed to "ordinary" mechanical mass. Fortunately the two kinds of mass differentiate themselves rather clearly. The mechanical mass is supposed to be independent of the velocity of the body, following the principles of mechanics laid
down by Newton, whereas the electromagnetic
function of the velocity and approaches as the infinity velocity of the electric charge approaches that of
mass
a continuous
is
light.
The experimental problem was resolved by Kaufmann-f, who measured the value of e/m for the electrons emitted by radium 10 bromide, some of which have velocities as high as 2'89 x 10 centimetres per second. He showed that the mass of these electrons varied with the speed, and in fact in a manner very similar to that predicted by Thomson. His final conclusion was that if there was any part of the mass of an electron which was ordinary mechanical mass it was very small in comparison with the part which was of electromagnetic origin.
The Negative Electron.
We
have now succeeded
whose mass *
Phil.
is
May. V.
much vol. xr. p.
t Ann. dcr Phus. vol.
less
in the isolation of a
than that
229 (1881). 4H7 (11)06):
xrx. p.
cf.
of any
also
charged particle
known chemical
H. Starke, Ver. der
I)entsch.
this
mass arises from the
charge the particle carri this is the negative electn structure of the positive electricity which goes to make electric
There can be no question but that
The
the remainder of the uncharged chemical atom
is
still
uncertai
but the results of experiments so far point to both the chai and mass of the positive electrons being different from th<
of the negative. There is no evidence, so far as the wri able to observe, which supports the view that the posit
is
electron
The
is,
as it were, a reflection of the negative.
foregoing considerations enable us to define more precis In future we shall restrict the te
the use of the word electron. to
particles
which
consist
of a
geometrical
configuration
electricity and nothing else, whose mass, that is, is all elect magnetic. For a particle which is a charged molecule or at< that is to say, a molecule or atom which has lost or gained one more negative electrons, we shall use the term ion. A wi meaning than this is currently attributed to the word ion, in
sense of any charged particle which is considered to have a separ existence or which behaves as a dynamical unit. Some of th are comparatively large bodies and contain very many atoms molecules.
To
distinguish
them from the smaller
ions alre;
Stark has suggested the use of the terms molion atornion. As, however, we shall not have to consider the la molions we shall simply use the word ion instead of atomic
referred
to,
;
According to the view we are developing,
all
interact!
between material systems result from the electrical char which make up their ultimate parts. The space in the nei bourhood of an
electric
charge
is
to
be looked upon as hav
properties different from that some distance away, since an elec charge of the same sign is repelled wifch a greater force in
former case than in the
by saying that the
latter.
This state of things is descri is surrounded by a field
electric charge
force.
It
is often convenient to attribute this field of force to turbances produced by the electric charge in a medium, the aet which fills all space. Looked at in this way the real electron, part which acts, is the surrounding aether which is outside geometrical boundary and the electron theory is the science ;
O
7
Jr
Jr
edifications.
Different Elements of Electricity.
Our ignorance
of the geometrical distribution of the electri-
ation constituting an electron is almost complete, but this is not serious disadvantage in considering many applications of the
be sufficient to regard an electron as a point In such cases a inertia coefficient or mass. definite arge having e mode of distribution of the electric charge, whether it is a eory.
It will often
int, line,
surface or volume charge and whether
it is
distributed
th spherical or linear symmetry or not, is unimportant, provided at it is confined to a minute region of space. Although this is ;en true there are some investigations for which the ultimate
ometrical distribution of the electrification
is
important
;
as for
stance in the case already mentioned of the calculation of the ictromagnetic mass. These two distinct classes of cases require
a rule quite different methods of attack.
These remarks will make
clear why it is necessary to have of electricity in different investielementary portions In the first place we ,tions dealing with the electron theory. on or arising from a forces have the consider to acting ay it
fferent
The elementary quantity of portion of the electron itself. be denoted concerned here by p dr, where p is the may sctricity lall
lume density of the nsideration and dr dtesiinal in
electricity at the point of the electron under is an element of volume of the latter, in-
comparison with the
size of
the electron.
In other
vestigations our element of electricity will be the charge on a igle electron whose value e is determined by the equation
,-f]f P iere the
*r,
volume integral extends over the volume occupied by In still another class of investigations the volume
e electron.
sment of electric charge will occupy a region of space containing enormous number of electrons both negative and positive. In L
.is
conveniently represented by p dr, or If n likely to occur with the first case by p dr.
case also
nfusion
is
it
is
when is
the
imber of negative electrons per unit volume at any point and and is the charge carried by each, being the corresponding lantities for the positive electrons, then p dr = (NE -f ne) dr.
N
E
structure of the electrification
and from the
electronic structure
c
be observed that each of th ihree elements of electricity is of a successively higher order c magnitude, both as regards distribution in space and in referenc
matter are smoothed out.
It will
to the quantity of electricity it contains.
Force between Electrically Charged Masses.
On
between material bodie carrying electric charges is to be pictured in a manner somewh^ different from that usual in electrostatics. The forces act on eac single electron present in the two bodies, irrespective of whethc it
the electron theory the interaction
may be
whether
regarded as forming part of the free electric charge, simply one of the constituent electrons in the matt*
c
it is
carries the electric charge. The field of force arising froi a single electron is assumed to obey the same laws as that arisirj from a small charged particle in the theory of electrostatics.
which
Let us consider the forces acting between two charged materi P and Q at a distance r apai that the matter at P of JVj positive electrons consists Suppose
particles situated at the points
charge
E
l
%
and
negative electrons of charge e l9 the matter at
being composed of N.2 positive electrons of charge JS2> and n s neg; tive electrons of on Q or vi< charge e%. The force exerted by
P
versa will consist of the algebraic sum of the repulsions betwe* all the like electrons and the attractions between all the unlil electrons.
The repulsion
1.
at
Q
P.
be made up of the four items which follov Any one electrc 2 on each individual E^^/r' positive electron
It will thus
exerts a force
The
total force
and acting on
P
of the positive electrons.
to all the electrons at 2.
3.
Q
The repulsion
amounts
to n^Zo
The
amount
1J
r
E 1 J52 /r 2
N.N,^^ r"
;
electron at
hence the force
di
.
negative
electrons.
positive
electrons
This
clear
.
J u~~
2
1?
r
JV
r2
attraction
to
is
be
of the
negative electrons at Q. to
way from each
arising in this
will therefore
of the
This
is
at
P
for
t
readily seen by similar reasoni]
THE ORIGIN OF THE ELECTION THEORY The
4.
of the
attraction
positive
electrons
at
11
Q
for
the J?
negative electrons at P.
Thus the
This
total repulsion
between
-
evidently equal to
P
and Q amounts
to
,,
,.2
,
l
is
= (N.E, + n, e ) (N*Ez 4- n x
If g ]? q% represent the
P and
Q
magnitude and sign of the
2
free electricity a1
respectively, then
^==^^4-^1^1 and
q2
NE
2
2
+n
2 e*,
so that the repulsive force is equal to jija/r3 the usual electrostatic law. ,
This result
may
easily
be extended to the general case where same sign provided
there are different kinds of electrons of the
with different charges. If
NE l
1
=n
l
e1
and
N E. 2
2
n 2 e 2) the bodies are uncharged
and
them vanishes. Thus the above formulation any room for the explanation of gravitational
the force between
does
not leave
attraction
between uncharged material
considered in Chapter xxii.
particles.
This lacuna is
CHAPTER
II
ELECTRIC INTENSITY AND POTENTIAL That branch of electrical science which deals with the properties of electrical charges when at rest is called electrostatics. It is the oldest branch of electricity, some of the fundamental phenomena of frictional electricity having been "
ancients.
By
another.
We shall see
charges
at rest
"
known
qualitatively by the at rest relatively to one no evidence for the view that
we mean
that there is the absolute motion of the charges affects their action on one another. When the charges move relatively to one another, important differences are observed which will be considered later.
From
the point of view of electrostatics the most interesting
properties of electrified bodies are their mutual repulsions and attractions. It is found that all electrified bodies can be grouped into two classes such that all the bodies of either class repel all the other bodies in the same class but attract all the bodies in the
other class.
A
body is said to be positively or negatively electrified to the class to which it belongs. The distinction is not according one of we as shall see in the sequel, there are merely sign, since, important qualitative differences between positive and negative electricity.
In the previous chapter we have stated that the force between two charged bodies of sufficiently minute size is proportional to the product of their charges divided by the square of the distance between them. This law of variation of force with distance was discovered by Priestley in 1767 and rediscovered by Coulomb in 1*785*
Wft
mav
F=
pp
'
express
it,
in
t.he
form
lc.
f
wh^rp k
is
a.
uoutibuu
wnust;
nua.iiiiJ.Ly
uicigiJULUuutJ
unit of electric charge. shall assume that k
e 5
uepeiius
In dealing with is
uii
uur ue 11111 UJLUIJL
electrical
a universal constant
ui
phenomena
independent
the sign or other quality of e or e', although we shall see that are to account for gravitation on the electron theory this
we
The omission will, however, make no practical the case of purely electrical effects: it is only len gravitational effects are concerned that the difference is nnot be the case. in
Bference
iportant.
The magnitude
w we
of our unit of electric charge will depend on determine the constant k which is of unknown dimensions
ice the
dimensions of
e
are
The dimensions
unknown.
of Fr* are
We
course perfectly determinate and equal to ML*T~*. shall fix .r units are the if the that convention separated charges only by
F is in dynes
ordinary space, then
'
=
This
1/4-7T.
is
and r in centimetres provided
equivalent to defining our unit of electric
charge
that which repels an equal and similar charge at unit distance th a force of 1/4-Tr dynes. This unit of charge was introduced
Heaviside and differs from the ordinary electrostatic unit which The akes k = 1 and which we made use of in the last chapter. r
iw unit has certain
raiulae
which we
advantages in improving the symmetry of
shall obtain later.
As the dimensions of k suppress
it
are
unknown, it is sometimes inadvisable we have given it a definite
in our formulae even if
particularly true in investigations of a fundamental character. In such cases it is convenient to write
imerical value. :ry
1
ee'
This
where
K
is
1
has the value -r- on the ordinary J electro4-7T
system of units, the value unity on Heaviside's electrostatic is stem and may take other values on other systems of units.
atic
K
metimes called the
dielectric
coefficient
or specific inductive
,pacity of the aether.
In dealing with electrostatics it is not necessary for us to itermine how it comes about that two electric charges attract or Two entirely different attitudes towards this pel one another. id the cognate question in regard to gravitational attraction have ;
3en
adopted
lopting the
w
of force
by
different
dogma
schools
of
of "action at a
between charges
is
thought.
One
school,
distance," holds that the
the fundamental thing and that
it
attempt to go beyond it. The other denies and derives the law of fo )ossibility of action at a distance charges from the effect of the charged bodies on useless to
s
1
'
the aether. There is no a priori reason one view rather than the other, although most of at xvestigators have been ranged against the action
g medium,
i
The great advantage of the medium view is t" the operation of a mechanism whose consequen ^6i> on foretelling themselves, whereas the other is mere d< For this reason the medium view has been m escription. school.
*es
iccessful in leading the great advances in electrical science, wh le strength of the action at a distance formulation lies in
.athematical
There is no absolute contradict Maxwell has shown that a system
simplicity.
itween the two views
;
^ossible stresses in the medium will give rise to the obser attractions and repulsions. Which is the more desirable is larg a matter of taste or convenience. This is particularly the case far as electrostatic
phenomena
When we
are concerned.
come
the consideration of electromagnetic phenomena we shall see t the medium view possesses important advantages, in cert directions, at
any
rate.
Electric Intensity.
We charge
is
now suppose
that the space surrounding an elec We do not need different from that elsewhere.
shall
how this is brought about. It may be that the cha produces a change in the state of the surrounding aether or charge may have parts which extend into the region about it consider
;
:
may be merely a manifestation of a hyperspatial mechanis or it may even be something which is incapable of descriptioi mechanical terms. The important point is that if another cha is placed at any point of such space it will be acted on by a ft it
and
accelerated.
The
force
acting on this
second charge
E de magnitude mined by the first charge. This is true provided that there no other charges in the field. In that case E will be compoun of the effects due to the various charges other than e. The ve< proportional jointly to its
E
is
e
and
to a vector
called the electric intensity at the point of the field ur
consideration.
WIM&I is Known as a vector poim/ lunciiou a function which for each point of space has )th magnitude and direction. function which has only one be called a single- valued point at each point may lagnitude iiuicuoii
L
other words
Li
it
is
,
is
A
The
inction.
electric intensity
E is single- valued in this sense where E vanishes.
;
at its direction is indefinite at the points
Clearly the electric intensity at a point distant r from a point = e'/^Trr2 This follows from the large e' is in Heaviside's units law of verse square force. The electric intensity due to a
E
.
L
>mplicated distribution of charge may obviously be obtained by le integration of the amount arising from each volume element, calculation it is necessary to integrate for each electric intensity separately and combine the the of >mponent isults according to the rule for the composition of forces. This i
making the
solution and subsequent composition of vectors is often trouble>me as well as clumsy, and it is not necessary for the calculation
the electric intensity.
'
may be dispensed with by the known as the Potential.
It
jction of another function
intro-
The Potential.
The Potential
is
defined as the work divided
hen an infinitesimal
electric
charge
is
by the charge brought from some stanThe standard position is
ard position to the point in question. sually taken to be a point at an infinite distance
away from
larged bodies. The value of the potential calculated in this way Lust be independent of the path of approach to the point under msideration, otherwise an indefinite amount of work could be Dtained by
making the charge move round a
closed
contour
assing through the point under consideration and the standard This would be contrary to the law of conservation :>sition. ? The potential is a function of each point in space and energy. Such a function is known assesses magnitude but not direction. 3
a scalar point function.
The
electrostatic potential is single-
allied.
Let
P
ich that '
1
at P. to
}
Q be two points at an
PQ
is
Let
dV
Q and
let
infinitesimal distance d$ apart
and
in the direction of the resultant electric intensity be the increase in the potential in passing from
the direction
P
>
Q be
considered positive.
Then
ELECTRIC INTENSITY
lfi
T
:-
,
- AW*.
*. that
A'- -
J
AND POTENTIAL
The value
of the
component of
whose inclination v wfc'iMtv at P in any other direction be less than
.fa
It
,
.
i,,,K *,i
!*
*C >i
/'
;
to
#
always J,-coB0. This laid off in the direction infinitesimal length an A*.***
7
will
t*
iil
with Pft we shall have
He angle
E. =
-
~
It is clear
the resultant electric intensity is *fe that the direction of diminution of F. Starting out from i*f ciuickest in the direction of the resultant 11-
off* length
hy
PQ
in the direction of the a moMty at >, then from Q length Q In this from on so and at point to point. Q .'*.)iuiit iiiMwity the that such in line tangent to space ** skill draw a curved
H4i
run
*[, *
at
any point
The curve
that point.
!*v*n<* intt unity at
the direction of the resultant
will give
will also
represent the
which moves under the ui'h
It
at tJvliivetiii cosines of the resultant electric intensity
EX =IE, Ey = mA Ez = nE, T
p
,:nt
an L /,
H,
we
shall
have
,
any where
If ds = (dx, dy,dz) is the' A, are the components of E. rkuiriit uf are of the line of force at the same point, A",,
AV
.
1
These
;ire
dz n= y-, m- dy -f,
dx
=^
.
the differential equations of a line of force.
an electric charge moves always at right angles to the lines iuree, no work will be done on it, so that all the points on the If
*f
Mir&ee on which
it
traced nut in this
moves
way
is
will
be at the same potential.
called
A surface
an equipotential surface.
It is
vitar that the equipotential surfaces
normally and that the whole field "i
prismatic
surfaces
In general
j-im
by means of a system of equipotential a series of tubular surfaces
intersecting orthogonally
remaining the
P
cells
always cut the lines of force may be divided up into a series
as
lines of force.
The
latter surfaces are called
tubes
we may write the value of the potential at any where E is the resultant electric
V = -lEcQsQds,
ELECTRIC INTENSITY AND POTENTIAL
E
angle that
makes with
infinity to the point P.
ds.
If
I,
The limits of integration are from m, n are the direction cosines of ds
E cos 6 = IEX mEv + nE
we have
4-
F= -
I
(IEX
l-
17
Zl
so tha-t
mEy + nEz ) ds = -\ (Ex dx
taken between the same
For a point charge
limits.
e the electric !
intensity at a point distant r is radial and equal to e .$nrr~\ so that the potential is e/^irr, as is clear upon integrating along a
V
radial path from infinity to the point
The
P.
following considerations enable us to find the value of the due to a number of Y
V
potential
point charges eQ ,e 1 e.2 sider the intensity at
,
,
etc.
Con-
Q (a,
y, z)
arising from one of the charges e Q at (a & c ). The resultant
P
,
intensity TV*
is
where
.
- ao) + (y - 6 )
=
3
The x component, #Q
X
2
Q3
4-
(z
~
GO)*
of this
is
X Fig.
1.
Similarly the x component of the intensity which arises from any of the other charges e n distant rn from x, y, z may be written
But the x component charges
X
of the electric intensity
g,^
So that the potential 4>7T
2
e ,
to all the
'
2
1
1
due
is
and
r
is
V
^4^7^
47T?*!
47rr2
/
d%
due to a number of point charges
to the equal ^
sum
of the potentials
due
is
to the
separate charges.
We may
now deduce the
differential equation
by the potential. Consider again the potential harge e Q at P. Since
<
which
F
at
is satisfied
Q due to the
ELECTRIC INTENSITY
is
lav
A
It
*,.:;!*
'
charges ^ Js
f
e,,
*h-
W*
etc.
+
Q
that due to
is
J^ +
It is therefore equal to j-^-~
thus find that the potential
r.2J
etc.
from Q,
them, and, by addition,
^
'^+^!^ +
tential I" at the point
p
f,,
f,,
points distant rlt
.'wtions will hold for each of
i
+ h r
etc. at
e,,
AND POTENTIAL
Fmust
we
= 0. ...) all 1
+
satisfy the
the charges
+
h
j-^
equation
vr v V 2 F=0 and sometimes A 7=0. This tvMih will clearly hold so long as Q lies outside all of the charges f etc. We shall see that when Q lies inside a charged body 2 fht< ili&'R-mial equation is modified. The equation V F=0 is This
t
,,
i>
i
.ft
en
written
:
kiMwn
u>
Laplace's equation,
and the operator
V =
+
2
dxits
Liipiiief's
2
+
dy
dz*
operator.
Tin luMilt
F=
2
? can
readily be extended from the case
>TK-S uf jwint charges to that of a continuous distribution of For the space tvrTrieity. occupied by the latter can be
-I
;i
mi-
split
;iii
indefinitely large
of these
number
is
up
of
volume elements dr. The p is the volume density of the
pdr, if 'nar^Mn ^ach ';^nhVan,.n. The potential due to the distribution, at a point
^i<-
it,
will
dearly be equal to
1jfjf.
^''^ 1il
dr,
'-x^-nds uver the whole of the electrified
where the triple" body and r is- the
element pdr from the external *^';*y point. ^'''
i-'
5
-
If there are
ELECTRIC INTENSITY AND POTENTIAL
add
an amount
to the r potential
~
1
1
19
- dS, where &
is
dS and
distance from
the surface
47rJJ r
density of the charge at the element the external point.
The Potential Further consideration
r
at internal points.
necessary before
is
its
we can apply
the
foregoing expression to the potential at points inside the electrified
medium. At such points the denominator r becomes zero, and we must be certain that this does not make the integral which
The matter may be investigated represents the potential infinite. as follows. About the internal point Q describe a sphere of infinitesimal radius e. shall suppose the density p of the
We
charge to be everywhere finite and the radius e to be chosen so small that the density of the charge varies continuously throughout the volume of the sphere. This condition can always
The
potential at Q will consist of two parts: from the charge outside the sphere, and (2) K, The former is clearly arising from the charge inside the sphere. finite. Let p m be the maximum value of p inside the sphere.
be
satisfied.
(1)
T7!, arising
Then
^
V^-^ffl -*4 7rJJJ
dr.
The element
of volume dr in polar F
S
coordinates
is
f dr smOd&d<>. 2
g This vanishes
when
r ['
e
So that
drT sin 8d6
is
made
[** d
^^
Pltl
sufficiently small
:
so
that
we
conclude that the charge in the immediate neighbourhood contriThe formula butes nothing to the potential at any point.
V= and
-
I
I
~ dr therefore holds generally both
for points inside the electrified
for points outside
medium.
The Derivatives of V.
The
V
dV electric
intensity
and
all
higher derivatives of
contain r to a higher order in the denominator than does They are therefore all finite at external points.
T
itself.
The o
/v.-n
>-i
electric intensity is also finite at internal points. n1 c?T-\r> CiTi r\f TO rl a ^ onH rliTnrJiri T' iri'K~v
-fl-/-i
01 >-k-v
1
m
Consider TTT-I-V
T\QT4-ei
AXD POTENTIAL
ELECTRIC INTENSITY
00
7)V I',
iml
V
t
As we have
as before.
can be
2
small
c
By
enough p/r *
with
0,
just shown,
made
so that the latter
is finite.
-^
as
large as
We
we
may be disregarded
Thus
in tin*
_ 3Jj } CT
vanishes
when
intensity
at
f rfr fsin 0d0 f Jo Jo
47rJo e
We
= 0.
internal
conclude
that
the
'obtained
by
therefore
may be
points
the potential
Gauss's Theorem.
The consideration of the distribution of the to
in
surfaces
in
space
any surface whatever Let
X
and
leads let
to
electric intensity
interesting
results.
dS be an element
of
it.
be the component of the electric intensity at dS, along the to the element, being reckoned positive if it is in the
N
drawn outward from the surface. We now prove that the integral JJNdS taken over any closed kv Is equal to e, where e is the algebraic sum of the charges
direction of the normal
within the surface.
ELECTRIC INTENSITY AND POTENTIAL Consider
P
point
where ~~
PG
is
(9
~ 2
first
the intensity
*S
the -angle between t ^le
S
^
angie
E
i
at
l
We
within the surface.
Q due
have
PQ
S =e l
If there are a large
number of
normal component of
to a charge e l at a l
^PQP- and
and the normal
dS
subtended by
far as the single charge e l is concerned,
21
at P.
Thus, so
we have NidS =
point charges e ly e,
etc.,
Now
to dS.
p -~-
da)^
the resultant
electric intensity
= e d^ -f e^do)* -f e^do> -f = ^TT^! ff do) = 4-7r&> etc. ff e = e +^ + ^ + ... = e ......... ...... (3). //JV dS
4*7rN dS
so that
Now
l
. . .
s
l
c?ft>!
,
e.2
.
,
r
So that
1
This result can obviously be extended from a series of point charges to a continuous distribution in the same manner as that
employed
in dealing with the potential.
It remains to prove that charges outside the closed surface It is evident that contribute nothing to the surface integral. element of solid doo conical angle arising from an external every
charge will cut the closed surface an even number of times. The value of NdS for the intersections of the cone by the surface will
be alternately positive and negative since the direction of the electric intensity is constant in space but alternates in sign with reference to
the successive normals.
The numerical value
NdS is the same for successive intersections, being So that the
surface integral
is
equal and opposite elements. conclude that the value
~
of
6
equal to
do>.
divided up into a series of pairs of Its value is therefore zero, and we
offfNdS over any
closed surface
is
equal
to the charge inside.
This result
is
known
as Gauss's
Theorem.
means of calculating the value of the electric intensity arising from various symmetrical Thus in the case of a uniformly distributions of electric charge. at any point external to the shell the intensity charged spherical
The theorem
is
of great value as a
It also follows with the shell. the point and concentric that the intensity must be inu the symmetry of the problem of radius r is 4an*> the electric Since the area of a sphere radial centre of the charged shell is the from r at a distance
E
= by 4ir^#
riven K
;
so that 2
#=e/47rr
........................... (4).
force vanishes inside
that the way we may prove
In a similar
case of a sphere These results may be extended to the volume so that layers equidistant from the throughout its Thus we may show, foi to are charged equal density. uniform solid a inside force sphere of electricity sample, that the
the shell
varies
the distance from the centre.
its
The
application
Theorem to the tubes of force As we have seen, a tube bounded by a surface which is the
of Gauss's
on p. 16, is instructive. a tubular region of the lines of force.
of
is
to
Let us apply Gauss's Theorem a portion of such a tube, terminated at each end by equiThe lines of force run along the tubulai surfaces. so that at each point
I
the component of the intensitj Over the ends the resultanl
to these surfaces vanishes.
!
*'leetric
at
E
Let it be and S2 being the
to the surfaces. intensity will be normal
the end where the cross section
is
E
S13
z
The value of ffNdfc corresponding quantities at the other end. I: nver the whole surface considered is clearly 1 S1 2 S2
E
E
.
a region where there are no electric = E^S2 thus the electric inthat so this vanishes, rharges at is as the area of cross section o: any point inversely Minify the tube of force
is
in
E&
the tubes
!
i
.f
;
force at that point.
Fwk-T the conditions contemplated in electrostatics the surface nt hi t r if electricity must be an equipotential surface other would be currents of electricity flowing from one par Lr w\rt nv tt another. The tubes of force must therefore star .
;
:\
.
i
;r..iii
ji *h
*;1* u*t '*
-
Now apply Gauss's Theorem t< a tube of force and its continuation int< the conductor and terminated
such a surface.
landed by if
by equipotentia ne in>ide and the other outside the conductor. Th< j:rh>ity vanishes over all the surface inside the conducto: ijnr.nal
component vanishes over the tubular surface
TIP \alue
of
jjNdS
is
thus
eual
to the val
&
A
ELECTRIC INTENSITY AND POTENTIAL end
this quantity over the
23
This is equal to the charge the surface the area of its intersection by the tube of inside
which
;
We
w here 7
is
section. a-
is
E& =
have
we make
the
end
section approach indefinitely near to the charged surface, $! so that
= &>,
force.
therefore
If
E=
(5).
a point just outside a charged normal to the surface and equal to the charge per
electric intensity at is
unit area of
it.
We may
This result
is
express Gauss's
known
as Coulomb's
Law.
Theorem
between a surface and a volume
analytically as a relation If p is the volume integral.
density of the electrification at any point inside a closed surface the total charge inside the surface will be Jjjpdxdydz. The normal component of the electric intensity outside the surface is JV
=
_.
Gauss's
Theorem may
therefore be expressed in the
form (6).
an
In a region where there are no charges, direction cosines of the normal to
JJ
A
vector
Jf(lEx
E
(IEX
any closed
whose components
Ex Ey E ,
,
vector.
We
z
I,
m, n are the
surface,
+ mEy + nEz) dS = NdS = JJ
+ mEy -f nEz ) dS =
solenoidal.
if
we have
0.
satisfy the relation is said to be
over any closed surface Thus the electric intensity in free aether
is
a solenoidal
shall see that there are other solenoidal vectors in
the theory of electricity and magnetism whose properties are of great importance.
Greens Theorem.
An
important theorem discovered by George Green* in 1828 us to express a volume integral taken throughout an enclosed space in terms of surface integrals over the boundaries of enables
This theorem, which is the space. a purely geometrical theorem, but
named it
has
after the discoverer, is
many important
appli-
cations in the theory of electricity. *
"An
essay on the application of Mathematical Analysis to the theories
"
,
,,nsickr
r M
thch Lybe
either internal
^tlTboundary
^
<
,
number of closed or external. As a particular
bounded by any any closed space
may be a sphere
of infinite radius.
function.
any continuous point
Then Greens
Tftt%r in states that
the space between
is taken throughout where the volume integral surfaces, surface integral over the bounding the and the to the normals the of ud I in and n are the direction cosines
which the volume drawn away from the space throughout
is
token.
Fig. 3.
PQRSTU
Let the two parallel lines represent the section, by of a of the whose section by the plane phne paper, right prism r w^iild be dydz. Consider the contribution to the integral If
p
. f
^
dsd'tdz throughout the space
*rti<'ii
^
"f
r
^
arising from this prismatic
This will clearly be
it.
dyds ( r*
A
.
ftc.
in t
up + UQ
-
UR
are the values of
the values of
Hi^i ls is
due
/
+ u s - U T 4- %), u
and
at P,
Q
cZS at
to the fact that
3
etc.
Now
P, and so on.
The
the normals alternately
ELECTRIC INTENSITY AND POTENTIAL
make acute and obtuse
25
angles with the positive direction of
So
sc.
that (T).
Similar equations hold for
XT Now
.
i
oz
cy
u
let
so that,
|
JJ
-^~
ox
by addition,
fffT
= -f
+ ^r~
+
*ox
and w,
v
[In l
+ MV -f w w
Ja
dS.
, .
\
.(8).
-
-r-,
y
)
cz
oy
After substitution we have 2
f|T rr
fa
F
^1-5-r 2 JJJ [9^
+
cT-F 5-;2 ?7/
a 2 F)
+ -o^ 3s-
-
dx ^dx
1
cz
cy cy
,
,
+
+ -5-
-a-f
,
dxdydz
c>z\
OTT
where ^
denotes the rate of change of \
If
we
substitute
o
TTT
F^9y which U
obtain similar expressions in Subtracting the two equations
the normal at dS. o 7T
TJ"
for w,
F-^9^
F along for r,
and
o^
F
and we have
F -5-
for w,
we
are interchanged.
-V^U} dxdydz ......... (10. tin
The three equations preceding,
(8), (9)
on)
and
(10), are all different
forms of Green's Theorem.
An important (9 j
or (10).
We
case arises
fa-F
We may rirlp
a.
when we put
U= constant
inequation
then have .
a
>2
F
,
a-F]
^d ^ = +
ft
.
(ll
apply this result to a minute cavity of any shape
o.harcrftd
bodv.
We
have seen
23) that
(p.
ll^dS^
jlone
this equality is
follows that
it
cavity
the shape and size of the independent of
327
327
327
.................. ( 13 >
+ ^-i^-/9 -^7+yr oz da? 2
2
dy
It is the This equation is known as Poisson's equation. the satisfied differential form of the potential by equation
At pints where there are no that we get Laplace's equation
Ex ,Ey,E
z,
This
0, so charges we have p as a particular case of it.
the components of E, are given
often
is
=
electric
V 2 7=0
written
the equation
is
We may
E=(EX9 Ey9 18g) = -gradV.
V 2 7=-/>
or
dEx dEy + Hx Hy This equation
by
in the form
dEz __ ~dz~ P
'
often abbreviated to div
E = p,
the operator
acting upon any vector denoting the sum of the results of the of the operator on each of the com-
grad
corresponding
jxn*'nts of the vector.
Transformation of Laplace's Operator. In
dealing
p-^iiig
with
certain
4 .
P
problems,
spherical or cylindrical
is
in
cases
not desirable
Poissun'sand Laplace's equations in rectangular coordinates.
ir
or cylindrical coordinates are
much more suitable. The new coordinates can very
truiM. Tiiiatnui uf the equations to these i-ily k- effected by use of the *
making
,
'
,
cF
'
ck
V'JM] 1
it
ij1
1
particularly
symmetry,
of the
'
if
over an >' closed snrface
il-ustnit^ '*
theorem that the surface is
e qual to the
volume
volume density inside (Gauss's Theorem, p 23) tht method we shall first apply it to the simpler case
tn^iiur ewrdinates.
ELECTRIC INTENSITY AND POTENTIAL Consider the element coordinates,
volume
of
whose centre
is
27
in rectangular coordinates of the
dxdydz,
The
x, y, z.
angular points of the element are x \
i dy, z
\ dz, etc.
Then the mean
z.
Fig. 4.
IcV
V+ =
value of the potential over the right hand face will be
d
CJLr
and that over the
hand
left
face will be
IdV F -~^
The mean
2 dx
value of the component of the electric intensity along the outward
drawn normal
the left hand face faces, as
hand
at the right
we may
=+*-
face
\V--z:
=
^-
#
is
dy dz x I
and dy ^ dz x
-- ^ v 2
-
9
2
f3 -<
2
-=
over these
"1^2
-^ + -j
OtJU
Ox
is
"~|
;
dx
(jiL
in the
one
I
The
total flux
therefore
Oy and Oz faces
is
F- &V 4- 3 F] 2
-I- -r
r
T-
-^r
F
Similar expressions hold for the
whole cube
TT
dx\ in the other.
-
over the pair of faces perpendicular to
flux for the
flux
)
d^;
rp\
case
c&c
the sake of brevity, the surface integral
call, for
of the normal intensity,
and at
~ -^
2 ex
The
-
2
F+
(
-r
>
,
,
dx dif
,
dz.
;
so that the total
W^"
_
Theorem By
thus derive
Equation
.
inside or to the charge equal equ
M. is HUB a
^
V7
P~5?+
in rectangular
V W
coordinate,
Coordinates.
the mdius
makes with
Fig. 5.
Let-
PQflSrWTIT be
the element of volume.
surfaces: the intersection of the following
}.v
and r + idr;
radii
plane passbg through about the axis
Oz and
c^axally 4. ^
th..-
tie respectively.
elements of
MIL
forrm
The mlius
(2)
of semi-angles
The coordinates
volume are
fl-irfi?
ai
of the angular points
P = r - %dr, 9 - 4 d9,
$
- ^ d^>, and of
by the intersection and the sphere of radius r is clearly r sin
of the circle formed
C'*n* of serai -angle
Thv*
is
two spher
two planes passn r-^dr with a fi 0^ making angles <J>-$(Z$ and ^+id
mitre
^
(1)
It
volume of the element
dr x rdd x
is
r sin
5
d$ =
2 ?'
*' sin
t
^
ELECTRIC INTENSITY AND POTENTIAL
uhe flux over the faces which are perpendicular potential at the centre being F, the mean potential
Consider to
The
r.
first
over the outer face will be
dr y_ 2_ dr and
-f
-
f
the normal
V
-T
7:
The
first order.
The
face
=- F ^ ^ dr or or\Z
being G
-f-
(
}
j
areas of these faces are respectively r
(r$dr)d0
C>
0'
\
and over the inner
dr) sin 6
or r (r
dO
d
to
the
fluxes are therefore
+
and
The
J
%dr)sindd(j>x
(r
intensities
dr}.
2 dr
dr \
19F V -\--~dr
total flux over these
~
*
x
<**) r
(*
sin
J
two
faces is thus
^ -**dra*0dBd^+-*J\. r or or'{
The mean
QRWT
are
~
potentials
F-
at
the conical
fiv
1
j
d0 and
F+
1 3
surfaces
PSVU
F
--^-d<9
respectively.
and
The
corresponding normal intensities are obtained by differentiation with respect to the element of arc r d9, and are therefore equal to
PSVU is x [sin (sin
to the first order.
<9
cos ^
-
-i
d0
cos <9^)
Similarly,
QRWT= rdrd<j> [sin The
fluxes therefore are
and
- dr cty
/
~ + ^~ d0
+ $ cos 0d0].
(sin ^
+
4 cos
cos
<9
sin ^
total flux is therefore
The
(13
The mean
m
potentials
Y+-~d(b 2
4
F
2
F-~
and
normal to the planes being
rsm'ffd(f>,
and
RSVW
The element
^rrdd) respectively. 2 o
d<j>
arc
PQTU
of the plane surfaces
the
of
mean normal
intensities are
8 f, r
1
The areas
1
,
7
of the
The
dr x rdd.
137 A
9
f jr
I
dV
.
each other and to
surfaces are equal to
fluxes are therefore equal to ,
sin 6
\d(f>
^
+ drdOfiV
,
alld
their
^
2
13 8 F
(^-2
7
A
to**)'
sum being
L_ r2 sin2 5
We
thus find for the total flux over
all
.
2
3c/>
the six surfaces of the
element of volume
By Gauss's Theorem
this is
equal to the charge inside,
which
is
pdr = p Si>
that
r cr
This i
TIH-
i
s
therefore Poissou's
nemtor
^-
-
r3r
_,
+
r3tf
'
1
U hich
Equation in spherical coordinates,
^ + ,-=
3fl
F
'
Laplace
s
operator Vi takes in this
ELECTRIC INTENSITY AND POTENTIAL
31
Cylindrical Coordinates.
other system of orthogonal coordinates similarly. For instance in the case
may be
Any
treated
dr
of cylindrical coordinates r, 6, z the element of volume is bounded
by (1) two coaxal r and r
cylinders of radii
dr with their axes coincident with the axis of z, (2) two 4-
Fig
6.
planes inclined at an angle dd to one another and passing through the axis of z, (3) two parallel planes perpendicular to the axis of z and at a distance dz apart.
The volume
of the element is clearly dr x rdO x dz. The of the element being V, the flux over the at the centre potential outer cylindrical surface will be
and over the opposite fr
V
face
d ^^} 2 / edzx^(V~l^dr} dr V 2 or J
}
the total flux over the two faces being *
*>
**>
The area
of the plane inclined faces
intensity over
_1-
them +
d
^ d6
^ V
r do
(
c)u
\
i
r or)
>*-
dr dz, the mean normal
is
}
and the
total flux over
J
them
The area
of the faces perpendicular to
total flux over
them =
r dr
dO dz
-=
Qz
is
dr x rdd and the
.
oz-
So that the
flux over the
L_ (
dr
4.
_13F r dr
Thus the form which coordinates
whole six faces
is
+
= rp
rj
__ ^^ 36/ 2
(
dz 2
x rdrdddz.
}
Poisson's Equation takes in cylindrical
is _
2
+ '
-
"
__~
.(16).
ELECTKIC INTENSITY
32
AND POTENTIAL
The Uniqueness of the Solutions.
and rectangular branches of
in spherical, cylindrical Laplace's Equation is of the greatest importance in many
mathematical physics. that the
first
It is clear
from the preceding discussion
derivatives of the solutions of this
equation represent
a vector which flows out from a series of points uniformly in all Its applicahility to the theory of radiation, of condirections. duction of heat and electricity, to attraction as well as to electrical
hydrodynamics and gravitational and magnetic attractions is at
obvious.
The
of
equations arises from the them we only need to be given the value
differential utility of the foregoing
feet that if
V over
we can
solve
certain surfaces in order to obtain the
complete
distri-
This result depends upon the satisfies the equation theorem, which we shall now prove, that if V 3 V = - p throughout any region of space and has certain assigned "button of electric force in
the
field.
V
over surfaces bounding the region, then it is the function which satisfies these conditions. For if not let
only
V also
satisfy
the same conditions and let us write
the expression for Green's
and since
RT
Bnt
Theorem
V^F = V4r = - p
the surfaces,
we
this
is
V V in
Then
throughout the space and
F-
V=
find
a sum of squares so it can only vanish if each We thus have separately.
integral
term vanishes
U= V=
in equation (9).
;
ELECTRIC INTENSITY AND POTENTIAL
(V -
Since fl
F')
~ (F-
F')
dS
33
when
vanishes
also
is
given over the surface S, it follows that F is unique except for an additive constant when the value of p is assigned throughout the
and that of space r also,
3F ^
over the boundaries
the electric intensity
It
so that, in this case
:
on is
would lead us too
determined uniquely. consider the functions
to
afield
far
(Fourier's Series, Spherical and Zonal Harmonics and Bessel's Functions) which are the solutions of Laplace's Equation approFor the development of this priate to particular problems. interesting subject the reader may be referred to Byerly's Fourier s
Series
and Spherical Harmonics. Total Energy of a System of Charges.
We may
find the total energy of a system of charged bodies and potentials as follows. Since the
in terms of their charges
potential at any point of the field
taken over
all
equal to
is
1
1
-
1
the charged bodies in the
dr +
J *j
field, it will
dS
I
'
jjj
j
f
be re'duced
to l/n of its value if all the charges are reduced in the ratio n:l. Let n be any very large number, and suppose that initially all the
charges are at an infinite distance from, one another. Bring up l/n If F is the final of each element of charge to its final position. the at will this operation any potential point change during potential '
F
The work done
from
to
will lie
between
.
and
Vp
dr.
will lie II n of all the charges &
1 up the element p dr
in bringing
The work done
between
and
/
bring up a second potential at
??th
part of
If this process
is
1
1
n-JJJ all
between
F/??, 1
;
(
1
Vp r
dr.
ZSTow
and the work done in
Vpdr and ^
1 J
j
Vp
continued the work done in the 5th stage will
between 1 ffr
Vpdr and
up
This will raise the
the charges.
any point from V/n to 2
this second stage will lie
1
in bringing
( If -, >>- iii
Vpdr. '
dr. lie
ELECTRIC INTENSITY
34
The
AND POTENTIAL
work done in bringing up the whole of the charges from
total
will a state of infinite dissemination
between
lie
n
Vpdr
-tJJ
and This
is
of the system, equal to the total potential energy lies
which
between ,
,
Tr Vpdr and
When Iff/
n
is
increased indefinitely each of these values coincides
This
Vpdr.
the part due to the
is
are surface charges
we
complete expression for the total Is
volume charges.
If
have to add ^fJcrVdS. The energy of any system of charges
shall
therefore 1
r r r
T
i
r
an The Energy in the Field. In the preceding paragraph we have deduced an expression On energy in terms of the charges and their potentials. the view that electrical actions are transmitted a
for the
through medium, we should expect that the energy would reside in the medium. It is easy to obtain from the equation (17) an expression which admits of this interpretation.
Since p
=-V F 2
and
a-
But by Green's Theorem, allowing from the fact that the normal considered, this
is
that
-m-nt
the ,,t
we
for
the reversal of sign arising
is
now drawn
into
the space
equal to
'lljl&fo S--.
= dn *'
k
(19).
energy of the system the field contained an
is the same as if each amount of energy $E> per unit
ELECTRIC INTENSITY AND POTENTIAL
35
Stresses in the Meld.
Maxwell showed that the forces acting on any system of charged bodies could be attributed to a system of stresses in the medium in which they are embedded. The necessary and be the case is evidently that the resolved part, in any direction, of the resultant of all the forces acting on the parts of the system, arising from systems external to it, should be expressible in the form of an integral over any
sufficient condition for this to
and isolates the system. The alternative that would imply possibility part of the force was not transmitted across the boundary, through the action of the parts of the medium on one another, but arose fro m so-called action at a distance. surface which surrounds
Consider any surface S surrounding and isolating the system Let be the x component of the resultant of static charges e^ l e from all external electrical on force acting 1 arising systems. and the volume density at any point Then if is the potential p
X
V
where the integrations are extended throughout the volume The volume integrals will be capable of transenclosed by S. formation into integrals over the boundary surface S* if we can write "*"
dx [da?
dP
.
., lorm in the f
,
-=r-
4-
ox
w We have ,
dVyV = 1
-~- *-; ox ox-
x
'dy-
dQdE TTh
d
/3FV
^ ^" "o~ 2 ox \ox ]
dy-
'by
^
"" d_
\
.
oz
cy >
dx dy )
/3F 3F\
dy \div dy)
'by
1
dxdy d_
fdVY
2 dx \ dy)
__ dz 2
dz \dK dz )
'
'
2
das
v\
\dz
)
01 P.han
TT
ELECTRIC INTENSITY
8fi
will
Thus the integrand
be
in the
AMU
form desired
if
we put
of the resultant force parallel considering the components at similar surface integrals axes we should arrive
By
y and
to the
involving
^
''
-Ijffl'-f V-f Yi \3/J \3*/ ^w~2(l3y/
( 23 )>
/3Fy_,9FY) T~ O..
(24) \^^J>
lj/3Fy -r = -*U
tl
f
3F3F
.(25),
pxy and ^2.
In the new notation in terms of the to students of elasticity,
jp's,
which
will
be familiar
we may write
(26).
The
last integral is
taken over the enclosing surface, and I, m, n drawn away from the enclosed
are the direction cosines of dS,
volume.
Similarly
dg
+Z =
anl
1
l
1
If
+ mp yz + np zz ) dS
(28).
we adopt the standpoint that the action of the electric
charges on one another tliiii
(lpxz
(27)
|i xjr ,
p
ini
,
is
transmitted by the intervening medium, are the six components of the stress
pa pyxt pZXi pzy ,
which transmits the action. ice these quantities,
From the point of view of action at on the other hand, have no physical
ELECTRIC INTENSITY" AND POTENTIAL
37
In order to obtain a more definite picture of the physical nature of the supposed stresses let us consider the case in which
dS
equipotential surface, so that its normal is tangential to a line of force. Let the resultant electric intensity at dS be E, then
part of an
is
37 - -_
dV --
7ET IE
da
_
=r
mE
,
and
%z
pxx = ^E- (I- - m" - n-), = m - n - '^ Pyy - -m
and
_ - dV = nE,
dy *
-
~
2
?i
The components
2
I*
the
of
pyz = E-mn, = -n, Pzx x = E*lm.
),
per unit area across
force
dS
are
respectively
+ np2x = ^lE-f 71^^ = i wi
\ 2
-
............... ( 29 ).
lp x
Thus the resultant
traction
Next suppose that dS surface.
Its
Vf
l
The
A?
f
since
(<*Z\*
T \l 7
__
-f
,W +
m' -5-
stress across
+
,97 = A
ft'
07/
/37y
-r-
It is therefore
a tension of amount ^E- per
is
W ^ d#
component of the 1
is
at right angles to an equipotential f cosines I', m', n will then satisfy the
direction
relation
normal to dS.
is
directed along the line of force and unit area.
___
, x
?7i
dS
is
/97V)
-
0.
58^r
,
3_7
, 7 ,
-f
?z
v-
h
^r-
== 0.
In a similar manner we may show that the y and 2 -ri E'\ are equal respectively to \ m'E' and
Thus when
d>Sf
is
at right angles to
3737
87 ^
z
components
an equipotential surface
ELECTRIC INTENSITY
38
AND POTENTIAL
them, they will represent a in the field not a tension. There is thus at every point pressure, of force and a .pressure of a tension equal to \E- along the lines at direction in right angles to them. amount
now have a negative sign in
front of
every
equal
It "may be
shown that these
stresses will
keep the aether in
are not the most general system of stresses equilibrium. They which are equivalent to the electric force on the system of charged bodies e^ For we may clearly add to them any distribution of
whose resultant
is
equal to zero
surface surrounding ^. in
They
which the stress at any point
electric intensity at
when integrated over any however, the only system determined solely by the
are,
is
that point*.
It is .well to point out that this interpretation of the forces as a system of stresses in the medium has only been shown to be a possible, not a necessary one. *
Cf. Jeans, Electricity
3rd ed. vol.
I.
and Magnetism,
p. 158.
p. 146,
and Maxwell, Electricity and
CHAPTER
III
DIELECTRIC MEDIA CAVENDISH, and Faraday independently, showed that when a condenser was charged so that the potential difference between the plates had a certain fixed value, the charge on the plates depended on the insulating medium between them. This proved that the forces between electric charges depended not only on the magnitude of the charges, but also on the nature of the material
These experiments are often regarded as disseparating them. the proving dogma of action at a distance. They are not capable, all that they prove with of however, establishing this inference ;
certainty is that electric charges act on a material' medium in such a way as to make it affect other electric charges. The quantitative
experiments of the investigators mentioned showed that the charge on condensers of different geometrical form to which a given difference of potential was applied always changed in a certain ratio insulating material was replaced by any other
when any assigned
assigned insulating material. Different insulators were therefore " said to be characterized by different specific inductive capacities."
The
specific inductive capacity of a
vacuum
is
now
universally
adopted as a standard, and its value is put equal to unity on the The specific inductive capacity of electrostatic system of units. In this book, for reasons differs from it. air only very slightly will appear, we shall use the term dielectric instead of specific inductive capacity.
which
The its
coefficient
potential difference between two electric charges is, by determined by the distribution of the electric
definition,
It is clear, therefore, intensity in the field surrounding them. that these experiments prove that the forces between charged
bodies are
not determined
solely
charges and their distance apart.
by the magnitude of their Those of our previous results
law of the inverse square will therefore which are based upon the account for the behaviour of modification if we are to require
in the neighbourhood of insulating materials. charged bodies
The
relation
V=fEds
is
of independent of the law
force, as
by means of tubes is also the device of mapping The ideas underlying the lines of electric force. generated by them can therefore be applied even when dielectric media are out the field
\
\
I
?
1
;
a tube of force starts from each suppose that These a unit of positive charge. embraces which element of area on to or flow another on end tubes must either infinity. charge
We
present.
shall
This follows since two different
j
equipotential
surfaces
cannot
intersect and since the lines of force are at right angles to the
;
surfaces.
|
equipotential
I
negative end at the positive end.
|
It is also
of a tube should
This
is
necessary that the charge at the
be equal and opposite to the charge required by the fact that the two
of a condenser acquire equal and opposite charges nature of the intervening medium.
I
whatever
Ithe I
\
\
The tubes which are determined in this way we shall call tubes of induction and we shall define the induction D at any point as the number of tubes which cross a unit area drawn perpendicular to the direction of the tubes at that point.
The induction
I
|
is
a vector quantity and its
components are
given by the usual rule for the resolution of vectors.
Fig.
1*.
wih
the
component of
th, direction
umu*r
OP
D
Thus
if
7.
along a line
OQ making
an angle
of the resultant D, it is clear that of tubes which cross unit area perpendicular to
6
the
DO
DIELECTJRIC
MEDIA
the same as that of those which cross an area cos to
perpendicular
OP, so that
evident that the cross-sectional area of the tube of induction the induction at that passing through any point is the inverse of
It
is
point.
Gauss's Theorem.
We
now in a position to consider the form which Gauss's Theorem takes when the field embraces dielectric media. Consider the value of the integral JfD n dS taken over any closed surface, where
are
Dn
is
the component of the induction along the outward
Q
drawn normal
at an element dS of the surface. If OPQ represents = 6, the direction of the tubes crossing dS, and if the angle = then D cos 0, where is the resultant induction at dS. Now n
RPQ
D
D
dScosd is the projection of dti on a plane perpendicular to the direction of D, so that
D x dS cos 6 = D n dS is
the
number
of tubes which cross
outside of the surface.
When
to the inside of the closed
value.
D n dS
Let us
first
dS from
the inside to the
the tubes cross from the outside
surface
D n dS
will
have a negative
consider the effect of those elements for which
has a positive value. Each of the tubes crossing them will from a unit positive charge beyond dS on the internal side. This unit charge may either be without or within the surface. If start
DIELEUTKIU MJIUIA
^9
D
an equal and opposite element n 'dS'. The tubes to the integral will be zero. Each such of total contribution dS in the positive direction and having tube crossing an element a unit positive charge, within the surface will start on
and
it will
give
rise to
its origin
of all so that the total contribution
the elements J) n dS which are
of the positive will be equal to that part positive to tubes of force which the surface which gives rise
charge within do not end on
similar manner we can a negative charge inside. In a precisely the elements n dS which show that the total contribution of all within of the that to charge is negative are part
D
equal negative the surface on which tubes of force
end which do not start from
inside the surface. positive charges It follows that the value of JfD n dS=JfDcos0
any
the algebraic
closed surface is equal to
dS taken over sum e of all the
the surface. charges enclosed by
We
Gauss's
thus see that
Theorem can be extended
media provided the normal electric intensity induction. normal the by
By
to
is
replaced
applying Gauss's Theorem to a cylindrical region
bounded
dielectric
and by two equipotential surfaces, one just inside and the other just outside a conductor, we find that the
by a tube
of force
induction just outside the surface of a conductor is along the normal to the surface and equal to the per unit area of
charge
it.
This
is
the general form of Coulomb's
Poi-sson's
Law.
and Laplace s Equations.
Let us now apply Gauss's Theorem to the rectangular parallelepiped whose angular points are the combinations of x ^doc, y
1%
being tht-
and
z
\dz.
The induction at the centre of the element the mean outward normal induction over ),
D (D z D y Dz ,
,
faces perpendicular to the axis of 4.
z
j^D <>f
+
C -^Jj.
_:
x
will
ty Taylor's Theorem.
j
induction are
n - [D x (
3-Da;
-=-=
dx-
x -z
^-
\dydz. y
be
-
(])
The corresponding
DIELECTKIC MEDIA faces is therefore total flux of induction over this pair of
The
we
of faces similarly, Treating the other two pairs
^dxdydz. 9#
see that the total flux of induction over all the six faces is
This
equal to
is
the charge inside, which
is
pdocdydz.
therefore get ,.
div
_
We
m
3D,
3A, 3D, ................. v 1 / D = -5-+ -?r- + -xr=P oz d% oy
as the general form of Poisson's Equation. we put p 0. Laplace's Equation follows if
The general form of
=
Induction and Electric Intensity.
From
the definition of the induction
it
follows that in geo-
metrically identical systems of conductors, furnished with identical distributions of electric charge, the induction will be identical at
every point whatever the nature of the intervening dielectric, provided it is homogeneous and isotropic. The experiments of
Cavendish and Faraday show that the intensities at corresponding points are inversely as the dielectric coefficient, since for geometrically identical systems the forces are as the differences of potential It follows that the relation
between the induction and the electric
intensity at any point in a dielectric
medium
is
D^tcE .............................. (2), where K
is
the dielectric coefficient.
also write the
Since
E=
preceding results in the following
grad V, forms
we may
:
Gauss's Theorem.
Coulomb's Law. d a
Poissons Equation.
dv\
or,
when K
is
d
/
dv
constant throughout the space under consideration,
DIELECTRIC MEDIA
44
The Energy in the Field.
The theorem (equation
Chap.
(17),
IF of any system of charges
is
11)
that
equal to
the potential
_
of force between two elements of depend on the law dielectric media are present. when true therefore is anl
ii
*hi*i)(<'.
>ujtv paEflivZ) and
= Dn we ,
Fdiv
have tfr
+
ox is
I
w-
w,
the
by
x
direction cosine of the
similarly
j{TilivDdr by
The
parts,
energy- per unit
the
integrating that
volume of the
normal to
other
Since
c
components
of
field is therefore
3V l!'
,.
ijiay
",,
are the components of E, since
be written V
+
3
#=-grad
F, this
.................. (8).
The sum of the products of the components of two vectors ukvn in this way is called the scalar product of the vectors* lr
>
;;ti:t
-teen written
(D),
so that in this notation the
volume of the medium
energy of
DIELECTRIC MEDIA In isotropic media
= /cE,
D
E
and
same
are in the
so that the energy per unit
This reduces to the result in Chapter
Conditions at the
45
and
volume of the medium
II
when K =
Boundary between Two
It is very important to determine
direction
1.
Dielectric Media.
how the induction and the
change as we pass across the surface separating two different insulating media. Consider first of all an element of electric intensity
area
dS of the
surface, so small that its curvature
may be
neglected.
Describe the prism generated by lines passing through the boundary of dS and at right angles to its plane. Let the prism be terminated
by planes
parallel to dS,
and
let
the height of the prism be a small
quantity of the second order, if the width of dS is of the first order. Then one of the ends of the flat prisin thus constructed will be in the
first
medium and
partly in one medium theorem to the prism.
the other in the second, whilst the sides are and partly in the other. Xow apply Gauss's The induction is necessarily finite, so that
the normal flux over the sides of the prism vanishes, since their Let l be the area is negligible compared with that of the ends.
D
magnitude of the resultant induction at the boundary in the first medium and let it make an angle d I with the normal, Do and 6.2 being the corresponding quantities in the second medium. The areas of the two ends being each equal to dS in the limit, the flux of normal induction over the first will be D cZS = D 1 cos ^cZS and over the second D no d$= IX cos 9dS. The sum of these two is total to the the prism, which is crdS, if a is the inside charge equal r
/il
charge per unit area of the boundary. AVe see therefore that the induction at the boundary between two media will always satisfy the relation
D
l
cos 6 l
D-2
cos 6
=
a.
In the majority of cases there will be no charge on the boundary surface, and thus the component of the induction normal to the surface will have the same value on both sides of the boundary. In other words
:
when
tion the normal
there
is
no charge on the surface of separais continuous from
component of the induction
DIELECTKIC MEDIA
||J
L
rj,
limn to the other;
t
M*
A
by the tangential component of Consider the rectangle which forms a the axis of the prism just considered.
siiiilar relation is satisfied
!retric
tfci
intensity.
iu;.i! Htetiii through
L
r
tin th*'
i4 lit i,
M
faleiuate the
\i
,*ji'i
i
work done in taking a unit electric charge
4iks of such a rectangle. The force being finite everymurk done along the short sides vanishes in comparison
h tlur
,<
The long
along the long sides.
e
Ii
\ii lui^th f
r
charge per unit area of the
to the
an amount equal
liv
this is not the case it is discon-
when
in the limit,
and
if
T T 1}
sides are
of
are the values of the
ii^ niidl components of the electric intensity in the two media, \\rk tloiie in taking a unit charge round the rectangle will
<
fh
F - J&
J
This must vanish, otherwise
,
,!i
finit."
i
h'liuW of times. \v4i^k that the ,
We
therefore conclude that 2\
component
=T
2J
or,
in other
of electric intensity tangential to
the
continuous in passing from one medium to the other. c^tr that this result holds whether there is a charge on the
i-
f
*i,
could obtain an
is
urfa.**'
If
we
amount of work by repeating the operation an indefinite
il.uy
Thtse
iU-
r
nt
results enable us to obtain the
law of refraction of the
induction at the boundary between the two media. Let > ^ipp.se that the interlace is uncharged and let xl9 *2 be the u-; :tn. coefficients of the two media. Then, since the normal
:
1
f
'i^' lien:
<
f
the induction
D
l
is
cos 6 l
continuous,
= Do cos
we have
2,
mi, Miiiv the tangential electric intensity is continuous,
Di rlKV
sin a 03
=
D.
.
-sin &,
tan
he tangents of the angles which the tubes of induction make in,- normal to the surface are directly as the dielectric meiviii> of the media. n
W,
the results of the last two sections that in dealing involving dielectric media the induction must be a ::,n M! the equation divD = p The further condition has -n>ii,d that at the surface separating any two media the i
^
iron,
proems
.
DIELECTRIC MEDIA
two
47
it, by an amount equal to the charge per unit area of the surface at that point. In addition, the tangential component of the electric intensity must have the same value on both sides of the surface. There is only one solution of the differential
sides of
equation which satisfies the conditions in anv assigned case so that any particular solution of the differential equation which can :
be made to
satisfy the
boundary conditions
will
determine the
field
The boundary conditions which we have established
uniquely. dielectric
media include those relating
or in
dielectric, as particular cases.
any
to conductors in a
for
vacuum,
Poissoris Theory of Dielectric Media.
There
is
a certain view of the behaviour of dielectric media
the mathematical development of which
French physicist Poisson.
According
is
to this
largely
due to the
view the modification
of the electric field arising from the presence of dielectrics is due to the substance of the dielectric being thrown into a peculiar electrical condition
by the external field. This condition arises from the displacement of electricity in the ultimate particles of the medium in such a way that each particle acquires a positive charge at one end and a negative charge at the other. When this medium is said to be polarized and we shall see that the
occurs the
polarization
is
measured by the product of the displacement and
the charge per unit volume.
We
shall first of all consider the nature of polarization as
it-
from the point of view which regards electricity as presents after the manner of a fluid, throughout distributed, continuously all bodies. Consider two equal spheres, and let one of them be itself
filled
with a uniform distribution of positive electricity and the
other with a distribution of negative electricity, identical with the first except for the difference of sign. Imagine the two spheres coincident in position;
we
which
no
will give rise to
shall
then have an uncharged body
electrical effects.
Xow
suppose that,
whilst one of the spheres is fixed, every point of the other is given a certain equal displacement, so that this sphere moves a small
distance after the
manner
of a rigid body.
The region
of space
where the spheres overlap will still be free from electric charge, but there will be a layer of positive electricity over the surface of the sphere on that side towards which the positive electricity has been
of negative electricity over the and a similar distribution measured parallel to the The thickness of the layer write face of the surface, will be the same at every point
Hisolaced
2aof displacement
measured parallel to the normal at any point the displacement multiplied by the the magnitude of makes with the direction of normal the that
so that the thickness is
equal to
wine
of the angle
the displacement.
the body shape of
This result will
may
be.
By
clearly hold
whatever the
making the density of the
and the displacement indefinitely small, charges indefinitely big the same result their that product remains finite, in such a way without altering the shape of the body. A body may be obtained
Fig. 9.
whose to is
electrical
behaviour can be represented in this
way
is
said
be uniformly polarized. The direction of the axis of polarization the same as that of the relative displacement of the positive
To facilitate discussion the charges which are imagined to arise in the dielectric in this way will be referred to as "fictitious" in contradistinction to the "real" charges which occur for example on the surface of conductors. The to the negative distributions.
legitimacy of this distinction will be considered
more
fully later.
Another way of regarding polarization is to suppose the body divided into equal cellular elements with the axis of each element the polarization. To represent uniform we then suppose each cell to develop a positive charge on one face and an equal and opposite charge on the opposite face, the faces affected being those normal to the axis of polarization. parallel to the direction of
polarization
DIELECTKIC MEDIA
49
A
body constructed in this way is identical electrically with a body of two uniform distributions of positive and negative This is electricity which have undergone relative displacement.
made up
because in the interior of the body equal and opposite charges coincide at the contiguous surfaces of opposite elements, so that there is no volume charge at any point whilst at the external boundary there is a charge, positive at one end of the clear,
;
body and negative at the other end, whose magnitude is the same for each portion of the surface in which it is intersected by the cell. But the area of this intersection is inversely as the cosine of the angle that the normal to the surface makes with the Thus the density of this fictitious surface layer axis of the cells. varies as the cosine of the angle that the normal makes with the
sides of a
and the distribution is completely identical with that obtained by displacing originally coincident electric
axis of polarization,
charges.
Fig. 10.
In some problems
it
is
more convenient
to regard polarized
as displaced charges which were originally coincident, whilst in others the point of view which considers them as made up of
media
cellular elements
having opposite charges on opposite faces has
advantages. If we suppose a polarized body to be intersected by any surface, the two resulting portions of the body will still be polarized. It is necessary therefore that there should be developed at any such
surface of separation a double distribution of electric charge, of equal amount and opposite sign at- any point of the interface.
DIELECTRIC MEDIA order to make the algebraic sum required in see that the of the body zero. /ill the charges on each part can be tltniHits which we have considered previously
Ths
is
I-u'I<*pment
We
-Vi'i'ir /.
Mi'l*
'Jitiuvh
i^iu ..!
& arising in this way by a series of fractures
I
pmllel and perpendicular
of the
body
to the axis of polarization
an
So
fur
we have not defined polarization in a manner sufficiently
We
shall now define f numerical expression. as we shall, for the or of polarization, tiir intensity polarization, moment electric the as call per unit it, the *ikf f brevity, usually our cellular of one Consider at vJmiie if the medium any point. faces be SS end normal the of area the t-lfim-nt* uf length &r; let
pruW
mid
admit
t
i
let
be the surface density of the charge over them. Then on each end is +
the charge element is
The
moment
electric
where Br
the volume of the element.
is
therefore proportional to the
element, and the electric
unit volume
moment per
volume of the
is
equal to the
Clearly
surface density
develop J.K.T a plane perpendicular to the axis of j*im under consideration by Thus the polarization may be defined either as the polarization.
electric
moment
per unit volume of the body or as the charge per
an interface perpendicular to the direction of polarizaThis interface may include the external surface of the
unit area of lit
iii,
polarized bixly as a special case.
The statement
in the preceding
paragraph
is
to
be regarded
the definition of the resultant polarization. The polarization, however, is a vector quantity and this of the case can be aspect
;is
provided fur by a slight modification of the definition. the polarization in any direction at any point
We now
define
as the volume of a thin slab of the polarized medium described about the given point and with its faces perpendicular ni"!n<..-nt
the
:
m.id.' i
n
per unit
direction.
ifiveii
up
t.f
Fi,:. !
tht-m
1
The
slab can
conveniently be regarded as
a series of the prismatic elements already considered If is the area of each face of the slab the
1.
will
AS
charge
be
AS
cos
where
the angle 1* 'tween the normal to the slab and the direction of the resultant |Hiirizan*rL
clearly
The
cr
6,
is
DIELECTRIC MEDIA of electric charge electric
slab
is
cos 9
&z?,
if &z? is
51
the length of a
cell.
moment
The
(perhaps strength would be a better term of the therefore a-co&ff&SSx. The volume of the slab Is
is
)
cos ff&S&v, so that the polarization along the direction
clearly
P
P
where
is
fl
=
normal
to the slab Is
Pcos0
(12),
the resultant polarization. The components of the by the usual rule for the resolu-
polarization are therefore obtained tion of vectors.
Fig. 11.
Polarization, Induction
We
are
now
and
Electric Intensity.
in a position to determine the relation between Let us first consider the nature of the electric
these three vectors. field
between the plates of a
parallel plane condenser filled
with a
of specific inductive capacity K. Then the induction the electric intensity are both normal to the plates and = o-, where a- is the charge per unit area of the plates.
dielectric
D
and
icE =
E
D
We shall make the hypothesis that the polarization is caused the electric intensity in the dielectric, and that the two vectors by This standpoint will be found to be are coincident in direction. fully
justified
when we come
to
consider the
phenomena from
the point of view of the electron theory, according to which the polarization arises from an actual displacement of the ultimate
The hypothesis electrified particles in the direction of the field. dielectric substances. In the is undoubtedly true for isotropic of rvrvsta.llinA snhstanc-es the electric inteiisitv does not in
DIELECTRIC MEDIA
5
Miril
X
al
I.
.!
.',**!,<
flit*
electric intensity is
from the external charged bodies and from the polarization at the boundary
charges arising In the present case the
amount
of this charge
It will be negative area of the plates. jier therefore have vice versa. and jmsitive
P
I*
will
&
mi
media the
wttliuni.
tlir
and their existence
vitiate the general principles involved.
any way
in arise partly
^niitMtMti
to discuss,
way
the theory of polarized
<>n
fix
t* far out of our
.1*
In
T.
Xi*w
nf
and a similar
of crystalmagnetic behaviour iv ut hi .Ms with regard to due complications which it Tht-M exceptions are i-.
t t.
\,.
direction with the polarization,
111
''HK.-ilt
unit-
where
We
i
F=
P=D P
(13)
tcE=D'
3
^-D
I
P^(K-1)E^
whence
(14).
between P, E and we have are jterti-ctly general, although only deduced them a that of a vrry special case, E and parallel plate condenser.
As mv
I)
/> an-
f4j
equal
a
at
see (p. 57) these relations
shall
the furces which would be exerted on a unit charge shapes made in the dielectric.
in cavities of certain
jhtint
Taking the
if
east*
E
first,
E
we observe that
the force which
is
wnuH
K/ vxt-nel
ti^rthrr with the Puissun distribution to
which the polarization
1
tlif
"I
dit-Ii'drie
iiiediiim is
*
equivalent.
This will be the actual
an actual cavity of indefinite length and i*S: ^-r.,a! rn^ *-ction, whose axis follows the direction of the * f int. For the charges which develop on " 'M"> s^ch a cavity, owing to the existence of polarization, "itiiMl : the two ends, and the contribution from these r*
*
-.uit ehar-.je
'j
in
?
.1
*ii
r
1
^
-
ins-i ir
^
:
the cavity will vanish in the limit,
ii^'le
to
diminish
; 1
r
V ^
''
:
t:
4l * t
,ij'l
.
'*
/*
4l
will
nuinc-d s*.lely by the real 1 1- identical with E.
^ :
a
Cilvit
and
^
i.^h
^
;
force will
fictitious
charges in
y whose cross section
;
ulthuiigh both
when the
The
indefinitely.
is
great compared Let the end
are infinitesimal.
e.i;ny be noniial to the direction of the polarization i
...,*'
...
i
..
-i
_
-1-
.
DIELECTRIC MEDIA
53-
P
charge due to the polarization and equal to per unit area. The force on a unit charge in such a cavity will arise partly from the charge on the walls and partly from the charges, real and fictitious, in the rest of the field. latter to E.
The
The former
total force
is
therefore
part
equal to
is
P + E = D.
P and
The
the
resultant
D at any point is therefore equal to the force which would act on a unit positive charge placed in a very flat cavity cut
induction
perpendicular to the direction of the lines of force at that point. It is easy to
show that the component
of the induction in
any
the component, in that direction, of the force which would be exerted on a unit positive charge placed in a similar flat
direction
is
cavity with its end faces normal to the direction in question.
The foregoing
specifications
the
of
induction and
electric
intensity satisfy the conditions which we have already laid down for them (p. 45) at the interface between two media and 2
K
of different dielectric coefficients.
and
B parallel
to the interface
For, consider two
and
indefinitely near to
normal component of the induction just inside K^
K
l
flat cavities
will
it.
A
The
be equal
to
K
the force on a unit charge in A and that just inside 2 will be to the in B. force on If the resultant a similar equal charge acting
two media are E and E.2 and the resultant and P2 and if they make angles 9 and 9.2 with the surface, then it follows from the results of the
intensities in the
polarizations are
the normals to
P
1
l
,
l
,
preceding paragraph that the normal force in
A
is
(E
l
-f
PI) cos
d*.
,
DIELECTRIC MEDIA
54 another can
seen by considering the charges, real and fictitious, arises from force inside they arise. The IK*
A
than 1
1
charges at a distance,
1
r2l
the jiolarization charges on the ends of
fill
the fictitious charge over the
Th
la^f
is
made up of a
boundary
A
}
0.
from the
positive charge arising
charge arising from the j r on G is equal to the The
and a
different negative
.
}
i/^iia* charge on C? is equal to the positive charge on an equal i^4 4 /;. The total force in -4 is thus the same as that which 1
H
,IS!M"
ttjli
1
(
fpilll
charges at a distance,
1
A
(2)
the positive charge on the right of
(3)
the negative charge on the left of G.
we can show that the
In a similar way
force in
I4l
charges at a distance,
<5
the positive charge on the right of C,
(*)
the negative charge on the left of B.
But
of
spK'ification
(2)
<4),
the
= (5) and induction
(3)
= (6).
makes
y
its
It
B
arises
follows
normal
from
that our
component
coiitinnniLs
The
tangential
component of the
electric
intensity
iinnini> in
is
crossing the boundary, since the only change in viectne intensity which occurs is that which arises from the
con-
the
change relative to the fictitious in^pHshiMn charge on the boundary, and his ;rnvs r is. n iy tu a force norma i to t jle i n t er f ace>
i
,,
Variable Polarization.
SM
lar
have confined our attention to uniformly polarized eas^ in which the has the same polarization
-.
P
;
m^e an.1 >tnv V !>
!
The number
of such cases verF important to the behaviour study n whleh the P^nzation varies from point to noint
mm
mi
ilnvcti,,n at all points.
,
f
and
!t is
DIELECTRIC MEDIA
DO
oppositely charged coincident distributions, if the original distributions of electric charge are not uniform. We shall, however, look at the matter from the point of view that the polarization
from the development of charges on the faces of the ultimate elements of the body in the manner which we have already considered. Consider any elementary rectangular parallelepiped of the material whose sides are Sx, Si/, z, and the coordinates of
arises
whose centre are
x, y, z.
Let the resultant polarization at the
P and let its components parallel to the coordinate axes Px Py and Pz The charges which develop over the faces of
centre be
be
.
,
the parallelepiped owing to the polarization itself will
P of the parallelepiped
be
Py 8xSz,
Px SySz,
Pz &xSy
and
These are the mean values taken over the whole of each respective face of the parallelepiped. If we consider any one of these faces, for instance that which is determined by x = # 4- ^ 8%,
respectively.
we
see that the next element, in this case the one to the right,
will give rise to a
Px 4- -^
x
is
the
charge over
mean
it
gp
/
=
(P^-f
\
&&) SySz, since
-3-7
x component of the
value of the
tion in the next element of volume.
This face
is
polariza-
therefore to be
regarded as caiTying a charge due to polarization equal altogether ri
P ~
to
SucSySz.
to the next to the
One
half of this
is
to
be considered as belonging
element of volume, so that onlv
element under consideration.
gives rise to
The
~
-T~ SxSi/Sz belongs *
2 ex
face for
an equal amount, so that the
from the laces perpendicular to the axis of x
which x = x
| S$
total charge arising is
1~ SzdyBz.
In
a similar manner we can show that the pairs of faces perpendicular to Oy, Qz cany charges which contribute
-f^Sf.gig,"
.
md -
'
respectively.
there
is
r
C
?y
Thus when the medium
cz is
associated with each element of
\-*r
+
-^~
-r
polarized non-iinifonnly,
volume a charge equal
-^} SxSuSs
(15).
to
fictitious charge like the charge which charge is a the polarization surface of dielectric substances on develops on the behaviour of dielectric media. It is to be definitely theory of the and other from the true charge carried by conductors
This
distinguished
in the field. charged bodies between the fictitious shall now consider the relation
electrically
We
when volume and surface charges which arise in a polarized body fictitious the is not uniform. charges Denoting the polarization
by dashes we have
can evaluate the integral on the right, taken throughout the If I, m, n are the method of Green's Theorem. polarized body, by
We
the direction cosines of the external normal
at
any
point,
we
see that
fff
with similar expressions for the remaining constituents of the therefore have
integral.
We
p dxdydz
= -(W* + mPv + nP ) dS z
(16),
where
Pn
is
the component of the polarization along the outward This is equal to a', the surface
normal to the surface at dS.
density of the fictitious charge arising from the polarization at that point. Hence 1
1 1
or the algebraic
sum
p'dxdydz
+
I
la-'dS^
(17),
the volume and surface charges arising from polarization is zero. This result would have been obvious had we developed the properties of non-uniformly polarized media of
by the sliding of originally coincident equal and opposite distributions of charge.
According to the polarization theory of dielectric media the of force which arises when such media are present is the same as that which would arise if the medium were all field
aether,
provided the true charges p and
a
are
accompanied bv the
fictitious
DIELECTRIC MEDIA
and
charges p
cr
of the
intensity will therefore
polarized
57
distribution.
The
electric
obey the form of Poisson's equation which
obtains in the free aether, provided the density at any point therefore have supposed to be equal to p-f/>'.
is
We
~(E v dx
so that
x
+
Px) + ljr (Ey + Py) + ~(E* + P ) = rp oz v
'
~
z7
y ....(19). v
dy
Comparing this with equation (1), we see that the vector whose components are Ex 4- PX} Ey -f Py z -h P,. is identical with ,
E
the induction D, so that the identification from a particular case
on
p.
52
is
perfectly general.
integrating both sides of equation (19) throughout any enclosed volume we see that the true charge inside is equal to
By
JfD n dS over the bounding The
surface, in
agreement with
p.
42.
Fictitious Charges on the Electron Theory.
necessary, in discussing the results of electro" " static experiments, to distinguish between true charges like
Although
it
is
those which are communicated from a conductor to the plates of a
condenser and the "fictitious" charges which appear to reside in the dielectric, there is no very profound difference between them.
According to the electron theory one
is
just as true a
charge as the other, although its reality is not so readily made obvious by experiment. The electron theory regards a dielectric as a certain type of distribution of electrons in space, and in this space the true electric intensity satisfies the equation div p.
E
This equation is assumed to be true when the element of volume is a small When the element of volume is part of an electron. enlarged so as to contain a great many electrons the equation will = p, where the bars denote average values. Thus p
become div
E
equal to the p-\- p' of equation (18) and p is just as real a part of the average density of electrification as p. This point will be considered more fully in the s.equel. is
CHAPTER IV THE ELECTRON THEORY OF DIELECTRIC MEDIA Potential due to a Doublet.
CONSIDER the doublet formed by equal positive and negative s. Let V be e placed at Q and respectively, where OQ charges the potential at a distant point P, where QP = r is large compared with
,9.
Then
__ ___
~* OQ
_ __
cos 6
-QP OP~~QP*' = in the limit
moment
when
s
cos d
=~
cos
...
vanishes compared with
r. /jt,
is
called the
of the doublet OQ.
Fig. 13.
This result may be mitten rather Let the axis of differently. the doublet be given a small displacement parallel to its length, so that Q moves to Q', where have
QQ'=Ss.
'*
= PQ + QQ'* - 2PQ 2
.
We
QQ' cos
6,
THE ELECTRON THEORY OF DIELECTRIC MEDIA
59
so that to the first order
a cos 6
and
=
;
'
ds
1
_ HL 3 !"
where
(2),
denotes differentiation along the direction of the positive
OS
In this differentiation the
axis of the doublet.
doublet
/I
moment
of the
be constant, so that the increment Ss
will supposed be determined by the displacement of the centre of the doublet. Let the coordinates of the centre be a, b, c, then s is to be regarded
to
is
as a function of
and
a, b
v-" -^-
whence
follows that
it
A
(-]^. \r)ds *3b \r) Fs*
OS
= JL iji fly we r
(3).
1)|} ^ J
m, n of the axis
s
So that
of the doublet.
If
-
^- are the direction cosines L OS
.
,
OS
;
(-}
~fa |da
Now
c
resolve the
moment
.
dc /JL
parallel to the axes, these will
of the doublet into
be /^
=
/-&,
jj*>
= mjn,
v
O
/I
\
components = U/M and
/^ ^k
(5).
Thus the
potential,
from the doublet
is
of its components. (see
Fig.
14).
and therefore
also the field of force, arising
the same as that due to the
For
sum
of the effects
at once obvious geometrically the resolution of the doublet 00 into its
This result
is
THE ELECTRON" THEORY OF
DQ
and opposite charges at each of the points R and S. This clearly incapable of changing the field in any three components must be the way, so that the field due to the the placing of two equal is
same as that due
to the original doublet.
Potential due to Polarization.
have seen that a polarized medium can be regarded as up of a series of cells having equal and opposite When the axis of the cells is over opposite faces.
We
being built
charges spread the direction of the polarization, only one pair of the parallel to faces, those which are normal to the direction of the polarization,
be charged, and each cell will behave like a single doublet. The moment of this doublet, being equal to the product of the will
length of the
cell
by the charge on
its
end
to the product of the resultant polarization
faces, is clearly
equal
by the volume of the
When
the faces of the cells, supposed to intersect orthogodo not bear any simple directional relation to the axis of nally, polarization, charges will develop over all the faces of the cells. cell.
These
and
be equal and opposite for each opposite pair of faces, be equal in magnitude to the area of each face multiplied
will
will
by the component of polarization normal to it. The moment of the doublet to which the cell is equivalent will thus be equal to that of the doublet whose components are the components of the polarization normal to the faces of the cell, each multiplied
volume of the
by the
cell.
Since every element of volume of a polarized medium is equivalent to a doublet, this result enables us to write down the from an element of volume of expression for the potential
arising a polarized medium. Let a, b, c be the coordinates of the centre of the element of volume, its sides being equal to da, db, dc. Let /, ?, n be the direction cosines of P, the resultant polarization. Then the polarized element is equivalent to a doublet, the components of whose moment are (IP, mP, nP) dadbdc. The contribution of this element to the potential at a distant point x, y, z is therefore '
-
1
f\
1 (') + ft I
+ ft I
(1)1
*.-*>
...(6),
DIELECTRIC MEDIA It follows that the potential arising
polarized
medium
01
from the whole of the
is
In these expressions
Px P y ,
polarization at the point a,
Pz
and b,
c,
are the
components of the
and not at
#,
y,
z.
The preceding formula can also be obtained by a transformation of the usual expression for the potential of a distribution of electric
charge
If this is applied to the case of a polarized medium, p and cr will represent what we have called the fictitious charges of polarization.
Thus
CP
CP X
and
o-
=
(Ps cos
,tPg\
n^o- 4-
Py cos
n f b 4-
P*
cos ?^c)
3
where cos?^a, cos 71^, cos /i^c denote the direction cosines of the internal normal to the bounding surface, referred to axes parallel to a,
Thus
b, c.
=1
1
1
I
I
I
1
iLTrrv*
477?'
1
\
dadbdc T^ + -^r + V-*) QQ riff
fJQ
I
(7 1 JJ47T7<
(Px
cos n^a
4-
Py cos ??& 4- P
Integrating the volume integi'al by parts,
-
z
cos n^
we have
-^ dadbdc = JJjda\r ^- {-P^\ dadbdc\PX ^- \-\dadbdc J ca\r) JJJ I
1
1
r da
/ _ cladbdc da \7V ]
1
1
;.-'
-- cos r
??,-a c?>S.
Since similar expressions are obtained from the other two terms of it follows that
the integral
Polarized Shells.
A
shell is
polarized
a superficial distribution of polarization.
It may be regarded as a region bounded by two surfaces at an infinitesimal distance apart and carrying opposite charges on the
In general the direction of the polarization at any orientated in any manner with reference to the point may be normal to the surfaces, but the only case which is of any impor-
two
sides.
that in which the resultant polarization is always directed normal to the shell at every point. Such shells are the along said to be normally polarized, and we shall confine our attention tance
to
is
them.
are of great importance in the theory of electro-
They
magnetism.
Let AD,
EC
mally polarized positively
and
charged.
Let
point,
be a section of the surfaces bounding the nor-
AD being
shell,
EG
P
OP being
negatively
be a distant
equal to
r.
any point in the substance of the shell and is the nor-
is
ON
mal.
are
AD, QR, AB, DC, etc. infinitesimal. The angle
PON=0. Fig. 15.
Let
t
be the thickness and
P the polarization of
AD=QR = BC by equivalent to a shells it is
the shell at 0.
Denote the element of area
Then the element of the shell ADGB is doublet whose moment is PtdS. In dealing with dS.
convenient to introduce a
new quantity
called the
strength of the shell.
The strength of a shell at any point is equal to the product of the intensity of the polarization of the shell by the thickness of the shell at that point.
We
shall
moment
$
is
denote
equal to the
Now
by
moment
consider the
^
We
it
.
Then-^Ptf.
of a portion of the shell
have seen
(p.
whose area
Since is
PtdS
is
the
dS, the strength
of the shell per unit area.
P
potential at arising from the shell. 58) that the potential at a point distant r
DIELECTRIC MEDIA
due
to a doublet of
moment u
is
U3
-(-],
where
4?r OS \ rj
differentiation along the axis of the doublet.
element of the
shell
u,~6d>S and ^
= 3s
~
denotes
cs
In the ease of the ,
~
where
denotes
en
8/2
differentiation along the direction of the outward normal to the Thus the potential due to the element positive face of the shell.
dS
is
and that due
to the
whole
shell is
)dS where the surface integral
is
(8\
extended over the whole of the
positive surface of the shell.
The most important case which arises is that in which the has the same value at every point of the shell The strength
then said to be uniform or of uniform strength. a case may be taken outside the integral, and shell is
In such
<
-//*where
a>
is
................................. l91
the solid angle subtended by the entire shell at the
point P.
We
shall next calculate the potential energy of the shell in the First consider the potential energy of a doublet OQ which Let e at 0. carries a charge 4- e at Q and Q be the field.
V
and Q
potentials at
the doublet
Then the
respectively.
,
V
potential energy of
is
eVQ -eV = e(VQ - 7 = e.OQ*?=p*Co C6 )
7
ar -~-
ex
where
p. is
the
moment
cosines of its axis
5.
+w ar + n ar\ cz
)
cy
of the doublet
Now
apply
1A
............ (10),
;
and
Z,
m, n the direction
this result to the case of the
THE ELECTRON THEORY OF
04
dS
area Considering the element of
shell.
polarized
of the shell
x ~- so that the potential energy of the whole shell ,
tis
-
Since
is
the force outward along the normal from the
tf/i
of the
side
positive
number of lines
which thread the surface from the positive is the It is thus equal to Ny where
N
to the negative side.
number
the surface integral represents the
shell,
of force
of tubes of force
which leave the
shell
by the positive
side.
Polarization on the Electron Theory.
The
electron theory furnishes a very natural explanation of The chemical atoms out of which matter is built polarization. are regarded as consisting of a large number of electrified particles. The behaviour of these particles is considered to be different
according to whether the substance
quite a conductor or an insulator
In conductors, part, at any rate, of the electrons are very smallest electric field is sufficient to cause
is
loosely held that the
them In
move about we shall
to
as
fact,
in the substance
see later, it
is
conductors uiiieli
extremely probable that in moving about inside,
many of the electrons are always in the same way as the molecules of a
a state of continuous motion.
in
from one atom to another.
The
gas are believed to be an external field is
effect of
simply to superpose on this haphazard motion an average flow in the direction of the field. This flow is what constitutes the electric
current.
When an i
hi-
electric field is
phenomena are
e..mse. be aftected
frum
iiinve
-'
by the electric field, but none of them are able atom of the substance to another. In the
uf an electric field
tlie
;il*,:it
applied to an insulating substance The constituent electrons must, of
uiie
!*j
aWmv
different.
atoms
we regard the
ni^iiKv.s the material exhibits .-n-
1" *
**
no
electric
Under these polarity. When an
tv],l I>
applied, the ultimate positive charges are pulled in th. field, and the negative charges in the opposite lilt' iliMilrtcwnHnt rf tK* /%k .--,..,->,-, i,_ ,,
>i* i-tinn
'
electrons as distributed
in positions of stable equilibrium.
1
'li.
,,f
/
DIELECTRIC MKD1A they are pulled back by forces of the same nature as those which held them in equilibrium before the external field was applied. In the position of equilibrium which finally results, the force exerted on the electron by the external force tending to restore
it
field will just
balance the
to its original position of equilibrium.
It is clear from what has been said that the displacement of the ultimate electrified particles, which occurs in a dielectric when it is exposed to the action of an electric field, is equivalent to the
creation of so
many
doublets, one for each particle.
We
have
seen that the polarization which occurs in the dielectric under the same circumstances can be represented as due to the development of doublets in each element of volume. We shall now consider the whole matter from a more quantitative standpoint ; as a result of our investigation we shall see that the results of the polarization
theory can be obtained just as well from the properties of the doublets which develop from the displacement of the electrons.
Actual and Mean Values.
We
have already pointed out (p. 9) that in the electron theory we have to deal with different elements of electric charge A somewhat similar in different classes of problems which arise. distinction arises in connection with
many other physical
quantities
which determine the nature of the electric field. For instance in the discussion of this and the preceding chapter we have regarded the induction, the polarization and the electric intensity as vectors whose magnitudes changed only very gradually as we moved from one part of the
field to another.
We
have always thought of them
as though any alterations in their magnitudes which might occur, in a distance comparable with the distance between two molecules,
could safely be considered as negligible, provided the two points compared were both in the same medium. This method of treatment obviously becomes inadequate when our view of the phenomena is so highly magnified as to take into account the effects of individual electrons or even atoms.
between two
So
far,
parallel planes filled
we have considered the space with dielectric, when the planes
region in which the electric intensity has the same magnitude and direction at every point. It is clear, however, that the actual electric intensity, are
maintained at different
potentials,
as
a
THE ELECTRON
gg
an infinitesimal a unit charge occupying the force exerted on in both magnitude and direction volume will constantly change At places which are to another. from one part of this space actual force will be enormous close to an electron the sufficiently
with what
cornpirecf
may have any
we have
and
called the electric intensity,
it
direction whatever.
what can be the use of a conception of tempted to ask so much at variance with what we the electric intensity which is The answer is, of course, that most of believe to be the reality. with the our methods of experimenting are so coarse, compared these enormous differences atomic scale, that they do not detect of atomic magnitudes. order the of distances within which occur for the most part measure only the Our experimental arrangements a large number of atoms. values over
One
is
spaces containing
average
is not because possess validity as such far so values but because, experimental they are the true enable us to detect, happens as if the
The reason why our average values
everything
arrangements
were the true values. average values It remains to specify the average values we have been dealing Let < reprewith more accurately than we have done hitherto. sent one of the scalar functions or a component of one of the
which determine the state of the electric
vectors,
field.
For
fxiimpk $ may be the electrostatic potential at a point. Let r lie any small volume so chosen that its linear dimensions are large distances but small compared with the ei.anpared with atomic 1
,
distances within which changes in
are perceptible by the usual Then methods. the experimental average value of may be Ivfin ti! as the value of <
i
(12), win-re the integral
evidentl
taken throughout the small volume
is
r.
We
have
(13),
where f is any independent variable such as time, distance, which 4 limy be a function. Since the actual potential satisfies the relation V 2
V
* >lK.w* that
V-
F=-p
;
and since
=-grad
V,
etc.
of
F=-p,
it
#=-grad
V.
Thus
DIELECTBIC MEDIA
67
the average forces and potentials are the same as those which would obtain if the actual charges were replaced by a distribution of density equal to the average density at every point. It is clear that the induction, polarization and electric intensity in a dielectric are average values in the sense indicated, and that the results that we have deduced are valid if this is understood.
Potential due
to
the Displaced Electrons.
We
have seen that in the presence of an electric field the electrons are displaced, the positive in the direction of the field and the negative in the opposite direction. shall see that the displacement thus
We
produced of
is
moment
equivalent, for each electron, to the creation of a doublet = es, where e is the charge and s the displacement, p,
This doublet will contribute to the potential at
of that electron.
a distant ^ point
P
an amount ~~ ^~ ;
4-7T
)
,
and
if
there are v such
ds \rj
doublets per unit volume the total potential to which they will will be give rise at the point
P
i
err
d
n\,
^ os \rj \\\vp,[-\dT. 47rjJJ In general the different electrons in the atom will be variously situated so that they will not all undergo the same displacement a given electric field. We may divide them up into classes, the electrons in a class being characterised by a given value of for a given field. Suppose there are n such classes and let vp
s in all
s
,
p
jji
and
class.
sp
denote the values of
Then
it is
v,
p, s for the electrons of the
pth
clear that
(14).
We
shall
now
consider the relation between the
moment /^
of
The the doublets and the electric field which produces them. exact form of the relation between the restoring force and the displacement will depend on the arrangement of the electrons in At present our knowledge of this arrangement is very the atom. limited but, in any event, the restoring force must be a function of the displacement, which vanishes when the displacement is zero
THE ELECTRON THEORY OF
g$
from Taylor's Theorem that for small displacements the force must be proportional to some odd power of the
It follows
restoring
displacement
and since the frequency of the natural vibrations of by their optical properties, is independent of
;
bodies, as exemplified
natural to suppose that this power is the first. We shall assume, therefore, that when an electron of the pih class is displaced a small distance sp the restoring force is equal to
the amplitude,
-f
X p Sp,
it is
where \p
a positive quantity which
is
librium
constant and
When
the state of equiattained, the pull of the external electric field on the The oc component of balanced by the restoring force.
characteristic of this class of electrons.
electron
is
is is
3F the force on the electron
is
ep
where
,
^
V is
the actual part of
the potential at the electron whose charge is ep which arises from If scp is the # component of sp the presence of the external field. then, provided the reaction to the displacement is independent of Its
direction in space,
measured
the x component of the restoring force
in the positive direction of
%
is
--
os
p
and the equi-
Xj>
libriiini
value of this displacement
is
^--X^ Xow
..................... (15).
a moment's consideration shows that
when a charge
placed a distance xp the electrical effect is exactly the that which is produced by the creation of a doublet whose
For the displaced system e^,. which is obtained when a doublet
is
distance a
way
c.
.incidt-s
IN
^
that
apart
xp
is
ep is dis-
same as
moment
absolutely identical with that ep at a consisting of charges
is
superposed on the original system, in such displacement^ and the charge -ep
coincides with the
with the original Thus the displacement (15) charge +eP Mjuivalent to the creation of a doublet whose moment is .
dV 111
!h " "!>*-"<*
>*'
an
electric field the
medium
is
unpolarized charges forming the *\> .line SJMVIU.S is zero. Thus the potential V due to the p polarized ni^miii is that which arises from the totality of the doublets which
"ii'J
-
th- jMtt-niial due to the distribution of
DIELECTRIC MEDIA
From
equation (5) for the potential due to a single doublet we SFp, the part of Fp which arises from the doublet the x component of whose moment is given by (16), is
see that
T^TT
{(j3Cp
OXp
Now in addition to being made up of electrons the matter with which we are dealing possesses a coarser type of structure which we may term molecular. Each unit of molecular structure, which we may fying
it
by the
refer to as a molecule without necessarily thereby identi-
with the more definite chemical molecule,
is
characterized
fact that, referred to its axes of
electrons satisfy
symmetry, similarly situated identical structural conditions. In considering a
structure of this kind
clearly absurd to
it is
endow the
electrons,
we have
done, with the property of suffering a hypothetical restoring force which, for a given displacement, is independent of the direction of that displacement in space. As we are confining as
ourselves to the case of non-crystalline substances this difficulty may be overcome by taking sp pj yp ,zp to be the average
^x
displacement of all the electrons which belong to the class p in a given small region and recollecting that all directions for the axes
symmetry of the molecules are equally probable. Suppose that each molecule contains n electrons, so that p has all the values from 1 to n, then P the average contribution to P which arises from a molecule at the point a, b, c is of
V
&V
*-*
A
4"7r
^"D p "M p
p=1
i *-\
f\
l
/
*
(dap da p \rj
^"7
If the dimensions of the element of
pared with r
=
coordinates of
{(x
-
P and
molecules, the part
volume dr
is clearl
a)-
+ (y
if it,
dVp
+ (z
2
&)
^T"
~
'
I
cbp cbp \rj
'
^ ccp ccp \r
volume dr are small com2 where x, y, z are the }-,
c)
nevertheless, contains a large of
VP
which
\
arises
number
of
from the element of
where v
is
number
the
of molecules in unit volume.
'(37 3
/T\
.
373
Thus
fl^
r'--** 3 (17).
with By comparison
components
equations (7) are
of the polarization
=-!/
2
and (16) we find that the
P
o y
^
.(18).
P =-v 2
2 3_F
z
=
*
P= thus equal to the sum of the moments of all The dielectric coefficient doublets in unit volume.
T/ie polarization is
the equivalent
K
given by the relation
is
.(19).
Thus da
For crystalline media, \p will take different values for the different and c because the axes of the molecules are definitely
directions a, 6
orientated in such substances.
Xow if we
average over a large
number
?>V
of molecules, -
under
dap the sign of summation in (19) does not become equal to the value of in the
cif
same
region.
these depends
I'K-etrons in
for this.
The
first
upon the defimteness of the arrangement of the The electron whose type we have
the molecule. the suffix
indicated
There are two reasons
by p always subject, owing to the definite structure uf the molecule, to certain geometrical relationships with the other electrons in the same molecule This fact is not taken iiei-i
is
unn of in the definition of
F and
of -5-
oa
in*k'j*ndenT ni'.k'cule
and
of is
the
.
The second reason
is
arrangement of the electrons within the caused by the molecular rather than by the
DIELECTRIC MEDIA
71
electronic structure of matter. The nature of this second factor can be most readily brought out by considering the dielectric properties of an ideal kind of matter whose imaginary molecules
are so simply constituted that the
first
shall therefore consider the case of
We
factor does not occur.
a substance whose molecules
are monatomic and whose atoms give rise only to a single doublet each, under the influence of the electric field.
Case of the Ideal Simple Substance.
We
suppose each molecule of the substance to consist of a and that, under the influence of the external field, each atom, single atom develops a single doublet placed at its centre. The forces acting on one of the electrons whose displacement gives rise to this doublet consist of (1)
the restoring force called into play by
its
displacement,
the force arising from the charges in the field, including (2) the doublets of the polarized medium not situated in its immediate
neighbourhood, the force arising from neighbourhood of the atom. (3)
When
It is clear
is
equilibrium
the doublets in the immediate
established
from our discussion of the
electric intensity in dielectric
media that
(2) is equal to the electric intensity
remains to discover the nature of consideration
as
centre -^
cs
comparable with
enough
it.
=
dV .
It
Co
About the doublet under
describe a sphere
,3F
small that the value of
(3).
-
whose radius
is
so
in a distance does not vary appreciably rr
At the same time the sphere must be big number of molecules. The force (3) will
to contain a large
be equal to the force exerted by the doublets in this sphere on the electron under consideration. This will only be true provided the
dimensions of the sphere are within the assigned limits otherwise this force will not be independent of the radius of the sphere. :
To
calculate the
magnitude of
(3)
we suppose the
spherical
THE ELECTKUJN
72
On account of this, and the centre of a small spherical cavity. fact that the doublets behave on the average like the to owing the equivalent polarization P, the ^doublet at the centre will be a force, additional to E, whose amount is determined acted on by
on the walls of the cavity. by the equivalent polarization charge at any point, on the surface area cos This is equal to per unit an angle with the makes of the sphere, the radius to which
P
direction of P.
The
resultant force
over the spherical surface
is
/Y
thus
1
1
due to the whole distribution
&P -
j
2
2
cos 0dS,
where the integral
This extends over the surface of the sphere whose radius is r. of radius the of the is and to sphere. independent equal |Pe
is
The remaining part of (3) consists of the force which would be caused by the doublets which we have removed, if they had not been removed. This will depend very much on the geometrical arrangement of the atoms among one another. In certain parA doublet situated at a point ticular cases this force vanishes.
whose coordinates are
#, y, z,
with respect to the centre as origin,
and whose moment has components equal to to a force at the centre, whose x component
JJL
X , /^,
^,
will give rise
is
*
,
4TT
where If the
r2
=# + 2
2 ?/
4- z*.
atoms are arranged fortuitously so that any one position
in the sphere is as likely as
mean
another the
values
Hey
= ~xz =
and
- r2
2
__ -
:3t/
-r
2
'
_ 3^ - r 2
-V
3 L =-
(ofi -f
It
u~ 4- z-}
J
2
3r2
=-
follows that the force
arising from the doublets which we out of the cavity vanishes on the average, if the atoms are arranged fortuitously. The same is true if they are in regular cubical order* It follows, in either case, if thv atoiti* haw the .simple constitution we have that the
liaw
ivitiitYed
imagined,
*
H.
A. Lorentz, Theory of Electrons, p. 306.
DIELECTRIC MEDIA force (3)
When
=^Pe.
73
the molecules have a less symmetrical
distribution, the additional force arising from the molecules which we took out of the spherical cavity will still be proportional to the
polarization P, so that we can represent the more complex cases if we replace the factor -J by an unknown factor a depending on the
configuration of the molecules.
Returning to the more symmetrical distribution, we see that the total force acting on the electron at the centre of the atom under consideration is, on the average,
..................... (20),
so that
comparing with formulae
-P,
K
,
whence
If
-
we apply
(19), since
n=I,
/0 _
P=-
,
and
and
(16), (18)
*
x
..................... ^ } (21),
1
this formula to the case of a gas,
we
see that the
only one of the quantities on the right hand side which varies with the density of the gas is v, the number of molecules per cubic centimetre.
K K
This
is
proportional to the density, so that for a gas
1
-
+2
should be r to the densitv. proportional r
The
results of ex-
periments are in agreement with this formula within the limits of experimental error, although the experimental measurements of the dielectric constants of gases are not very exact.
When we come dispersion of light, which K is replaced
to consider the
we
phenomena
of refraction
and
shall see that a very similar formula, in
2 by n connecting the ,
refractive index n with
It seems the density, can be developed along similar lines. advisable to postpone the detailed discussion of the experimental evidence for and against these formulae until the optical phenomena
are considered, as the evidence will then be vA/r^
ol->oll
ri/-v\iT
fii-t-ii >v-k
-f/-\
/~/-\Y-
01
H
ti-v*
-i-T^ti
much more -h^or
/-\-T
f
n *^
complete. 4-Tt-/-v
voo O/-M-I o
THE ELECTRON THEORY OF
74
value why the average
of
is
^
not equal to
^
.
This one depends
itself and not on the on the complexity of the atomic structure constitution. atomic an The matter that fact mere possesses
by considering a very contain a very large to atom an Suppose exaggerated case. If such an atom is held. all number of electrons very loosely nature of this factor can best be realized
field it will behave like a conductor of the placed in an electric same size and shape so that there will be no field acting on the ;
The
electrons in the interior.
atom
will
move
electrons towards the outside of the
so as to shield those inside
from the action of the
The same effect will also occur to a smaller extent even when the number of electrons is comparatively small external electric force.
and their displacements are inconsiderable. It is clear that the a small volume of the average value of the force throughout material
is
different
from the average value taken over a particular
type of electron.
The
force acting on an electron inside a molecule will arise from the charges outside the molecule and partly from the pirtly doublets inside the molecule itself. We can regard each molecule
as equivalent to a simple atom possessing the same average electric moment, so that the force acting on an electron inside a molecule arising from external causes will
be
e
(E -f aP) where the constant :
depend on the geometrical configuration of the molecules, when the distribution is a fortuitous one as taking the value in a fluid. The way in which the second part of the force on the a will
internal electron depends upon the external field may be realized by considering the conditions which are necessary in order to
change the displacements of
all
the electrons in a given ratio. so that
The displacements are proportional to the forces acting, this means that the force acting on an electron in the field changed in the same ratio at every point.
will
be
Now
the force arising from a given doublet is proportional to the moment of that doublet, so that the part of the force acting on the given electron which arises from other doublets in the same atom will be altered in the
same
ratio as the total force at
that the difference
part of the force
between
which
is
any point in the field. It follows and the total force, which is the
this
of external origin,
must be changed in
DIELECTRIC MEDI^
75
the external field may change, the force acting on any assigned electron will always be changed in the same proportion. This result may be represented by putting M -'pv"p
o
CSp
where
Lp
is
v-*-*
"r
*--
................(
i
Lo
)
5
a constant characteristic of the pth class of electrons.
Comparing with
p.
73 we see that
tf(E + aP)
=P=
(
K -l)E,
so that
and
~~
q
1
1
- a2z/ \pLp ep
z
_
= av 2
\p Lp e/
('25).
P=I
\p Lp and
ep
,
but they
will
vary for different electrons in the same molecule have the same value for corresponding electrons in
may
different molecules of the
The expression on the
same substance.
may therefore be represented by a summation over each molecule multiplied by the number of molecules of the sub-
right hand
side
We
stance in unit volume.
therefore find that
(26),
where k
We
is
a constant and p
shall
significance
find
that
is
the
when we come to The investigation
the density of the substance. coefficients
\p have an important phenomenon of optical
consider the
leading up to formula (25) will not apply to optical problems without modification, as the displacement of the" electrons in such cases is not necessarily always in phase dispersion.
with the corresponding
When we
"
force."
are dealing merely with the electrostatic behaviour f*xr\ a-flvYrrl fn -npo'lpr flip r-mnnlir^f irm<; in^f
nf rH Alar'f.vi^G TVO
1
!-,
THE ELECTRON THEORY OF DIELECTRIC MEDIA
70
which arise from the mode of arrangement of the electrons in which have been the molecules. It is clear from the considerations to,
instead of (16) brought forward that
where
? is a
cr
new
constant,
we might have put
and thus obtained
*-l = OLV 2
(28),
OL
n
2 o>e/= 2 \p Lp e/.
that
cr p and the natural frequencies of not an obvious one without further investigation. of importance in dealing with the corresponding
However, the relation between the material
This point optical
is
is
phenomena.
CHAPTER V MAGNETISM
WE
are not yet in a position to enter into the theory of magnetism in any fundamental way. We shall have to defer that until we have considered the phenomena which are grouped under
the head of electromagnet-ism.
seems desirable, however, at
It
this stage to enumerate some of the elementary facts and principles of magnetostatics, and to consider to what extent they
resemble or differ from the corresponding phenomena of electrostatics.
One first to
A
of the most striking of magnetic phenomena, and the is the occurrence of intrinsic magnetization.
be discovered,
body exhibiting
this
phenomenon, usually called a permanent
magnet, possesses polar properties. When two magnets are compared it is found that the ends A, A' of the first always exert forces Thus if A and of a certain kind on the ends B, B of the second. f
S
repel one another so do A' These relations attracts
B
and
',
whereas A' attracts
B and A
f
between the two magnets are in fact the same as those between two similar portions of matter endowed .
with electric polarization.
Although
this formal
resemblance between magnetization and be borne in mind that there is no
electrification exists, it is to static reaction
between a magnet and an
electric charge.
behaviour of magnets enables us to speak of the magnetism at the one end as positive, and that at the other end as negative. And here, at the outset, we meet one of the It is clear that the
most striking differences between magnetization and electrification. It is impossible to separate the magnetic charges from one another. There Every magnet carries equal and opposite magnetic charges. between two small magnets which varies inversely is no force acting
MAGNETISM
75
attracthe square of their distance apart except gravitational is intrinsically polarized. which a is a Thus magnet body ting
si>
a
common
in electro-
phenomenon Although polarization under the action of an statics, and occurs in every dielectric is
external electric
field,
intrinsic polarization is
comparatively un-
in crystals; where it appears to be probably occurs iniportarit. iviiuiivd to explain the production of electrified surfaces by fracture It
It
ami the phenomena of pyroelectricity. intrinsic
dielectric
supposed
For unless a -substance
always cover is
just
polarization
itself
such as
is
is
possible, however, that
commoner than is
is generally a good insulator, it will
with a distribution of electric charge which required to annul the external action of any
intrinsic polarization it fore,
is
may
possess.
In
all
but a few cases, there-
the existence of such a property would be
difficult
to
detect.
we are unable to separate the the investigation of their mutual forces opposite magnetic charges, as the in so is not theory, corresponding electrical problem. simple,
On
account of the fact that
In a very elaborate investigation
Gauss examined the interaction
1>etweeii two magnets and showed that the forces were such as we mid arise if each element of magnetic charge repelled each like, and attracted each unlike, element with a force proportional to
the product of the charges on each element,
squares of their distances apart. considerable accuracy.
We
are
now
and inversely as the This result was established with
in a position to define our unit of
magnetism.
Consistently with our definition of the unit of electric charge we shall define it as that charge which repels an equal and similar at unit distance from it with a force equal to charge l/4nr dynes. To avoid the difficulty which is created by the inseparability of the magnetic we charges from the opposite that those
poles may suppose between which the forces are measured are at the
ends of infinitely The other long uniformly polarized magnets. ends will then be so far that will exert no influence. away they is defined, in an analogous as the force exerted on a unit pole at intensity,
Magnetic intensity
field.
It is
convenient to
way
to electric
any point of the use the term magnetic force rather than
MAGNETISM magnetization, which
is
the
name
79
usually given to the quantity in
magnetism which corresponds with intensity of polarization in
The magnetic potential at a point is the work electrostatics. done in bringing a unit pole from a point at an infinite distance up
to the point in question.
we have
with dielectric media., so in magnetism there are media which, without being themselves permanently magnetized, have the power of modifying the distriJust as in electrostatics
to deal
bution of magnetic force in their neighbourhoods. There is therefore a vector called magnetic induction analogous to the electric
We
induction D.
B
and
shall denote
the magnetic induction by the
its
components by symbol which corresponds to the dielectric permeability and denote by p.
Bx B ,
IJ7
coefficient
Bz
.
The
K we
coefficient
shall call the
The behaviour of different media towards magnetic force more variety than the corresponding electrostatic phe-
furnishes
For a few substances
nomena.
may have
very large values, and usually these are the substances which are capable of being permanently magnetized. Since iron is the typical example of this JJL
they are often called ferromagnetic. For all does not differ greatly from unity, on the electromagnetic system of measurement. It is found that, in addition to the substances for which JJL is greater than unity, and whose
class of substances
other substances
behaviour class for
is
p,
analogous to that of dielectrics, there is another large is less than unity. The former are said to be /JL
which
paramagnetic and the latter diamagnetic.
It is very probable that paramagnetism and diamagnetism arise from the operation of shall, however, postpone the consideration of separate causes.
We
the physical causes which underlie the varied magnetic behaviour of substances until a later chapter. At present we are concerned with the formal relationship between magnetism and electrostatics. If
H=H H H X)
y,
z
is
foregoing considerations,
the magnetic force it follows from the combined with the results of preceding
chapters, that the following propositions are true for (1)
in a
The
medium
force
between two poles of strengths m,
of permeability
p,
at a distance r apart
-, 4<
magnetism in'
embedded
is
(1). /
/ />*-
:
MAGNETISM
H (l 1
2
of magnetic intensity can be are of lines of force whose equations
The
1
iii*aiis
field
da___
(8
(4
The magnetic
1
The equation
1
dy
potential
satisfied
_
dz
fi at
by
ds
___
,^\
any point
O in
mapped out by
is
free space is
vn=o 1
5
If rf$
)
is
an element of
(4).
any closed surface not intersecting
a magnetic medium, that- is to say, one which lies entirely in a medium of permeability unity, and if I, m, n are the direction cosines of the normal to this element, then
taken over the whole of the closed surface
is
equal to zero.
This
the magnetic analogue of Gauss's Theorem for the free aether, and is true since each magnetic substance contains equal and is
magnetic charges. It follows that in free space the is a solenoidal vector. intensity magnetic c*
opposite
*
H
of a system of magnets may be obtained, as system of electric charges, by bringing up equal fractions of the final system, one at a time, from a state of infinite (0)
The energy
in the case of the
dissemination. se pirate
As we do not wish
magnetic charges
it is
to
contemplate the existence of
desirable to regard the disseminated
elements as magnets and not charges. This introduces the intensity of magnetization (IX) Iy Iz ) instead of the density of In this way the energy of charges into the final expressions. a system of magnets is found to be ,
the integral being taken throughout the magnetized matter.
It
follows from this expression, by a calculation similar to that carried out in the similar electrostatic case, that the energy per unit
volume of the
the free aether
field in
is
.............................. (6).
The '
points involved here are discussed at length by J Jeans --.--. -
MAGNETISM
The magnetic
(7)
be represented as arising from
forces can
the following system of Maxwell stresses in the aether
_
if/any-
(2Q\*
:
cQ\~]
Coy)
}
an an
= 80 8O d,x
"dz
fc
<&/
= - an an ?y a2
,
2
These are equivalent to a tension ^H' per unit area along the lines of force and an equal pressure at right angles.
The
(8)
magnet
at
potential at a point, at distance
a, 6 ; c
^
whose moment
i(3
"
I=
J2
is
p = ^x
/r
,
/ju,j,
3 /i->
.
due to a small
r,
/t, is
a
.
/r
1
,^
the intensity of magnetization (not magnetic intensity) the potential to which it gives rise at a point If
(9)
at distance r .Q
=1
/a;,
{(x
/, p
a)-
is
+ (y
ffk i (1) + 7. 4 (i) + c6 da
4?r JfJ J
''
[
/ being the value
V
2
2
6) 4- (2
V/V
?/
at the point a,
c)
7,
p
is
)l
cc (i \r/
dadbdc,...(9\ "
'
"
j
&, c.
The potential due to a uniformly magnetized shell at a (10) it subtends a solid angle &> is which at point
where
<$>
is
the strength, or magnetic
moment
per unit area, of the
shell.
(11)
The
in the magnetic field potential energy of the shell
W=
3
.(11)
is
MAGNETISM
3
of the magnetic induction calls for rather From the electronic standpoint it is consideration.
The behaviour
)
more detailed the induction as a vector flowing out in undesirable to define The reason for this, as will be tubes from the magnetic charges. clearer later,
is
than that of an electric
artificial
a magnetic charge is. more We are on safer ground if charge.
that the conception of
as the force in define the magnetic induction
we
since kind considered in Chapter IV,
we know
surfaces.
two modes It
other.
would be unfortunate were
make of
our magnetic measurements.
It
clear that the
is
cavity of the
when magnetized develop on the new
Chapter iv that the
induction are consistent with each
use of the conception of a
all
flat
that,
substances are fractured, opposite polarities It is clear from the reasoning in of defining the
a
it
otherwise, as
we have
to
magnetic pole as the foundation
whole of the theorems which
we have
the heading of Poisson's Theory of polarized media grouped under may be transferred bodily to magnetism if we replace polarization
theory was originally developed by It follows that when the Poisson as a theory of magnetostatics. there are surface charges but no volume magnetization is uniform
by magnetization.
In
fact the
'
charges,
of
any
and that in any event the algebraic
sum
distribution of magnetization is zero.
are the magnetic force, the of magnetization at
and
of all the charges
Also
if
H,
B and /
magnetic induction and the intensity
any point of any medium,
B=H+I = {jiH
(12),
I=(p-I)H
(13).
These are vector equations and are each equivalent to three equations between the corresponding components of the vectors. Let us now describe any closed surface S intersecting the Let us cut away an magnetic media B, C, etc. in D, E, etc. infinitesimal layer of each of the magnetic media at both sides of the intersection. Then the surface lies entirely in the medium whose permeability is unity, so that if x z are the comy ponents of the force on a unit magnetic pole at any point of
F F F ,
the surface
til
A
,
MAGNETISM surface,
FX3 F Fz
BXi
Bz
B
are actually the components of the induction It follows that over any closed surface in spice in
fn
.
(14).
Fig. 16.
^
It follows that
~
da-
and the magnetic induction
~~
.(15)
CZ
CIJ
5
solenoidal
is
everywhere.
The
corresponding result only holds for the electric induction in regions which do not contain any electric charges. Where there are "true"
charges of volume density p x
_
~T"
<.
cx
^
~
cy
,-
cz
Force on a Magnetic Shell.
We
have seen that the potential energy of a uniform magnetic T -A is the number of" lines of where
of strength $ is magnetic force threading
.shell
,
N
the positive direction. force on the shell in the calculate the to enables us
.shall
see that
it
may be
the boundary of the
becomes
N + SN.
it in
This result field,
and we
represented as so much per unit length of Let the shell be displaced so that
shell.
The diminution
N
of the potential energy due to
the displacement will be SN, and this will be equal to the work done on the shell by the magnetic forces. The value of 8-V is easily calculated.
Consider the prismatic figure, bounded by the
MAGNETISM
S4
%
thf
*!#
We
shall suppose
tlitMlispkeeiiieiii
this figure the total Sinn- ilwiv aiv no magnetic charges inside flux of force over its
y
4.
8JV
is
boundary
-y+
1
1
//"
cos
Hence
zero.
nH sin ^s& 8*' = 0,
ex
Fig. 17.
where If *r$
is
is
the magnetic force at any point s of the boundary,
the angle between the direction of the displacement S%
the element ds of the boundary, and ?^ff direction of
H
&Y = - &r JH cos
if
A'
is
the angle between the
and the normal to the element of area
Thus But
is
and
?z^& sin
the force acting on the whole shell in the direction
A
Sac
A
H cos nH sin xsds. .
)
.N
the force A"
is
equivalent to a set of forces of A
(f>H cos
amount
A
nH sin xs
Since x is arbitrary this IKT unit length of the edge of the shell. To find the direction of the gives the force in any direction. ivrsiihant
force
A direction sin xs t
(is.
It
is
ilie
It
is
we
=
also
notice that
normal
Therefore along the
magnitude of
when
and
as
s
are in the
same
0, so that the resultant force is at right angles
to
H
}
since
it
vanishes
common normal
this resultant let us
to
H
when and
nH =
7r/~2..
To
find
ds.
suppose that the displace-
MAGNETISM
So
H
and nH = Tr/2 Hs, where Hs is the angle between and Thus the force on a length cfe of the edge of the shell Is
ds.
A (f)H sin
and
is
result
H and
at right angles both to
ma
be written more
If the force at a point
Thus the
-.
boundary of the
In vector notation this
a single magnetic charge m placed acts along r and is equal to
H
r from ds, then
7? it
-r
c?,s\
as
briefl
H arises from
P distant
Hsds,
force exerted
m
by
on an element ds of the
j-^ sin
shell is equal to
?$(?$.
This must be
equal and opposite to the force exerted by the element of the shell on in. Hence the force on the pole is equivalent to a series of forces of
amount
-r-
of the shell.
i
*2
sin rsds arising from each element ds of the
The magnetic
of the boundary
is
intensity at
$
is
due to each element ds
thus
TJ
and
P
boundary
A
-
j sm rsas,
perpendicular to both r and
ds.
These results have important applications in the theory of electromagnet-ism.
CHAPTER VI ELECT ROM AGNETISM The Magnetic Potential due
to
an Electric Current
IN 1820 Oersted showed that an electric current gave rise to forces acting on the poles of a magnet placed in its neighbourThus an electric current gives rise to a distribution of IiiMicl.
magnetic intensity. between two points
Since the difference in the magnetic potential is the work done in taking a unit pole from
one point to the other, it follows that the magnetic intensity is The potential is the spice derivative of the magnetic potential. It is clear therefore that the a scalar quantity. essentially mag-
any point due to an electric current is a quantity which possesses magnitude but not direction, and which depends the point relative to the circuit only on the position of carrying on the and current the magnitude of the current. If we can deternetic potential at
mine the potential at every point of space arising from the electric current, we can deduce from it the distribution of
magnetic
intensity.
We
shall
now
consider
how the magnetic
potential
may
be calculated.
We tions
shall base
our demonstration on two empirical generalizaThey are
which are the result of experiment.
:
That the magnetic force arising from an electric current round a given circuit is flowing proportional to the magnitude (1)
of the current.
That any circuit carrying a given current can be replaced one which carries the same current and by continually zig-zags across it to a small distance on each side, without altering the magnetic force to which it gives rise. (2)
The r
first
JTroflilT" OiYirl
generalization was established ilici c*c\r.f\t-\A
I^-TT
4-"U
,-,
.-.I?
A
^
by the experiments of T
,
/
11
n
,~^
ELECTRO 31 AUJN JUTISJH that two equal and opposite currents flowing in the same circuit same branch of a circuit give rise to no magnetic force. This must be the case since the total current is then zero. or in the
Let us take a large number
of pairs of points on the of and each circuit boundary any Suppose join pair by a line. that along each of the lines there are two opposite currents flowing, each equal in magnitude to that flowing round the
The
boundary.
lines
we have drawn have divided the
circuit into a series of small areas
bounded by the
lines.
entire
Round
each small area there will be a current flowing equal in magnitude to the original current and circulating in the same sense
about the normal to the effect of circuits,
It follows that the magnetic circuit. any circuit is the same as that of any number of small bounded by the same contour, into which it may be
completely subdivided, provided each current as the original circuit, and that
circuit all
carries
the same
the currents flow in the
same sense about the normal to the imaginary surface in which they lie. The potential due to the whole circuit must therefore be equal to the sum of the potentials due to the constituent circuits.
We
shall now show that the potential in a plane is zero at every point in its own plane. First consider the potential at the point due to
due to any
circuit lying
the circuit A BCD (Fig. 1 8) bounded
by
arcs of circles
at 0,
0. field
and by Since are
whose centres are
radii passing
the
all
reversed
through
forces
when
currents are reversed,
it
in the all
the
follows
that the potential at every point will reverse when all the currents reverse.
The
current
A BCD
will therefore reverse if the potential at the point is reversed in direction. But the direction of the
current can be reversed by simply rotating the circuit through the angle TT about the line of symmetry through
A BCD lying
in the plane of the paper.
But
and not
nothing which enables the potential at tell which way UD the circuit
direction, there
the point 0, as
it
is
were, to
since potential has only
magnitude
AECD
ELECTROMAGXETISM
SS
must therefore have the same value after Its value must therefore be zero.
Tiit* |H>tential
lie*. is
it
iwrrsnl as before.
Xnw
A BCD
if
is
by tlrawing a large
bounded by
tijuivaK-nt circuits
contour can
IK*
original circuit
replaced by a coplanar circuit of any shape, it can be of radii from replaced by
number
radii
and
its original
contour.
The
replaced by a series of concentric arcs since the will give rise to the same field if it is caused to
Each of these constituent circuits zig-/ag about its original path. zero at the to rise point 0, so that the whole circuit potential givi-s at rist- to zero 0. potential gives
X w c<
nsider any circuit
A BCD (Fig. 19) lying in space.
Describe and whose apex is at 0. Also flesi-ribe a sphere of unit radius about 0. Let this cut the cone in the cnnv EFGH. Let i be the current round ABGD. From draw a series of lines OEA, OFB, etc. intersecting the curves in th- cone on which the circuit
E, F, A,
B
t
etc.
lies
Imagine equal and opposite currents
i
to flow
Fig. 19.
alung each of these ar.iiin.hliP
thr only opposite i!
curve
lines.
GFEH
Suppose
also that a current i flows
in the direction
from
new current that has been added; currents along FB. etc. cancel each
-w
F
to
R
This
is
since the equal and other. The system
contemplated resolves itself into a series of currents i round such as ABFE. Since lies in the plane of each of these circuits the due to the system of currents is zero. potential at The potential at due to i round is thus equal and circrais
F^EHG
OUlU'iSITH
Tit
flint vrmnrl
J
vD/~m
mi..
_.
.
,
,
i
i
ELECTROMAGNETISM is
therefore the
Now
E-*FGH.
tf
same as that due to the same current
up the whole of the surface, cut off from the sphere of unit radius by the curve EFGH, with equal contiguous and superposable small areas. Imagine a current This will equal to i to flow round the boundary of each area. leave the original current i flowing round the boundary, and two round
let
us
fill
The equal and opposite currents along every line inside it. to the will therefore be circuit due to the equal original potential all the Since small circuits. sum of the potential due to the small circuits are equal and have the same geometrical relation to It 0, the potential due to each one of them must be the same.
must be proportional
to the current
i
since the force
due to any
system of currents changes in the same ratio as the currents when The argument all the currents are changed in equal proportion. is
circuits, and the only that the potential due to the of those due to its parts. It follows that
independent of the shape of the small
other condition to be satisfied
whole circuit
is
the
sum
is
the geometrical factor to which the potential of each constituent circuit is proportional is So), the solid angle it subtends at 0.
We
therefore conclude that the potential due to each elementary is a constant which circuit is AiSto, where depends neither on
A
the magnitude of the current nor on the geometry of the circuit. It follows that
when the
current subtends a finite solid angle
a>
the potential to which it gives rise is Aico. This is universally true since we have proved that all circuits carrying equal currents
and lying on the same cone whose apex
is
at 0, give rise to equal
A
The magnitude of the constant potentials at 0. depends on the unit in which we measure the current. The usual electromagnetic unit of current is defined by making A=l. In the units used in this book A = (47TC)" 1 where c is the velocity of ,
light.
(See
later, p. 112.)
In our study of the properties of polarized shells
we saw
that
the potential of a shell of strength was equal to (f>o)/4<7r, where a> was the solid angle subtended by the shell at the point where the An electric current is thus equivalent in potential was measured. its
magnetic action
4>7rA
to a
magnetic shell whose strength
is
equal to
times the intensity of the current.
There
is
an important difference between the field due to an and that arising from the pinmivalpnt, rna.o-nAtin
electric current
!l<
KLECTROM AGNETISM
I
taking a unit pole Minn! ^ i-Instil curve which pusses once through the shell is zero, shell is just .inn- flit- work done against the forces outside the In the to tin- work done In* the forces inside the shell. the
In
-lirli.
Inttt-r
ens.-
the work done
in
f*f|iid
C.MM-
of
I
lif
|H*iti.!i
eleetrk'
ticiu^ it> Ixiimdarv.
nothing which fixes the space except the current which
current there
the equivalent shell
iff
It
ill
is
follows that there cannot
be any region in
\vhk-h llie titrtv has the peculiar distribution characteristic of the The force due to a current shell. interior of an actual
magnetic
i!uTelire he continuous everywhere,
will
and the work done in
the current once will i^'iiiM- round anv closed pith which embraces 1" t/|iu] in 47rAL If the pith followed by the unit pole circulates
round the current
m
times in the positive sense and n times in
the negative sense, the
work done on
it
will
be 4tirA(ni n)i. determined not
Thus tin* magnetic *J'ly by the relative position of the current and the point, but also by the number of times the path of the point has previously encircled the current circuit. It is what is known as a multiplevalued function of spice. The complete expression for the magpotential dtie to a current is
netic potential at
any point due
Ai [47r(m ~ The
force
due to a current does
to a current >?)-!-
may
be written
ft)].
not, of course,
depend on
m or n
Inn only i.n ft), so that it will be single-valued and will depend only n the intensity and geometrical distribution of the current and the position of the circuit relative to the point.
The fuivguing
result that the work done in taking a unit magnetic p:Je once round any closed path embracing a current is proportional to the current embraced is not confined to linear
currents, but is
Minions
true if the currents occupy a finite volume. This we divide up the whole current i into linear conbounded by tubes of flow. Then the work done in is
if
stituents ci
taking a unit pob round a path enclosing one of the constituents will !e 4-TrAdi and if the path encloses the whole current i it will
!
477.4 Itie
S Si* =
consideration of non-linear currents enables us to express il] rather different Consider any analytical form. ^; rlace /> in snace. tr^v^rser] In- r-nwanf^ o-nri i^^-.-,^^^--i i^, _ ,.i,. __i is
lvs
}
ELECTROMAGNETS! If the current
9J
flowing in any specified direction at of electricity transported across unit area direction in unit time is called the current that to perpendicular It is a vector density at that point. quantity; let us denote it be the Let z X) ) at y by j. magnetic
contour
s.
any point, the
is
amount
H~(H H H ,
point.
Then the work done
intensity any in taking a unit magnetic pole around
the boundary s will be
H,dy +
What
the total current
is
between the normal
to
i
across the area
S
ILdz). If
?
an element dS of the
is
surface,
the ano-Ie o and j the
resultant current density at that element, the current Si across the element is j x dS cos 9. But J cos 5 = j n the component of the current density normal to the element. The total current i throuo-h
the entire area
may
therefore be written
i
I j
j n dS.
Hence
(2).
we shall prove an important geometrical theorem, due to Stokes, connecting the line Before discussing this equation
further
and surface integrals of vector point functions.
Stokes' s Theorem.
Let
R
be any vector point function which is continuous throughout the region considered. Consider the value of the line integral
A
1=
RcosRsds taken |
along any path
PTQ
from
P
R
to Q. Rs is the angle between and the tangent at a point of the path. Let us find the variation of the integral when the path of integration is changed by an infinitesimal amount, so as to lie along the curve PUQ. PUQ from PTQ, and the terminal points
is
P
only slightly displaced and Q are not varied.
ELECTROMAGNETISM
= s|
it'
-V,
F and
Z an-
Let us integrate
the components of R, since
| .
p
X^ (8#) cs
rfs
by
The integrated
parts.
--
.
A'2.r
p
^ N*w
and the unintegfi'ated part & r
cX = cX -^
.........
'A-:
-r-
G/' ^ -f .....
is 1
cX cu dX 3z ^- ~ -f Sr u*v
( I
p\
.
t.btain
cs .
,
-
1 "* ?:
cs
piveisely similar expressions for
and
^
so that
\dx \\
part
^
4. T^
^
"2 4. T ?: dz cy os
"^r
<">
_ ds,
"l I
o j w/o.
is
93
ELECTROMAGNETISM
Q
Now SX =
Sx +
dx
~ ,r
ex ;r-
= dX -^-
cs
and
similar
f
+
ij J
so that
8*,
dz
cy
8
A
^
^
d
5
d5
c
obtained
are
expressions r
SY-J*
for
and
cs
$Z -^
.
cs
Collecting together all the terms which contain 8# as a factor, and and &z, we have similarly those which contain
%
^
-
}+* The terminal points P and Q are fixed, so zero at P and Q. Thus the integrated part the terms rather differently we get
that
BJL\
&y and Bz are
vanishes, and arnmgiiig
:
*T SI-
I
fiY - dX\ -^(~~ \^
d/
)
cZ\
To
Sxdy Syrfj? consider the pamllelogitim of and which ds in its original displaced positions forms two opposite The angular points of this parallelogram are defined by sides. 5, s+ck, 5 (=s + S5) and ^ -fc^ (==s-r Ss-fds). Let ABDC of this parallelogram on the plane (Fig. 21) be the projection AB corresponds to ds and CD to d$j. Draw D-ST, of xOy. find the value of
1
5T
AH
parallel to Oy and 5/i, and to AH, produce DB perpendicular
parallel to
DK = %,
^A" =
8^,
BH =
7
(Ar, CJ?
to
rf?y
parallel to
AH in E. AH =
AH, CG
meet and
f/.r.
Also the parallelogram
-
=
.4
(AH - EH] - 5Ar BH = 8yd*
EFC =AEx CG = CG
Sydx
- CG
.
EH = Sy
c/ar
x
.
Then
ELECTROMAGNETISM
94
dS and the axis of normal to the original parallelogram that normal being drawn downwards. So Similarly
= cos nz. Sydx Sxdy -. dS so on. and = dS cos nx Szdy tydz t
iZ
idX
cY\
E
z,
the
Thus
~~
--
coswy
H
Fig. 21.
any closed contour PRQS (Fig. 20) and let P and two y any piints on it. From P to Q draw an indefinitely large number of paths such as PTQ, PUQ an infinitesimal distance apart. Xi'xt cnnswlf r
!:K"
The
between the value of the integral / taken along and along PRQ will be the sum of the differences along the
difference
PSQ
infinitesimal piths.
This
integral.
that along PJ?Q, starting at rtjiial
t> the
iin/i'tion t|n;fi
to
Hence
since integration"
from
P
to
Q
P
along the same path adds nothing to The difference between the integral along PSQ and
ami kick again from Q to ill*-
is clear,
P
and ending at Q in each case, is all round the circuit in the
same integral taken
PSQ:
I from Q through R to
since the value of
/ taken from
tor the entire closed
cY
P
to
P.
contour we have
cl
oZ
Q
along
PRQ
is
ELECTROMAGNETISM
9o
where the surface integral extends over any surface bounded by the contour, and where the cosines are the direction cosines of the normal to the surface
at the .point of integration.
It
will
be
observed that the positive direction of the normal is towards that side of the surface from which a right-handed screw would move if it
were turned in the same sense as that of the integration
round the contour.
The First
Law
of Electrodynamics.
Let us apply Stokes's Theorem to the case in which whose components are x yt magnetic intensity
R
is
H H H
H
,
the
We
z.
then have
|
(Hx dx -f Hydy
H dz)
-4-
2
dHy 2 = n\(cH M -7=r~ --
^
nx
J](\dy dy
cH x --^-~ + fdff
cos nu J
\cz
oz )
cH
- -^x + fcH -^ to / 1t
\ I
\
But we have seen
tj
is
cy
7 ~,
J
that
\Hx dx + H where j
)
cos nerdS.
dy
+ H2 dz = 4
1
1
jcos nj dS,
the current density at any point of any surface bounded A
by the contour and nj is the angle between the resultant current Xow density and the normal to the surface at the point. A
.
j cos nj =jx cos nx
where cos
-f j y
cos ny -f J2 cos nz,
n#, etc. are the direction cosines
of the
normal.
It
follows that
dy
02
cH.
2TTz dH
.
7rA/ 1f
Ox
!
V
(4)
cy
everywhere.
These
differential
equations are the most general expression
ELECTROMAGNETISM
iji;
They are one form of the first of To distinguish Maxwell's two famous electromagnetic equations. refer to them sometimes we shall ilit'ii! from the second equation distribution of electric currents.
as the statement of the First
Law
of Electrodynamics.
which we have already become relations between cornfamiliar, these three equations represent shall frequently find it We of vectors. derivatives and jHHients convenient to represent them by the abbreviated notation Like
many
of the formulae with
rot
H=
4>TrAj,
H
we mean the vector whose components are the where by rot taken in succesquantities on the left hand side of equations (4), This equation is a vector equation, that is to say, of rot independently for each of the components sion.
it
H
holds
and j
respectively.
The Electric Current.
The electron theory regards the electric current which flows It supposes that in a conalong a wire as a convection current. duct ir there are a number of charged particles which do not 1
execute small displacements about a position of equilibrium, ias in the ease of those electrons which we considered when we were discussing the behaviour of dielectric media, but
which are able
move
In a freely from one part of the conductor to another. metallic conductor these particles are believed to be electrons and to
them from the bound which only undergo small displacements from their position of equilibrium when an electric field is made to act on thrin. The free electrons in a metal are believed to be in much tin.- sune condition as the molecules of a When we come, gas.
aiv culled "free electrons" to distinguish electrons,
later MIL in consider it
is YI
the evidence for this belief,
TV strong and that the resemblance
we
shall see that
a very close one. In th-cji>eot liquid electrolytes the charged particles are of atomic "i- nmircular dimensions, and in many cases of electric conduction
thi'Mi^li
In
gases this
all
is
is
the case also.
these cases the charged particles, of whatever nature,
aiv K-ln.'Yvd to be
moving about
irregularly in all directions,
they are not subjected to the action of an electric ^hyii fill- liMl-'ll does Tint fiUKi* !im' fi-i-now, ^f .-.r .,1,-,,,4- , :.L .-,+,*.-,
even field. _.
EL fcXTROM AGX ETISM
on the be moving in any average, the particles are just as likely one direction as in external electric an of other. effect The any field is to on definite drift, so a motion the superpose irregular that on the inuve in the the average t.<
positively charged particles direction of the electric field, and the negatively charged particles
move in the opposite direction. In be different. general there may kinds of ions of one the unit of in that sign present. Suppj.se volume there are
N
p positive ions of type p, that their average of drift under the applied field in the diivction of the velocity field is Let the corresponding p and that their charge is p
U
E
,
.
quantities for the negatively charged carriers be denoted by small letters. The current density at any point will be the total amount. of positive to electricity transported across unit area perpendicular the direction of the field at that direction, in the positive point,
plus the total
amount
opposite direction.
Up
of negative electricity transported in the
This
is
clearly
the volume of a cylinder whose axis is parallel to the field, whose sectional area is unity and whose height is p the In the case velocity of drift of the particles under consideration. of solid and conductors the JV's and J s are independent of since
is
U
,
5
liquid
the electric intensity whilst the U's are proportional to it, so that the current in these cases obeys Ohm's Law. The same is irue in the case of very small currents in gases at moderate pressures. In general, however, in the case of gases both the JV's and the U's
may force.
vary in a complicated way with the applied electromotive It is for this reason that the relation between the electro-
motive force and the current in gaseous conduction
is,
generally
speaking, quite intricate.
The current whose
properties
we have been discussing
is
often
There is another kind of electric current called the displacement current, for the conception of which we are indebted to the constructive imagination of Maxwell.
called the true current.
The Displacement Current. Consider an electric circuit consisting of a battery JL a condenser B and a one-way switch G. When the key C is depressed, int.-, rV~ ,0 a current flows through the wires from t"hp
Wt^-
.-,+.-..-
ELECTROMAGXETISM
|.H
This current
the v'ii
<,!'
iht'
way round weiv replaced
In*
rtiinlt'iiM-r
emiilt'iiser. i-tfift
matfnftk-
however, continuous
a wire, since there
Maxwell
such a circuit
f
is
if
all
the
no actual trans-
between the plates of
the
forward
put
would be
it
hypothesis that the
was identical with that arising
and there replaced by a suitable conductor, in the circuit. This thf siiu* runvnt flowing along the wires
wftHi is
not,
the dielectric I'Iretricity across
!'
jmrtaii"!! ill**
is
same way that
the circuit, in the
tli*'
c'Mitfieiiser is
owing to the existence, when the field is changing, of Maxwell called a displacement current in the dielectric.
conies alnmt
what
Tlif tiisplatvmeiit current density at
the electric
I) is
wiit-re
any point
is
equal to -j~
,
induction.
This value for the displacement current makes the current round For if
etjual
||
,
point, the current i in the wires is
dS, the integral being taken over a plate of the
But,
CHiideiiser.
any
D
if
is
the value of the induction, the displace-
niHit current close to the plates is
We
surface.
current
is
have seen that
equal to
i
D=
1
dS taken
1
-^~
so
cr,
and the current
is
that
over the
same
the displacement all round the
continuous
circuit.
This result
may
be proved to be quite general as follows
Consider any closed surface in space. the charge e inside this surface
is
By
Gauss's
:
Theorem
equal to
Thi- current flowing into the surface is
de
But
if
jx
,
/,
f |7,
W,
u
.
W,
jz are the components of the true current density
=~ d?
Hence
W
.
jfa*
+
m J + n ^ dS
'
So that the vector
3D
i=j + -^CL
Is
Like the
solenoidal everywhere.
tubes of magnetic induction, the tubes of flow of the total cunvnt are always closed regions of space. They have no free end*, unless at infinity. It would be very difficult to devise a direct experimental proof of the magnetic effect of displacement currents. The best proof of it is an inductive one. The hypothesis of displacement currents forms the basis of the electromagnetic theory of light, and the
way in which the requirements of this theory have by experiment shows that it is built upon a solid
extraordinary
been
fulfilled
foundation. It
is
often
desirable
consider the
to
displacement current
dD
density -^- as inadt- up mf t\vu pins, (1) the aethereal displacedt
dE
iiient
2)
..,,.,.. i
.
-
dP <
i
i
rrf o-r^H ~J + h ii y\i-io it ryfit-i, ri r*inT r and current (2) the polarization current -T-. -j-
= E -f P,
~y- is
I
v
\
k
>
:r'ni
A
.
bince
always equal to the sum of the aethereal current
Ci b
and the polarization current. On the electron theory, as is obvious from the discussion in Chapter v, the polarization current corresponds to an actual displacement of charged electrons, and is to that extent very similar to the true current.
Convection Currents.
of electric currents which flow in wires, electrolytic cells and so forth, and which are carried by extremely minute
The kinds
not the only ones particles, are Rowland* showed in
detected.
whose magnetic effects can be 1ST 6 that an electrostatically
a sufficiently high speed charged disc when made to revolve at affected a suspended magnet in the same round the disc. Effects of this kind,
way that
as a current flowing is to say, effects
on a large scale, may depending on the movement of electricity Let p a of means be summarized very simple formula. by easily be the net volume density of the electrification at any point, and
and direction by T.
be given in magnitude current density of the electricity at
let its velocity
pV
is
the
*
Ann.
tier
Phy*.
vol. CLVIII. p.
87 (1876).
the
Then
point
in
ELECTROMAGNETJSM This expression may be made to include not only cases or a small number of charges where the motion of a single charge flow of electricity the like a but also those
question.
is
contemplated, which the
wire in
we have this
is
along
number
to take the
of carriers
is
enormous.
In such cases
of p V, and it is quite clear that average value of the true electric current as magnitude
to the
equal on p. 97.
specified
it is necessary or advisable to distinguish between the of electric current which may occur, we may write kinds different
When
in the form equations (4)
Tt)
W-
we shall require the these of vectors values just as in the average, not the actual, theorv of the behaviour of dielectric media. Where we are dealing with material systems
Induction of Currents. In an electrostatic field the
work done in taking a unit charge
round any closed path, to the point from which it started, vanishes. The electrostatic potential is a single-valued function of the
space and the electromotive force round any closed circuit is This is no longer true in a region in which the magnetic
coordinates, zero.
induction field
is
changing.
inside a closed
current
was caused to
Faraday* showed that when the magnetic conducting circuit was made to change, a flow round the circuit. He also showed that
the electromotive force
round the circuit was equal to the rate of
diminution of the flux of induction through the circuit in the This statepositive direction, multiplied by a universal constant.
ment First
is very similar in form to that in which we expressed the Law. If we put it into analytical form we shall see that the
resemblance
is
very close indeed.
The electromotive force round any closed circuit is equal to the work done in taking a unit If & positive charge round the circuit, denotes length measured along the circuit this is
J
E cos Es ds=l (Ex dx + Ev dy + Ez dz\ *
E.r.n.
Rfs
S.llfi
ELECTROMAGNETISM
101
E = Ex ,E ,E
Let B be the z ) is the electric intensity. ( magnetic induction at any point of any surface bounded by the circuit. The flux of induction through an element of area dS of
where
tJ
B cos nBdS, where nB is the angle between the B and the normal to the surface at dS. But A B cos nB = Bx cos ?u* -f $,, cos wy 4- B cos H
this surface will
be
resultant induction
2
.?,
where cos nx, cos ?n/ and cos nz are the direction cosines of the normal The total flux of induction through the surface is therefore II(BX
B
cos nx 4-
f
ti
cus
-f J5,
ny
cos nz ffo )
and the rate of diminution of this is proportional motive force round the circuit, so that
A But b
Stokes
J
s
l
-r
I(BX
cos
nx
4-
By cos y^
+B
2
to the electro-
cos
j) rfS.
Theorem
-
1
I
,
1
[V9y
- -^r1 3^
cos
nx
~ -=^
-
+
cos
/ vf
_
^
,
cos nz> ^s.
3y J
\
Since these relations are true for any surface bounded by the contour, the surface integrals must be identically equal so that ;
cE1 _cEI = _ cy
l
ct
a^_a^ = dz
x 1 cB_
cz __
4
ex
ct
^_^ = _j
or
cB^ l
ML* l
'
ct
cy
1-01^=-^^ The value
E
of the constant
A
l
is
..................... (8).
determined by the units in
These three equations are measured (see p. 111). When it is represent the second group of Maxwell's equations. which
and
jB
ELECTROMAGNETISM
tary
f
with
tin-
them a name we
to give
the
First
Second Law
shall refer to
them
as the ex-
Comparing them
of Electrodynamics.
Law,
..,e that, since %v see
kind of reciprocal
B
is
H and D to E, there
to
proportional
relation
is
a
Also they are not quite
between them.
in front of =the same in form on account of the negative sign
.
Not onlv do these equations enable us to deduce the whole of of electromagnetism, but, as we shall see, they are the
phenomena
flit/
basis of the science of optics as well.
The Dynamical Theory of Electromagnetism.
The Second
Law
may be
of Electromagnetism
looked upon
from two different standpoints according to the attitude we take If we regard electrodynamics as more towards electrical science.
Law
We may
we must regard the
than dynamics proper, then
fundamental
Second
as a fundamental law of nature empirically given.
however take the standpoint that the aether, which we medium in which all electrical actions occur, will in
pustulate as a
the last analysis prove to be a mechanical basal laws of dynamics.
Provided
system subject to the this assumption, even
we make
though we know nothing of the nature of the mechanism, we can show that the Second Law is a consequence of the First Law. The view that electrical actions are ultimately dynamical is one
whose development in the hands in the science, and it
advances at
any rate until quite recently,
of
Maxwell led
to
notable
view towards which, most authorities have leaned. the
is
it is equally logical to accept the Second Law as an ultimate fact and then, later on, to consider what we can make of the laws of thus dynamics from the
Nevertheless
standpoint
"fte shall
now proceed
to consider
adopted.
some
of the consequences
which follow from the assumption that every electrodynamical system is a dynamical system subject to the operation of the first law and of the fundamental laws of electrostatics and magneto^
statics.
The energy A A:
(E* 4-
of the field 77
2
/? 2\
,
is I
equal to (
TT
2
.
TJ
2
I
TT
2\
fC\\
ELECTROMAGX ETISM per unit volume. Since a change in the electric int^nsitv accompanied by a corresponding change in the magnetic force, it
is is
natural to assume that one of these terms represents potential and the other kinetic energy. Moreover the presence of magnetic
energy
is
always associated with charges in motion whereas electro-
energy is present when all the charges are at rest. The obvious conclusion therefore is that magnetic energy should be static
identified
energy.
with kinetic energy and electrostatic with potential
A
more rigorous proof of the necessity
magnetic with kinetic energy
and Magnetism,
will
for
identifying
be found in Jeans's Electricity
p. 483.
Assuming that the magnetic energy of the electrodynainic field represents the kinetic energy of a dynamical system, let us consider the behaviour of a system of n circuits 1, 2, 3, etc. carrying currents ij, L, i., etc. In this case the magnetic energy can be written as a quadratic function of the n variables i^ L, ?V, etc. For
the
number
any
one, let us say the mth, circuit,
of tubes of magnetic induction which flow through is the sum of n terms each
representing a contribution from one circuit. This follows since the magnetic force at any point due to a current depends only on the relative geometry of the point and the current, the nature of the
intervening medium, and the magnitude of the current, to which it is Thus if m is the number of tubes of magnetic proportional.
N
induction which traverse the with circuit in the positive direction
Nm
L lm
L M L + i 3wl -f -f L IAm i m -f .('10V -f L iim i n The coefficients L etc. depend only on the geometry of the i\
-f
?'
litl
circuits
3
. . .
.
.
-
.
,
and the nature of the material
are called coefficients of self-induction
in
which they
when the
lie.
They
suffixes are like,
and of mutual induction when they are unlike.
The magnetic energy of currents,
is
equal to
of any system, including that of a system
^jjipH-dr, taken throughout the volume
of the system. Let us suppose that the .space is mapped out by unit tubes of induction. These are closed tubes which means of
completely fill the space and never intersect. If tS is the normal sectional area of a unit tube at any point, the element of volume
dr may be replaced by BS ds, where ds is an element of length of the tube and is normal to BS. But SS is the area over which the flux of induction
is
unity, so that
ELECTRON AfcLN JtUTlSM
j0 1 Tin-
of the
pot
volume integral which belongs to any one tube
we indicate by S a summation over becomes the magnetic energy f of the system
Thus
is
the unit tubes,
all
if
law of electrodynamics fHds = 4-7r-42 i, where sum of the currents which are embraced by /
Now bv
the
first
2V
is the* algebraic is taken. the tube along which the integration is of linear currents equal to the of a
T
system
with each circuit.
Hence
Xm
3
4-
+
ZLuiii*
.
+ Lnn in*}
......... (11),
the algebraic sum of the number of tubes thread the wth circuit in the positive direction.
At
the
which
is
u ii
where
all
over all the current equal to the sum, *f 2irA times the number of tubes which are linked
This
they are linked circuits,
in the case
sum, over
of the currents with
sum
tulK-s/of *lirA times the algebraic
Thus
first
conclusions
is
which
be inconsistent with the as to the equivalence of reached we which formerly to sight this result appears
We
see from equation (11), Chap, v, currents and magnetic shells. is which that the energy of the shell equivalent to the mth circuit r
where
Nm
the
is
number
supposed invariable, which thread the permeability of the medium. meability
= 4firA fii m
=
4tirAi m
;
it,
m
of tubes is its
of induction,
strength,
and
p,
is
medium of unit permedium of permeability ^,
For a
so that for a
This result follows since the magnetic force which a given current produces is independent of the medium, while it is the induction due to a shell which is of the f>m
(
.
independent
permeability of the medium. in a fixed field the energy
Thus is
for a single
-4nrAim
Nm
.
equivalent shell find the total
To
energy of the system of equivalent shells we have to imagine them created in infinitesimal steps, so that each increment is
prop>rtional
circuit.
The
to
the
calculation
final is
magnitude
of
the
corresponding
precisely similar to that followed in
ELECTBOMAGNETISM
1 (b
obtaining the tutal energy of a system of electroform of the expression for the potential energy- of the system. This calculation introduces a factor so that the energy of the system of shells which produces
Chap.
p. 33, in
II,
static charges, except for the difference in the
,
the same magnetic
the system of currents
field as
- 2irAI,im NM = - %irA (ZnV + 2^,22 -f The
difference
energy which
... 4-
Z, m / w
2 ).
and (11) arises from the within the volume of the shell itself*. This
between lies
is
this expression
'2T in the case of the equivalent shells, quantity, which is equal to is zero in the case of the currents.
The method of Lagrange's equations f enables us to find out a great deal about the behaviour of a mechanical system even when we have no means of discovering the precise nature of the
We
mechanism.
apply that method to the problem we have to express the and the potential energy TF as functions of the shall therefore
under consideration.
As
a preliminary
kinetic energy T generalized coordinates
#u
We
#2, etc.
energy in the
xlt x, etc. and the corresponding velocities have seen that if we identity T with the magnetic
field
we get
T= + %TrA
V+ s
n
(
12*1*2
+
13*1*3
+
-')>
and this will be a quadratic function of the velocity coordinates Thus the etc. etc. if we make ^ = 0*,, i = d^ i 3 = #3 0, l5 it'
,
the generalized displacement xs becomes fia dt and is equal to quantity of electricity which has flowed round the circuit after
In the present case TF = since the system fixed instant. does not possess electrostatic energy, the capacity being regarded We also notice that the i's are functions only of as negligible.
some
the geometrical arrangement of the circuits and of the nature of the intervening medium. They do not involve the generalized
displacements If
X
8
sponding
is
x.
the component of generalized external force corre-
to the generalized displacement
a fdT\ -,
ct *
t
Of. Jeans, Electricity
For an account xvi.
\cuy
- dT = A,
and Haynetism,
may
(12),
cj-g
p.
433.
of the part of generalized chrnamies
present discussion the reader
Chap.
1
^, we have
which
is
germane
to the
be referred to Jeans's Electricity and Magnetism,
1
ELECTRON AGNETISM
1 if]
motion being tincniLstrained. An equation of this type holds coordinate xs whether the system is a JTIHH! tnr each generalized tin-
einsrrvati\v oiu* or not.
may jKThaps be worth while pointing out for the benefit who are unfamiliar with this branch of dynamics
It |
uf those readers
A"s are not actually forces nor the #'s actually displacecondition that they have to satisfy is that the work
that the
The
ments.
'dour luring a small change in the state of the system due to a value of the generalized coordinate changf, let us say &/*, of the = in the Thus is A^S/v ./; present case, since &?; s means a change If"
<
.
in the quantity of electricity
which has flowed round the circuit,
He the effective electromotive force in the circuit.]
A', will
T= +
Since
'2-rrA
[L n Xf
+ L^x^ 4- L n ^x
+...],
s
o ;-
ct
^.
ct
CJ'J
(1,3),
where X*
is
sih circuit.
t
number
the
As
T
of tubes of induction
which thread the
fiT
does not contain the
o?'s.
r
f
=
0.
so that
Snppi.se that in this circuit there is an intrinsic electromotive E, the work done during a small change $xs in the quantity f electricity which has flowed round the circuit will be >ree
;
where R# round circuit
the resistance of the circuit and
is is the current term represents the reaction of the matter in the on account of the Joule Since is dt = heating effect.
it.
is
The
last
.,
=E
s
(14). If there is
current
Law we
is all
E
no intrinsic electromotive in the circuit, the force, 8 to the From Ohm's electromagnetic induction. ,
due
see that the electromotive
fmv^
wl-nV.1-.
o-m'c^c,
A.^,^
-<-i^,
10
ELT3CTROMAGXETISM n electromagnetic induction
jR 7x
is
fc
5JV
=
4?r J.
~~ ~
Thus reverting
.
to our former notation
\(Ex
dx +Ey dy +Es dz) =
It follows that the
Law
- lirA t
Second Law
(IEX
\
\
a consequence of the First
is
provided we admit
that the magnetic energy in the field can be identified with the kinetic energy of a mechanical system
:
and provided
also that
A = 4irA. l
The
following investigation which leads directly to the second of Maxwell's equations is instructive. We shall suppose that
there are both electrostatic and magnetic forces in the field, so that if T is the kinetic energy, and the potential energy, pel-
W
we have
unit volume of the sstem,
and
IF
We
shall restrict
=
*
(Ej
4-
E* + E/).
our proof to cases in which there are no free to satisfy the first law of electrodynamics.
Our system has
charges. so that
rot
HTT
.
.
4-TrJ.
cD =
.
.
4?rJ. tc
and we
also
Now
have
div
cE ~ ,
ct
ct
D = 0.
the changes occurring in every dynamical system
subject to the principle of Least Action,
which
may
tire
be put in
the form
**(T-W)dt = Q
.........
.
........ (15).
This means that in any natural motion of the system, from a given configuration at time ^ to another given configuration at
time
the actual motion
such as to
make the lime
integral of \\ a minimum, and that any slight variation from the actual motion of the system, subject to the conditions being unchanged
T
t.2
,
is
r
for the initial
and
final configurations (t
=
and
t
=
t.
2 respectively), shall variation of the integral is zero. sufficient to establish the Second Law.
t^
We
must be such that the see that this
T
is
the kinetic energy, the jETs will be velocity coordinates. Let us put =0 so that X =0 X ff = llt Z S where the ffs are generalized coordinates. We may now write the
Since
is
H
H
,
il
H
=0
,
KI.ECT1U iMAGKETISM
P;M Liu
as
,,,!M!i.n *
utnuiiif> t*
,,i
\ari->
Iff
ly
SD
we
shall
extend this
true of any change of the dependent whethtT in respect to the time or not. Thus if the variation to be conditioned by sliall
luake
/>
we
^ = 477^1^, and
rot
ii,riii
T,,
ami
tf
v,inaii.n
\V-
the
in
it
suppise rot SO accunling to the tVirmula an element of volume,
f
havt-, ifc/r is
y>
+
e t se v
= 4<7rA SD.
+ 8,M,)}
drdt,
'
/t
'3
dt dS \M (nE - mE J^ ./,;["
;'
i
-
r
.
, .
,
dr
7 at
.....
t
4:7.1
X
I
f
477.1
.,.
f,, fSE bt? -^ x
i
\
I
.'
z)
)f
?EZ \
lt
-
-^
&
cy
/
Thus e
1
r/T
T
i
T
I!'
I
rff
=
*
=
dr I
p,
(6 X S0X
+
y
S0
tJ
+
9 2 BO Z )
oy ''
dS
\
ELECTROM AGNETISM
1
uy
The present analysis is restricted to the case where the media are at rest and where there are no free charges. Moreover the actual and the varied system are to be identical at t = ^ and t = L. This makes the limits S6 X
first
= S0 V =
92
volume integral vanish, since at both time
= 0.
The
surface integral will vanish
if
we
suppose the surface to be a closed surface at an infinite distance from the region in which the actions are going on. For dS is although
2 proportional to r both
E
follows that the second
volume integral must vanish.
and 6 are of the order
-, at
most.
And
It
since
the SB's are perfectly arbitrary both for each element of volume and for each element of time (except for the limits of time) the coefficients of each of them must always be zero. therefore find
We
cE cE z _--~! = CZ
cEx
cEz
Jnrjipt! ^
},
^-rrAfJi
Q U2
\
..,
M
or
J
,
\
E=
Thus the Second Law
-
,
477,1 -^ct
follows if
4-77 J[
__
,f
-J
'
ct
=
cy rot
4jrA -^~ Ct
^
CX
?E ---?EX =
=-
2
tf
CIJ
cBx
i
j
B?
47T-d -^~ ct
,
.
we assume the truth
Law, and make the further assumption that the dynamic field is a mechanical system. First
of the electro-
The foregoing deduction is practically identical with one given by Larmor (Aether and Matter, Chapter vi), who has shown that the laws of electrodynamics can be built up by giving the aether a mechanical constitution.
For the further development of
this
theory, including the natural extension to the case where the presence in the field of electric charges is contemplated, the reader may be referred to Larmor's Aether and Matter, Chaps, vi, vir,
and
x.
Electrical Units.
We
have seen that the work done in taking a unit magnetic a path situated in free space which encircles a current round pole of strength i once, is given by
1
1
l*i
KI.KCTK*
i
,
,
t
,
,,f
j.
]
iii-
in]- -ii
p
Miiivt-rsd constant -i will clearly
the mills in which
'..i^jiTv!.- ui ',
M AOXETISM
units uf length
ill*-
//and
i
depend on the
are measured.
It also
and time.
hav- already defined our units of magnetic pole strength Frm either of these units, although not triV ehaix'. li-Miu
ma\
d'ri\rd
ht-
other electric and magnetic quantities we keep to the same imfqiiivoeally provided
units nf
1Mb, thr
all
Thus, t< length and time. iiia^n'!ir intensity // we only neel t> flu- fiuvi" in dviie.s nil a pole of strength f
jinit> f JN
//
i*
Hut
determine our unit
!ii,i>>,
determined lv the unit
f
m
the
remember that /w,
and of
mff
force.
aH>!^ii!M!
i
m:
the product
tin> of
electric
evident
is
mav
arbitrarily be
must always have the dimensions
The dimensions
of a mechanical force.
It
are independent of the
dimension* whicli, since they are unknown, it
mH
so that the unit of
of
many
other combina-
and magnetic quantities are predetermined in the instance, Ee has the dimensions of force and pH-
same way; fur and icE- have the dimensions of energy per unit volume, and so Tlie unit of current
on.
charge and time, and
is
clearly only involves the units of electric therefore determined in our case since we
i
have already fixed the units of electric charge and time.
When
// and
eMiMant iaetr
is
i"
477
are measured in this way, the value of the
J =
I c,
where
c
=3
x 10
1
"
cms. per
The
sec.
quantity r, which has the dimensions of a velocity, and thus has a numerical value which depends only on the units of length and liiiif, is nut.- of the most important As we physical quantities.
shall see,
other things, to the velocity of light and other forms of electromagnetic radiation in empty it
is
equal,
among
space.
When units
It
is
the stipulation above as to the character of the derived understood we can write equation (7) in the form
remains to consider the value of the constant
A
l
which
the expression for the second law. This may. be isc vered by making use of the principle of the conservation of en rgy, and for this purpose the of the simplest ers
into
possible example induction of currents will suffice as well as another. Consider a .-ingle circuit earning a current V, the self-induction of the circuit
ELECTKOMAGX ETISM
Then the magnetic
being L. :>.TrALir
= ^-ir.
If the
III
or kinetic energy of this circuit
current
i
is
changes, the aniuunt of this
and it is a known physical fact that there will be a self-induced electromotive force consequence acting round the circuit. In general the circuit may lose energy by radiation or in energy
will alter
in
other ways, but under suitable conditions these losses are negligible. In such cases the rate of loss of magnetic energy will be equal to the work done against the resistance of the circuit less the work by the battery of electromotive force E. Thus
supplier!
L
ci .
= Jb
p Ki.
c ct
The
electromotive force due to induction
is
therefore
_
-
'
c
c ct
where
N
number
the
is
A =
-
=
l
47r^4,
of tubes of induction, measured in our
which thread the
system of units, It follows tion.
circuit in the
positive direc-
by comparison with the equations on
when the
electrostatic quantities are
c
.
ct
p.
101 that
measured
in
the modified electrostatic units and the magnetic quantities in the modified electromagnetic units.
The
following example is also instructive, since it brings out quite clearly that it must be the magnetic induction and not the magnetic intensity whose rate of change determines the magnitude of the
Consider the behaviour of a bar of
induced currents.
magnetizable material encircled by a solenoid having n turns of The bar is of uniform cross section a, and wire per unit length. length L maintains a magnetic
A
of indefinite
relation
between
H
current
field
and
i
i
flows in the solenoid
of intensity
H
in
the bar.
and
The
is
H = l?irAni = -i c
..................... (16).
The work done consists of
two
in establishing the magnetic field in the bar is the intensity of parts, (1) IzfHdl, which, if /
work in magnetizing the bar, magnetization, represents the actual
Y
ELECTROMAGN ETISM
*> j j
an. I
(,2l
tii-M
in
the energy of the magnetic latHdH, which represents the bar. In a small the aether of the spice occupied by
jntrrvul of
time dt the increment in the magnetic energy in the
clone against the back electroThis must be equal to the work is work This motive force of induction.
Sothat
(I
+
t
B= H
Thus It
is
4-
well
/
in
to
we
H^A-.
accordance with Chap. VI and
A =-= l
4-TrA
c
understand clearly the difference between the are using and the two systems, the electrostatic
system of units system and the electromagnetic system respectively, which are most frequently used in books dealing with the theory of electricity,
The
unit of electric charge in the
electrostatic
system
is
V4?r
times our unit of electric charge; but it is not this difference so much as the difference in the units in which the magnetic quantities are
moment. obtained
On
measured which
from the unit of
electric
quantities are then obtained unity in the equation I
This
fixes
desirable to emphasize at the system the unit of current is
it is
the electrostatic
charge,
by giving
A
and the magnetic the arbitrary value
Hds = 4i7rAi.
the unit of magnetic force and so determines the unit
magnetic charge. In our units the measure of i is V4?r times greater and A is 4-Trc times less than in the electrostatic system, so f
that our unit of
magnetic force is c\/4?r times greater than the Since has the same value on all systems which have the same unit of mechanical force our unit of pole
electrostatic unit.
.strength is cV4-7r
mH
times smaller than the electrostatic unit.
The electromagnetic system of units sets nut
also
makes
A
1,
but
it
by denning the unit magnetic pole as that which repels an equal pule at unit distance with a force of one clvne. On this
113
ELECTROMAGXETLSM
system the unit of pole strength is therefore \ 4-jr limes, our unit and 1 c times the electrostatic unit. The electrostatic unit of pole thus greater than the electromagnetic unit by the on cms. per sec. It follows that the measure of
strength
is
factor 3
x 10
H
1?l
the electromagnetic system
is
1
"e
times
its
measure on the
electro-
Whence, by considering the equation \Hds = Tri true on both these systems, it follows that the measure ot
static system.
which e
is
on the electrostatic system
is c
times greater than on the electrounit, of electric charge
Thus the electromagnetic
magnetic system. is c times the electrostatic
unit,
and
c \ 4ir
times our unit of
electric charge.
when the unit of electric charge is defined as which that repels an equal and similar charge at unit distance with a force equal to 1/47T dynes, and the quantities which can be derived from it without making use of the two laws of electroIt follows that
magnetism are measured
In terms of units
this unit of electric charge
:
which are based on and when in addition the unit of
magnetic pole strength is defined as the strength of that pole which repels an equal and similar pole at unit distance with a force of 1/47T dynes,
and when the quantities which can be derived
from this without making use of the two laws of electromagnet ism are measured in units which are based on this unit of magnetic pole strength: then the two laws of electromagnetic induction
become
:
(1)
rot .0
(2)
=-
~ c
In the sequel
and
we
........................... ilSi.
ct
shall always use (17)
and i!8l rather than
(
(8).
Magnetic Force due
We
have seen (Chap,
shell of strength
to
an Element of Electric Current.
v, p.
85) that the force due to a magnetic can be represented as
at a distant point
P
shell. Since a arising from each element of the boundary of the current of strength i placed in a medium whose permeability is JJL
causes the same distribution of magnetic intensity as a shell of
1 1
= pi i\ it fallows that each element ids of an can be Actnr eiinvnt regarded as giving rise at every point in magnetic intensity of amount
*,} i
ELECTHt M AGXETISM
4
!^'!ii
*
*l
i
*
^=
JarA/jLi
;<
47TT
r-
P
to r/s and rs is the angle radius from the point of the direction The k-twt/v!! r and civ. intensity is normal to r whether the and <&. This result is true space is empty or filled
wht'tv r
\\iih
i* tli*-
matter,
hoiiiMi^i-neous
meability
and
is
independent
of
the
per-
/u.
Furce on
Element uf Current
(in
in
a Magnetic Field.
\Ve have seen that the force on a magnetic shell of strength can be when placed in a magnetic field where the intensit}r is <
H
resultant of forces equal to -$[Tcfe] acting ivpivseriUHl us the on each element ds of the boundary of the shell. It follows that the force exerted on an element ids of electric current i by a field
H
i>
Where
the current
due
is
motion of an
to the
electric
charge
of vnlnine density p with velocity v, the current per unit area is pt\ and r he force on unit volume due to the field is
H
(21).
If the charge e will
is
by a
carried
be given by
e=
particle
whose volume
c..iiipnents
Fx Fy F ,
,
r the charge
pdr, and the force acting on
|||
""-[*"] The
is
z
it is
<>
of this force are
(23 ).
CHAPTER
VII
ELECTBOMAGXETIC WAVES The Equations of Propagation.
LET us for the moment confine our attention to the application of Maxwell's equations to regions in which there are neither electric nor magnetic charges. Strictly speaking, from the point of view of the electron theory, this should restrict us to the case of the free aether, since all matter is supposed to be made up of electrically
In material media, however, the mean density particles. of the charge at any point, if we average over a volume containing an enormous number of particles, Is in most cases zero; so that we
charged
examine the consequences, Incidentally, of supposing that with material media the peculiarities of the individual electrons -can be left out of consideration. We shall soon see that the results
shall
at which
we
arrive, while
exact for the free
when applied
crudest kind of approximation
aether, are 'the
to material media.
which we have been led in the preceding = are chapter in the case in which p
The equations
to
^
rot
u IcD =-KcE JT=^r -TTT
rot jb
IcB = c
,
c ct
c ct
ct
pdH m
c
ct
We have seen that the two foregoing equations are really an abbreviation for six equations between the six components of and E. These equations can therefore be solved for each one of
H
ELECTROMAGNETIC WAVES
Ilf |
T
i
hc
variables
.h-pi-mlt-nt
tiiiit
nut
in
foil
dy
{
H H H Ex E x,
z>
iP
>
v
and
E
Writing
s.
we have
_ dz
dt
dt J
_
_
dz\d% ^
'
a
dEx
^ Now
-^ r.c
since there are
In the same manner
We
an*.l
z,
,
show
7)
8y
a.
dz
dE 3^
Thus
we may show that each of the z satisfies the same equation,
H H y,
variables
investigate the solution of the equation
the general equation of propagation of waves Let us take the integral of both sides of equation
throughout a closed volume, then
f
dy
+ -~z = rp =
thcit it is
with velocity 1
,
now
shall
dJSy -5-2
\fa
no charges in the medium.
Ex Ey E Hx .
4-
dz / j
*-*
ELECTROMAGNETIC WAVES
III
by Green's Theorem (Chap, n, p. 24), where the surface integral extends over the surface Let S be the enclosing the volume. surface of a sphere of radius r whose centre is at the P, and point
be the
let da>
and
jjf
solid angle
subtended by dS at the point P, then
I dr -fj'f'g *
Differentiating both sides of (8)
i*z
1
1
Wr rfw
by the upper
=a
2
r- ^-
!'
cr
cf'JJ
1
1
crJJ
-
x
limit
u,da>
\
r,
.
Xow ffu r d
P
by
it,,
then C~ -
->
or
-
f
f
,
'
,Clly\
'
*M( ct
Xow let ru = v and = p at + r and q = at r,
new independent
introduce
r
then
.
o.-
""
'
8p 3^
8f
9 ~
ft
2
of-
cf
S^
'
dpcq
\cp
*
'
dr 2
c'
r
dp cr
_
r?"-
c~v
dp-
cq cr ^ ~
cp r-?'
c'-v t "
cpcq
cq-'
^- =
so that
cq
0.
cpcq
The general
solution of this equation
is
clearly
where/1 and/^ are arbitrary functions. Since w r
is
never
infinite, v
=
= ~/l (at) jo (aO
when
r
= 0.
so that
for all values of
f.
variables
ELECTROMAGNETIC WAVES
J18
but one
Therefore /, and j are not entirely arbitrary, Thus in sign to the other.
r"^=/;(^ + r)-/ (^-r)
is
opposite
............... (9).
1
Differentiating by r
Whenr=0, But
^
is
=2
wi
mean value of u over a sphere of infinitesimal it is therefore equal to the value u p at
the
radius about the point jP;
P itself.
So that up
Again we have from
and
(r!Tr)
Thus and
for
t
I (r
(9)
=// (a* + r) -// (a* -
,)-
I (r,)
for
u r and
-'
we
shall take as the
are given for every point
Let
OOe=o=^,2/
But when ?= af,
2/' (r)
3
and
/>
= UP
,
~
=^'
so that
Thus the value of uf at any time t subsequent to the time at which the values of F and F' are given throughout space obtained by describing a sphere of radius at about the point P
= 0,
is
= 2// (a* + r)
a certain instant, which
origin of time, the values of
t
r).
=
Suppose that
in space.
= 2// (at).
ELECTROMAGNETIC WAVES
The value
119
we know
of UP will then be determined, provided
&u '
average value of as the rate at
=
t
over the surface of this sphere at
which the average value of
changes as the radius
The meaning
t
=
5
the
as well
u over this sphere at
time
increased.
is
of this solution
is
perhaps seen more clearly
consider a case in which the value of u at
t
=
except in a certain limited region of spice T. it at some external Let the shortest line from point P.
of
=6
if
we
zero everywhere Consider the value is
P to
r be
and the longest line c. Then until u, ?/p will be zero since the sphere of radius at will not intersect the region For similar reasons for which w was not equal to zero at t = 0. b
when Thus
the value of
is greater than c'a the value of up will again be zero. represents a disturbance which is propagated in all direcThe foregoing solution, which is due to tions with the velocity t
u.
.
Poisson, shows that electromagnetic effects are propagated with
In
waves.
finite velocity, like
tact the equation
the general equation of wave motion and contains the mat hematics underlying all the different kinds of wave morion contem-
is
plated in physics. J
Before leaving Poisson s solution it very simple concrete example to which
may be
Suppose that by means of two ctirrent sheets AB and it
applied.
CD
perpendicular
the
paper,
we
field
magnetic the arrow. t
What
will
produce
plane of a uniform
the direction of
in
Suppose that at a certain
=
time
H
the
to
the
currents
are
stopped.
be the value of the magnetic
P
afterwards ? intensity at the point will be no at there field until Clearly
P
, t
= PR = -b a
.
where a
.
a
propagation
.
;
that
is
is
.
,
1
the veiocitv of
to say, until the
sphere of radius at cuts the plane CD. To find the value after the sphere has
may be ^
well to consider a
D
ELECTROMAGNETIC WAVES
120
CD
cut the plane Mi-iii.ii
t
CD
with
be
S and
-.uvpt in a circular strip
Even
diameter. tiiti-irntl
H
cv-vt-
SljT
in
SD
t-0
at
p\TTt
-^
=
everywhere
of infinitesimal width of which
ST is
the
OZT
=
F
which involves
(10)
f
so that the
is finite,
is
zero
when
t
= 0.
Also
every point of the spherical cap of which 1 __ -ffxQ^ = */Y^ n^rfo) = a section, and Thus
frniii i-ijuation
Q
Then
in this infinitesimal strip -r
.F at
i
is
2T.
the points of inter-
let
ut
constant
t
AB,
but not the plane
ji.ing
as
(10)
J5T^
lies
it
ratlins at intersects
^
=H -^
w at-PR H
^
-^pQ-
,
and
independent of the position of
is
between the planes.
both planes
-
^7T
\\Fda
J j
When
= ---, "Zi
the sphere of
where d
is
the
Ctt
distance between the planes, and
It is clear that
intensity
is
H
d in which the magnetic two slabs of equal thickness in which the J5T/2, and these are propagated in
the slab of thickness
splits into
magnetic intensity is opposite directions normal to the faces of the slab with the velocity a.
The moving
slabs of
magnetic force form only part of the This is clear because the energy of the in the two slabs taken moving together is only half
solution of the problem.
magnetic
field
of the energy of the original magnetic field. to consider the electric as well. intensity
E
At
*
= 0,
E=
rl
everywhere, but
Sy in the plane of the
It
is
necessary
J? is
not zero.
Consider any
paper such that & crosses CD Then, by the first law of electrodynamics, confining ourselves to the case of propagation in free space for which * = 1, ^ = 1 and a = c, rectangle to
S.r,
which Sy
is
parallel.
ELECTROMAGNETIC WAVES
121
is independent of y and the work the along right-hand side of the rectangle where the angle SPQ = 9,
since everything
,
= c
ct
Thus f /
~
rf*>
= 2~ at
JJ dt
I
J
EP = nj+
_
When
ct
&r and, from
the value of
I
-^- S^c
to
Thus the
= 2?r- cE
if
,
{/,/-.
ct
!,
t
H
ffoE do) = 1 icE cir = 5-y. z 2a ct
-^ 47rJ J c?
j
I
-TT
J
PQ
11
cuts both planes \ve
ip
J ct
and opposite
)
(
10
<
,
7
a l Cos
But
0.
ct
the sphere of radius ,-a
zero.
^
cE
2-7T -TT:-
a-t-
vanishes
HSy
H-
contributed bv *
that from the
first,
the second plane so that
slab of
Ep
force
H
see is
that
equal
again becomes is
moving magnetic accompanied by an equal electric force at right angles to and of equal magnitud e with H. The electric force in the slab which moves to the right is in the opposite direction to that in the slab which moves to the left, whereas the magnetic force is in the same direction in both
cases.
Velocity of Propagation.
We
have seen that the components of the
intensities satisfy the equation
=a V% 2
Ci"
electric
and magnetic
so that their
changes
time and space are such as would arise if they were propagated from every point with velocity Fur a vacuous space \JJLK. = = 1 and so K that it follows that 1, JJL electromagnetic disturbances in
Vc
are propagated in vacuo with a velocity which is equal to c. the ratio of the electromagnetic unit of charge to the electrostatic unit.
At the time that
this
conclusion was
first
reached (by
had not been shown that electromagnetic disturbances Maxwell) were propagated with Unite velocity, so that there was no experiit
mental material available, by means of which the conclusion c mid be quantitatively tested. Maxwell however put forward the view. which Faraday's instinct had previously led him to express although his mathematical limitations had probably prevented him from being able to deduce adequate experimental evidence in favour of it, that light itself was really an electromagnetic
ELECTS BIAUXETIC WAVES
I**
-\vm, Maxwell predicted, its velocity of r of the two kinds of K>h.uM equal to the ratio lr,.jfcitfi!i"n of the value determination a mit M,i\\\t-ll hinisrlf carried ;UH!X In that
n..m.-ii"ii.
|.fi.
be equal to the velocity of light through At a much within the limits nf his experimental error. -|kuv, much a in was stronger Ittt.T dale ilSKS) Max\\vlFs Theory put flit- n-searches of Hertz* who showed that vibrating jHMiinii ly rlrctnVal sysifiiis Mich as could be set up in the laboratory Hertz investigated the principal rinittrd rWmniiatfiietie waves. f iliese waves and showed that they were analogous M!
i-
and
li'iind
it
t
jii'Mju-rnes I**
waves if
nf jmssrssiMii
Mnif
from which they differed principally in the
light, liiiieli
rn-eiit
longer wave-lengths.
determiimtions have shown a continually increasing
between the values of (1) the ratio of the two units charge, ("2) the velocity of electric waves and (3) the
a^Tt'euieiit.
f electric
velocity of light in
The agreement of the results of by the following numbers Electricity and Magnetism, p. 506.
space.
different observers is well exhibited
which are taken
froiii'Jeans's
Fir the value of results
c,
the ratio of the two units, the following
have been collected by H. Abraham f as likely to be most
accurate: Hiuistedt
3*0057 x 10 10
RMSI 3-0000 x 10 J. J. Thomsuii 2-9960 x 10 10 2I
>
Abraham
2'9913 x 10 10
Pellat
3-0092 x 10
lf>
Hurmuzescu 3*0010 x 10
10
Perot and Fabry 2'9973 x 10 10
The mean
of these quantities c
is
= 3-0001
x 10 10 cms./sec.
For the velocity of propagation of electromagnetic waves in Blondlot and Gutton J by
air the following values are collected
Blondlot 3-022 x 10
10 ,
:
2'964 x 10
10 ?
2'980
x 10 x 10 10 10
Trowbridge and Duane Mac-Lean
2-9911 x 10 10
Saunders
2-997
The mean
{"
3'003
2*982 x 10 10
these quantities
,
x 10 10
is
2-991 x 10 W cms./sec. * r
Ann. der Phys.
vol. xsxiv. p.
551 (1888).
Imports du Congrea de Physique,
Paris, 1900, vol. n. p. 267.
For the velocity of
light in free aether
Gornu* gives
as the
*0027 x 10 10 cins./sec. Dividing by most probable value 3'0013 1*000294 the refractive index of air referred to a vacuum, this gives for the velocity of light in air
-0027 x 10 10 cms./sec.
3-0004
The
velocity of electric
waves
known with much
is
less ac-
curacy than the other two quantities but they are undoubtedly all three identical in value within the limits of experimental error involved in each case.
Since the velocity of propagation light
in
magnetic
and
dielectric
is
c/V/4/c,
the velocity of
media should be inversely
proportional to the square root of the product of the magnetic permeability and the dielectric constant. Since it follows from the light that the refractive index n of a medium is inversely as the velocity of propagation of the light through it, it follows that for different media of the same magnetic per-
wave theory of
meability n-
oc
K.
This law has not been found to be even approximately verified In fact, a moment's confor the waves which constitute light. sideration shows that it
must be wrong,
since it
would make n
of dispersion shows that n is a function of fact that n* is not proportional to K is not to be regarded as an objection to the electromagnetic The theory on which it has been deduced is theory of light. constant, whereas the
phenomenon the wave-length. The
when
applied to the free aether but its scope is not wide enough properly to account for the optical behaviour of material media. The reason for this is that material media contain
exact
charged particles which are set into motion by the and magnetic forces of the light waves, and it is necessary consider the dynamics of these particles to account satisfactorily
electrically electric
to
for the optical
If
behaviour of such media.
we turn from
light
waves to the
electrical vibrations of
much
lower frequency emitted by the Hertzian oscillator the state of affairs is very different. The period of these vibrations is, as a rule,
great compared with the natural periods of the electrons that the motion of the
in the molecules of the substance, so *
Loc.
cit. p.
246.
ELECTROMAGNETIC WAVES
J-J4
rlrctn.ii>
ihi.-
i
!
hi-
the
same
magnitude as
.sum-
ihr
,.f
much
i>
litfht
wave.
as
it
the
would be under a steady instantaneous
field
value of that
Under these circumstances the material
nvated as u continuous medium of definite dielectric character the velocity of propac..rrtk-i'iit if, and fr waves of this be should media different inversely as the square roots gation in ran
In*
This conclusion
i.t'thr dielectric coefficients.
is
substantiated
by
Thus A. D. Cole* found the refractive
ivsuhs ofYxjieriinents. ind'X t* water t be 811 whereas tin-
its
dielectric coefficient
tc
= 80.
With other substances the agreement appears to be satisfactory within file nit her considerable limits of error of the determinations if ihe dielectric coefficients f. defer to the next chapter the consideration of the
\\\* >hall
of bodies towards light different than that predicted by the simple form of the electromagnetic
make the behaviour
causes which
There are, however, a theory which we have been discussing. of phenomena exhibited by electromagnetic waves in
number
matter which are partly true for light waves for true very long waves. The rest of this chapter strictly will be occupied with an account of some of these. their relation to
and
Properties of a Plane-Polarised Electromagnetic Wave.
A
solution of the equation
provided is
J-'-f
-M
2
-Hi 2
=L
-^ = a~-u
is.
"
The expression on the right-hand
side
a complex quantity, being equal to
The
real part
of a therefore represents a disturbance of
X and amplitude
which
wave-
propagated along the straight line .r/ = ?/'w = with constant .?/?? amplitude it Q and constant It is thus the velocity a. appropriate specification of a monochromatic train of plane waves of If we take wave-length X. the direction of propagation to be along the axis of z we shall length
is
t
*
*
IHVrf.
Ann.
vol. LVII. p.
290 (1896).
Fleming, Principles of Electric
Wave Telegraphy,
p.
320.
ELECTROMAGNETIC WAVES have
n
=
1
and
u
be equal to u Q e
will
125
T^~~>
an( j its real
part to u cos
Let the
will
for
(at
Ex
,
the x component of the
the wave front, and consider the train of
Ey = E
which
..................... (12).
2)
real part of u represent
electric intensity in
waves
^A-
=
z
The
0.
electric intensity in this train
be completely specified by the equations
>* ?
Ex = real part of
Ex = E
or
Q
It is clear that
ables such
cos
~(at-z) A.
~z
.................. (13).
any equation between functions of complex
as, for
vari-
example,
F
l
(a, iv)
= F.
2 (a!,
Real part of
Imaginary
part-
of
iy)
two equations
involves the separate truth of the
and
(at
^ = Real part of F
2
F = Imaginary l
part of F.2j
=V
1 would be equal to a real quantity, which is This principle effects considerable simplification in the working out of problems arising in connection with the propagation of waves, as it enables us to work with the complex solution
o therwise
i
absurd.
and then pick out the
The advantage
real parts at the end of our calculations. of this lies in the fact that the complex equations
are usually simpler than their real equivalents.
Suppose that we are dealing with the train of plane waves propagated along the axis of z. Each of the vectors x E>n Ez x z which serve to specify the state of the medium at any
E
,
,
H H H ,
, tJ
-~
~
(af r) ~* The values point at any instant must be of the form u Q e however are not independent but have to satisfy the six equations .
on
p.
116,
viz.
cH
z
cH,
= cEx ic
t
cy
cz
c
ct
dHx
cHz
K
cE
,
'
ELECTROMAGNETIC WAVES
12(5
>
As
eacli
9y
8^
c
?
?
?#
c
dt
C&y
C&x
?#
%
p d*2z
r ^ = ct
we have
.
i
9
2-7T
---
X
a,
x
e d
=
^r
9^
c
of the dependent variables const,
dy
is
of the form
^
*
= n0,
9
-~
=
oz
.
^
2-7T __
(14)
\
/C(7
, .
O IT
rr
L*
that e yrw
The
eqiicitinns
added
to
above
.(15).
may
all
them, but this would
have a constant of integration mean merely the superposition
H
E
are equal to zero, so that there is
no component either of electric u:r in the direction of propagation of the wave. magnetic intensity The resultant electric intensity and also the resultant magnetic = so that Also intensity lie in the wave front. x x+ y y
E H
E H
the resultant electric intensity is at right angles to the resultant magnetic intensity. In addition p, (Hx + y *) = K (Ex* + y *) so that the electric energy in the wave is to the
H
E
equal magnetic energy. In the free aether p, and K are each equal to unity in our units so that the electric and magnetic intensities are equal. If
along
we choose the it,
axis of x so that the electric intensity then the vectors which specify the wave are
lies
The
electric intensity will always remain along the axis of x and the magnetic The wave is thus intensity along the axis of y.
plane polarized
;
since its properties are not the
same
in reference
ELECTROMAGNETIC WAVES
127
two planes at right angles to each other pissing through the direction of propagation. shall see that the plane which is usually called the of plane polarization in optics is that which to
We
contains the direction of the magnetic vector and is perpendicular to the direction of the electric vector. (See p. 132.) It is evident that as the wave moves pist any fixed point the electric and magnetic intensities always remain propor0, tional to one another and both vanish and also both have their
maximum
values simultaneously.
In the case of waves of sound the kinetic energy
is
measured
by the squares of the time rate of change of the displacement of the particles constituting the medium, and the potential energy by the squares of the strains which depend upon the rate of change
same displacements in space. Both the velocities and the strains travel together or are in phase just as the magnetic energy and the electric energy in electromagnetic waves are in phase. This may be regarded as another reason for identifying magnetic of the
and
electric
energy with kinetic and potential energy respectively.
The following elementary method of deducing the velocity of an electromagnetic wave is instructive. Let the wave be prox be the x component of the pagated along the line Oz and let
E
electric intensity in the
when the wave
wave
front.
Consider the state of things
front lies in the plane perpendicular to
Oz which
passes through the point 0. Then the lines OQ and 017 (Fig. 23) are in the wave front. Describe the rectangles TOPE US and TOP VQ W, TP being small compared with PR\ud PV. TP, SE and TTF
PR
are perpendicular to the wave front, and ST are parallel to and Tlfare parallel 10 OQ, the axis of y. OU, the axis of ^ and Consider the work done in taking a unit magnetic pjle round
PF
the rectangle path vanishes
PTWV.
The work along the part
QFPO
of the
which constitutes the wave has not yet reached this part of the path. The work along Or is equal and opposite to the work along WQ by symmetry. x But this is equal The net amount of work is equal to y ;
for the electrical disturbance
TW H
.
the total current embraced by the path. In the present case the current is all of the displacement variety, so that it must be equal to the rate of increase of the flux of induction to
1/c times
through the
circuit,
in
the
units
we have been
using.
The
ELECTROMAGNETIC WAVES ,
u M s'e ,
,
,-ntiivly vrl'.ciiv tim.'.
amount of induction through the circuit arises wave so that, if a is the fn*m thr forward motion of the tcEx x a x to be will it per unit equal "f in
flif
TW
priig!ition,
'Tin*
Fig.
'23.
By considering the work done in taking a unit the circuit TPRS we find
electric charge
mund
It
at f
iVilluws
that a
any instant .r,
it
will
=c
\ IJLK.
We
also see that if the electric force
along any particular direction, let us say the axis always remain parallel to the axis of x, and the
magnetic force
is
will also
always be parallel to the axis of
y.
ELECTUOMAXET1C WAVES
ami Re fruition.
RejiesivH
We
129
shall n.\v consider the
behaviour of a train
waves
f plant"
incident on the boundary jTPt" between two insulating media whose dielectric constants and magnetic permeabilities are K^ /i 3
and K2s pu
components
(1)
The
Ex Ey ,
Hx H H !n
,
separation are
bt*
The
respectively.
by the components
z
conditions which have to be satisfied
E? of the electric intensity and the of the magnetic intensity at the surface of ,
:
must
tangential component of the electric intensity
continuous in the two media. (2)
The normal component
of the electric induction
must be
continuous in the two media. ^3)
The
tangential component of the magnetic force must be
continuous in the two media. (4)
The normal component
of the magnetic induction
must be
continuous in the two media.
Let us take the bounding surface to be perpendicular to the axis of z\ then these conditions will be satisfied of time,
E = J?x Ey = E Xl
,,
}h ,
,
xl
E = 2l
K.2
if,
EZ
at every instant _
)
{
'^^"
s-#^^<'&&&^&'s,'Z^ P
.
\
\ **
\i
. -*'
KLECTROMAOXETIC WAVES suffixes
the
whi-iv
2
1,
refer
two media.
the
to
For if the urv not nidi-pendent njiLttiMti* the since iitllnws equation thf sixth
first
These six two are satisfied
_
__ cU'
ot
c
3#
medium. The third equation follows always h>ld in each means of the relation similarly from the fourth and the fifth by iititsf
Thus
fur
uiilv
''Qv
_ ?^ =
~
Rr
?y
c
d
of the six boundary equations of condition are
ivuJlv imiejK'iident.
Wave polarized
We
shall
in the
Plane of Incidence.
now consider a wave propagated along OP and such force Ex in the wave front is per-
that the resultant electric
If the angle OPFis #1} of the paper. pendicular to the plane the vectors which define the to boundary, being the normal
PV this
wave
where
c/j
will all
is
be proportional to
the velocity and
&=
2-7T
--
They may therefore be
.
A-i
by
re|)ivsei)ted
since these expressions satisfy equations (1)
(C).
The simplest way in which we can hope to satisfy the boundary conditions that the tangential electric intensity and the normal electric induction should be continuous is to make the and z y
components of the
electric
intensity vanish for the refracted and reflected as well as the incident wave. The refracted wave will therefore be given by
Ex = X, e *" '
(**'
~
si
**
- ~ cos
Ey = 0, f
*->) ,
Eg =
;
ELECTROMAGNETIC WAVES '
x
=0.
=
H,;
- cos
.....
JLtfl
6X.,e*M-y* "*--< ~*.\
a., p.,
and the reflects
wave by
!
Ex " = X^''-'-?^---^ /// =
/?" = --
0,
E, = !
V'
0.
=
Ct;
c^0,X,e^^'y^^' s ^^.
......
-/*!
#,"= - .JL
s i ri
ft,
A>*3'.v-^*;-:r^
.
sA*i
Since the boundary conditions have to be satisfied at of the time, for all of
tile that-
factor in these expressions
Thu>
them.
A
=
Jic/i
determined by the medium,
= A^
'./..
that
.so
Alsiu
,.
;
=
t/j
ci, ;
.
boundary conditions must be values of
y,
the velocity
Hence
A=
so that the exponential factor in u must be Thus fii sin ft = fi* sin ft =/3- sin ft= ;
=
sin ft, so that the angle of reflexion in equal magnitude to the angle of incidence. Also ft
sin ft
/?2
ci
is
and
common 3 sin ft.
4
2
must be
(ij
.(17).
sn Thus the
^-
satisfied also for all
to each of the vectors.
Therefore sin
values
all
must be the same
retractive index or the ratio of the sine of the angle
of incidence to the sine of the angle of reflexion is equal to the ratio of the velocities of propagation of the light in the t\vo
media. of sin
$i
Also with the convention as to the signs of : and G S and and sin ds which is here adopted, in conformity with
general usage, cos
The boundary
#,
=
cos
X
.
conditions will
now be
satisfied if
X^X, = X, cos al ul
a
ft
(X,
AV) "
=
cos 0,
X
19),
1
a.2 jjL2
sin ft (X.-r '
1
-
'
(IS),
AV)= a. '
sin
ft, ~
J,
r20).
"
"
2
9
-i
ELECTROMAGNETIC WAVES
&=
Sim-.- sin ft sin
i
and (20) are identical. equations (18)
<*,>>
we find S,himr ^nations <1) and (10)
-
cos
ft.
ft!
+ -
sin
ft!
cos
ft,
/*a
cos
A>
cos
ft,
sin
0,
sin
--
ft-
sin
ft!
cos
2
J + -- sin
ff j
cos
ft,
*
.........
w
/^
For
all
wuiiltl
media on which experiments have been
serve to test these equations sin
(ft2
we may put
made which
^ =/t
2
,
so that
-#i)
Since the intensity of the waves is equal to the energy per unit volume multiplied by the velocity of propagation, and since the
a wave magnetic energy in
is
always equal to the electric energy,
the ratio of the intensity of the reflected to that of the incident
wave
will
be "
(TJ =
The values given by these expressions 1 /a the reflected and the refracted ray in terms of the intensity of the incident ray agree satisfactorily with those found experimentally when the light is polarized in the plane of where sin
ft^'sin
ft.2
ij
.
for the intensities of
This shows that the plane of polarization is the plane incidence. which contains the magnetic force and not that which contains the electric intensity.
Wave polarized
in
a Plane perpendicular to the Plane of Incidence.
In this case the magnetic force
is
plane of incidence (the plane of yPz). sent the incident wave by
to be perpendicular to the
We may
therefore repre-
cos
sin
ft,
&h( a J-y
^
&i
-*
cos ^)
ELECTROMAGNETIC WAVES
The
13o
expressions for the refracted and reflected waves are obtained suffix I by the suffixes 2 and 3 respectively.
by replacing the
As
must be
before, the exponential fact ore
since
oft and
when
identical
the boundary conditions are to hold tor
=
j
values bith
all
It follows that
y.
=
rtj
.;
sin
.
=
:;
6 and a
sin
From the condition we have
l
c/.^
2
=
& = sin
s
2
sin
#,.
that the tangential magnetic force
is
cm-
tintious,
Hi +
The continuity
H^H*.
of the tangential electric force gives
and the continuity of the normal sin Si
--
f
electric induction
---= sin
rr, ffl jflyl
ft,
(XZj-f
!
As
and
before, the first
^ = ^ = 1 we
/...
(I-,
last of
these equations are identical.
Fur
have K /e.= a./'Gf so that the second equation
may
I
be written sin
Solving for
e cos
0!
i
H
H,
(
H
and
5
- Hj = sin
we
//..
=
- sin
0j
+
value of
2
,
L
=
= TT '2, = 7r ^,
tan (0! 2
.
H.>.
sin
:
sin
0._.
tan 2
:
f
cos
H~
:
cos 0, 7
- tan tan
0j -f
+ $J = x
;
:
icot
L
:
icot
0.,
so that
,
~
tan
0_)
ran
0.2
__
)
for the particular
the intensity of the reiit-ered wave known as the polarizing angle. When 0o,
This angle is exceeds Jy H~ becomes negative so that there zero.
2
cos 0o Tr
=
1 -f tan1 -f
When
cos
0.-,
:
_ ~
cos
find
sn
TJ-
2
,
of phase at the polarizing angle.
is
is :
a sudden change
These formulae are found to be obeyed by light waves polarized to the plane of incidence for all angles except those perpendicularly of the polarizing angle. It has in the immediate neighbourhood that the and amount of Jaminf although been shown
by Airy* becomes very small in the neighbourhood of the
reflected light
polarizing angle
phase which not sudden.
it
never actually vanishes, and the change of
takes place in that neighbourhood
DrudeJ has pointed out
that
is
gradual and
these
differences
between theory and experiment disappear if we suppose that the transition from the one medium to the other is gradual and not discontinuous.
If there
is
a layer of very small but finite
thickness in which the properties of the medium gradually change from those of the first to those of the second substance, the amount of reflected light never quite vanishes and the change of phase The layer of transition may be small becomes a gradual one.
compared with the wave-length of
It
light.
is
probable from
the existence of other physical phenomena, such as surface tension and the tenacious retention of surface films of gas by solid
do occur.
substances, that such transition layers
Conducting Media. In those cases in which the
medium
affected
by an
electro-
magnetic disturbance possesses electrical conductivity, the displacement current will not be the only current which is set There
be the
true
current of density Maxwellian equations suitable for this case are will
also
a, /9\ (2)
j.
up Thus the
**rot+
r J=--
.
f)t
Now if the medium is isotropic urn! a tivity, j^aE, so that the first equation may
is
l>e
its
specific conduc-
written
dt, *
Comb. Phil.
Trail*, vol. iv. p.
219
(18IJ2).
t Ann. Chem. Phys. (3), vol. xxix. p. aiW (ln.<30) J Lehrbuch dcr Optik, Leinzicr 1900 n '><;/;
;
ibid. vol. xxxr. p.
105 (18SO).
Let us suppose that waves of frequency p impinge on the boundary of such a medium. We shall see that the boundary conditions of the problem can be satisfied by means of a transmitted and a reflected wave both of which have the same frequency The time variation of all the vectors can therefore be approp. priately represented
Let us
first
by the common
factor e Lpt
wave of this frequency and magnetic forces
consider the propagation of a
The
medium.
inside the conducting
.
electric
have to satisfy the equations
and
rot
E = - -c
-^
.
dt
<"\
Since
=
waves under consideration, these equations
ip for the
are equivalent to rot
H= c
,
.
and
rot
+K}
(
J dt
\ip
E = - -c -=dt
E = ~c -5dt
.
.
Thus the results which we have obtained for a non-conducting medium will still hold if we replace the dielectric constant of the medium K by the complex quantity /^ = + a lip. The com/c
ponents of the
and magnetic vectors
electric
still
satisfy the
equation
equal to c /p, (K + cr/Lp). which the waves are propagated in to case the ourselves Confining ? 1 the is of since z, proportional to e everything along the axis
but the constant
a? is
now complex and
2
1
,
equation of propagation
P ~ 2
The
for this case o
f
\
(ip
reduces to
= &u
.
solution of this, appropriate to plane waves,
where
(a
+
2
/3)
=
2
--
0"
(ipcr
p'
fc),
is
136
so that
2a/3
=
y
factor e&, the complete expression for u Restoring the time two waves, one propagated a2 #(***& will This represent is e in the negative direction along other the and in the positive to the one propagated in Confining ourselves the axis of z. vectors which determine the that see we the positive direction .
are given by u = $-* &<**-&. distance z in time 2ir/p and periodic wave-length in the conductor /3=27r/X.
This shows that the wave
it
= 27r/y8.
=
term
real exponential
az
e~
Thus
if
X
The occurrence
is
is
the
of the
shows that the amplitude diminishes c:
the law of diminution, or absorption," progresses, falls off in equal ratios whilst the the that amplitude being The quantity a is distance covered suffers equal increments. as the
wave
of absorption, but sometimes also the usually called the coefficient coefficient of extinction. I/a is the distance in which the amplitude falls off to 1/e
of its initial value.
Solving (27) for a and
a=
1
/3
we have
/up
cV
2 ............ (28).
e The formula
for a shows that the coefficient, of absorption is In other the higher the conductivity a of the material. higher of the medium the more the the words, conductivity rapidly higher is it able to transform the energy of the electromagnetic waves
traversing it into energy of other forms. cr the value of the electrical conductivity
If in (2S)
we
put, for
deduced from measure-
ments with direct currents, the values of a which result; are higher than those which would be obtained from measurements of the
Thus we have approximately
transparency of metals for light in the visible speetrmn. in the case of copper, = 1, p 3 x 10 10 c yu,
= a = 3'3
;
x 10 6 cms."
measurements* *
is
Cf.
and sodium
=
3
x 10
light,
lr>
= 47TC-
:
x
x
^
and
0-\ whereas the value of a dedueod from optical 2-8 x 10 s cms." The particular value a.ssi^ned cr
,
1
1
,
1
.
Drude, Lehrbuch dcr 0-ptik, 1st
eel.,
]_>.
li.HH.
A"
1
to K in calculating a from (28) does not exert any important influence on the result. The formula also does not the
suggest observed difference in colour between the incident and transmitted
These discrepancies arise from the fact that the current by discrete electrons, with the consequence that both and K are functions of p. The reason for this will be made
light. is
a
carried
clearer in the sequel
;
we
now
shall
phenomena which
consider the
attend the reflexion of light at a conducting surface.
Metallic Reflexion.
The problem of metallic reflexion is very similar to that furnished by the case of reflexion at the boundary between two The same conditions as to continuity of the insulating media. tangential electric and magnetic forces and of the normal electric and magnetic inductions have to be satisfied in both cases. The difference arises from the conducting power of the metallic medium, and we have seen that the type of theory proper to an insulating medium accounts for the propagation of waves in a conducting
medium
if
quantity #/ #2' its
we
replace the
icr^/p,
where
dielectric coefficient
cr.2
constant by the complex
dielectric is
and p
the conductivity of the medium, the frequency of the waves. It is
the boundary conditions cannot be fitted by the method previously adopted, the only change made being that the real quantity K2 in the former problem is replaced by natural therefore to see
if
f
'
the complex quantity K.2 where tcz is the real dielectric icr^jp, coefficient. We shall consider here only the case of waves polarized in the plane of incidence.
A
more complete discussion may be
found in Drude's Lehrbuch der Optik,
p.
334.
The incident wave being
Ex = X,e^ (<'it-ai-s<x>**i\
H
x
= 0,
H
y
=
cos
Ey = 0,
O.X.e^ (J- y *mo
sin 0,
E = 0; 2
l
X,e^ (it-v*
- a <x>*e
i ) j
01-^cos^
the refracted and reflected waves will be given by similar ex2 and 3 pressions with the suffix 1 replaced by the suffixes
ELECTROMAGNETIC WAVES
H38
We
respectively.
since K2 is
= af =
have of
complex so y#2 a 2
a.
also is
= & Oi
and
2
and af
c ///j KI
c
2
/^2 /c2
Thus
.
Since sin
ft l
^ = /> sin
2,
& and sin
0.2 are also complex. The factors involving y and being common, the two independent equations yielded by the boundary conditions become
y
t
A. l -T
COS
and
0j
v
,
Z
=
Z
=
*
a^d
3
A
-rr
2
.
9"F
TTT^ +
(
^eos^
a_
^
a 2 jUL2 COS
i
may be put
29 )>
v
ai^cosft
Now complex, and
COS #2
3)
j
is
2
-A^ =
j
real.
Ay
3
,
-(A!
Thus
Ay
;
COS
cos ^2
//^i^a
V ^/Cj COS ^
in the
+ ift,
form a
where a and
ft
Then
The complex
fraction
pe
where
p sin P
may be put
3
into the form i
sin
e),
^?
=
6
^kL-0
so that
= p (cos e +
Lf
+ /3*
4o^
+ a) +"^ a
(l .
tan
Thus
^
will
6
=
-2/3 ci a -/9 3
l-a
be the real part of
pX
t
e
^1
(
ai *~2' sine
i+CO
.
(ol)j
/oo\ 5 v'
^-
are
where fore
pXl9 /3l) l ,a 1} and e are all real quantities. There will therebe a difference of phase e between the incident and reflected
waves.
Experimental measurements of the ratio of the intensity of the reflected to the intensity of the incident radiation for infra-red radiation, in the case of normal incidence, have been made by In this case O l =
Hagen and Rubens*.
i
6%
= 0,
so that
COS $2
and ptc
Hence
a
"
/
JL
Even in the visible part of the spectrum a-Jp is a large quantity, 15 although p is of the order 10 and /m l} /^ 2j /^ and tc z are all comThus a/3 is much larger than a2 yS 2 To a parable with unity. '
.
first
approximation we may
therefore put
V
2/jio f)Ki
and, from (31),
= 1-2" The results of Hagen and Rubens are expressed in terms of the reflecting power of a metal. This is defined as the perof the incident is found in the reflected which centage intensity
R
beam.
Thus
in
terms of our notation
R=
3
100/>
,
^l
100- R=200XA/
and
J
1
(36).
The following table exhibits a series of values of (100 R) x V^a We observe found by Hagen and Rubens for a number of metals. *
Kitz. tier
Phys. vol.
Kon. Aktul. dcr WimteuKch. Berlin,
XT. p. H7:-J
(
190)5)
;
Phil.
Mag.
[(>]
vol.
p.
269 and
vn.
p.
p.
210 (1903)
157 (1904).
;
Ann.
tier
ELECTROMAGJV ET1U W A V US
140 that for
all
the metals investigated, although there is a consider100 - R, the product (100 - R) \faz is
able range in the value of
The confor each particular wave-length. very nearly constant the the marked more the on whole, wave-length X. longer stancy is,
The values
of
cr
2
are those at 18
at which the experiments
C.,
which was approximately the temperature
were carried out.
The numbers
in the preceding table were obtained by a direct This was comparison of the reflected and the incident radiation. furnished by a Nernst filament and the different
wave-lengths were obtained by dispersion through a fluorite prism for the 4 4 range from 10~ cm. to 8 x 10~ cm. and a sylvite prism for the 4 4 range 8 x lO" cm. to 14 x 10~ cm. For longer wave-lengths this method was unsatisfactory owing to the small intensity of the radiation thus obtained.
An
method was therefore used comparing the radiation, emitted normally, from surfaces of the different substances with the radiThis
instead.
ation from a "
consisted
indirect
in
perfectly black
"
at the same temperature (seo of the emission of radiation from hot bodies shows that the radiation emitted surface is to the
Chap. xv).
body
The theory
by any by a similar surface of a perfectly black body the same temperature as the radiation absorbed by the same
radiation emitted at
the radiation incident upon it. Since the reflected equal to the incident radiation less the radiation absorbed, this method enables the of metals to
body
is
to
radiation
is
be determined with accuracy.
reflecting
power
-N
JD,
1 1VJ
WAV
The following table gives the values of the product (100 H)a^ obtained by this method for the residual rays of wave-length 25*5 x 10~ 4 cm. which remain after successive reflexions from sur-
}
faces of fluorite.
ment
Considering the smallness of 100
of the experimental and
computed values
is
R the agree-
remarkably good.
There are two results of this investigation which are of special In the visible part of the spectrum it is known that the reflecting power of metals does not agree with the
interest.
predictions of Maxwell's Theory in its simple form, so that the present experiments determine the boundary of the region where other considerations have to be taken into account. The experi-
mental results are in accordance with the simple theory when the 4 wave-length of the radiation is equal to 25*5 x 10~~ cm.
The other The computed /u/1
=^=
1
point relates to the magnetic qualities of metals. values in the tables have been obtained by putting and #! = K.! = 1 for all the metals. see that the
We
the case of the magnetic metals and alloys such as nickel, steel and so on is just as good as in the case of the non-
agreement
in
magnetic metals.
It follows that for oscillations of the frequency
of those experimented with, the magnetic metals behave as
though
OHAPTEE DISPERSION, ABSORPTION
Medium IN the
VIII
AND SELECTIVE REFLEXION
containing Electrons.
we have seen that there are a number of Maxwell's Electromagnetic Theory which are
last chapter
consequences of
of very long period, but which by electromagnetic waves from being borne out by the waves of much higher frewhich constitute the visible and ultra-violet regions of the
satisfied
are far
quency
spectrum.
We
shall
now show
that the reason for this
is
that
we
have neglected the part played by the inertia of the electrons. When that is taken into account we shall see that the results of the experiments, even on waves of very high frequency, arc in with the consequences of electrovery satisfactory agreement
magnetic theory.
We
shall
up of a
number
of
similar
units
which we
shall
made call
Each molecule contains a considerable number of
molecules. electrons.
suppose the matter under consideration to be
large
In the absence of external electric force the electrons
are to be regarded as establishing themselves in fixed positions of stable equilibrium, or in configurations of stable orbital motion.
In the presence of an external electric field the electrons will be drawn away from the positions of undisturbed equilibrium in a manner similar to that discussed in Chap. IV. Owing to tho inertia of the electrons
with the time
their behaviour in
a field which varies
not so simple as that in the case of a steady field, which -we have already considered.
Our
first
is
concern will be with
substance which in stitution than
any
all
real
probability
is
the behaviour of an
somewhat simpler
substance occurring in nature.
ideal
in its con-
We
shall
suppose that under the action of an electric intensity different electrons in the molecule denoted "by the suffixes 1,
E 2,
the ...
n,
undergo displacements x1} x2) ... xn from the equilibrium configuration. These displacements are supposed to be all in the same
which
direction,
We
that of the applied intensity E.
is
shall
suppose that when a displacement #a for example, takes place, there is a force of restitution called into play which is proportional ,
and
xl) and independent of #2 x3)
parallel to
we
xn
...
,
tion that the force of restitution, which
This assump-
.
denote by ^/X 1} is independent of #2 #8 ...#n can only be justified as a rough approximation, since it is clear that each of the displacements en xnj ^,...cn is equivalent to a doublet of moment e2 #2 ^3^3 ,
,
shall
,
>
,
are the charges of the respective electrons, and it is evident that each of these doublets will give rise to a force on the
where the
e's
electron e l proportional to
approximation
its
The
moment.
case in
not made will be considered later
is
which
(p.
this
169).
In addition to the forces of restitution each of the displaced by one or more forces of
electrons may, in general, be acted on each of the following types :
The external impressed
This arises from
the
external electric intensity E, and it might be thought at sight that its magnitude would be given by Ee JEe^, ... Een
first
(1)
force.
.
l ,
The
however, identical with that discussed at the end of and the argument pursued there shows that the value IV, Chapter of the actual force of external origin acting on the electrons is
case here
is,
given by (E + aP) e ...(E + aP) e n where the medium, and a is a constant, which, l
ciently symmetrical,
There
(2)
is
the
is
the polarization of
medium
is
suffi-
equal to one-third.
will also
oppose the motion.
P if
,
be
These
forces of frictional forces
may be
type tending to represented by a term
proportional to the velocity, which in the case of the 5th electron for example we may denote by /3 s x s where /3 8 is a constant.
The
We
understood. is
mechanism
precise
accelerated
it
of these
shall see later that
rise
to
is
not yet properly electrified particle
emits radiation, and the emission of this radiation e
gives
forces
when an
a
reaction
2
--
U 7TG s
harmonic vibration proportional
'x
(see
p.
to e tpt this
266). is
For a simple
equivalent to a force
DISPERSION, ABSORPTION
144 (
- ?~k 3
'
AND SELECTIVE
be represented as a and may J therefore
fractional
term
67TC
provided ^
B8 =
^~ 67TC
It is likely
.
3
enough that
forces of this character
In most in the case of vibrating electrons. always be present for more cases however they seem to be incapable of accounting as the observed absorption of the term than a minute will
&*
part
in general,
is,
# 2<jD
much
to which the frictional force greater than that
2
- TT~ 07TC
& would give
3
rise.
Lorentz* has suggested that the frictional term arises through the disturbance of the motion of the oscillating particles caused collision the molecule by the impacts of the molecules. At each
that the regular forced regarded as so profoundly shaken up The average into oscillations are converted irregular motions. result will clearly be of the nature of an absorption of energy. is
In
if
Lorentz finds
fact
between two
terval
(loc.
collisions,
cit.)
that
if
r
is
the average inis the same as
the average result
there were a frictional force equal to
2m
x acting on the
m
is their mass. electrons, where Unfortunately this cause does not seem to be large enough to account for the observed effects. It is found that the values of r calculated from absorption phe-
nomena
are
much
smaller than those deduced from the kinetic
theory of gases. It
seems to the writer that the following view of the mechanism
of the absorption of light has much to recommend it. In the of cases it is majority probable that the resistance to the motion
due
to radiation,
and
to the effecfc of intermolecular collisions is
small and comparatively unimportant. It is probable that in the neighbourhood of one of the principal periods of the substance the absorption of energy by the electrons continues until the If the equilibrium of the vibrating system becomes unstable. absorbed is the electron will be omitted energy sufficiently great, from the molecule and photo-electric will be exhibited.
phenomena
Nevertheless this need not All that is necessarily be the case. necessary is that there should be a temporary rearrangement
among
the electrons inside -the molecule. *
Theory of Electrons,
p. 141.
In either event the
AJMJ
OJiL/HUJL JL V
Jl
JttJ^J?
JUJ1A1UJN
energy of the resonating system will be converted into energy of other types, and that particular degree of freedom will possess comparatively little energy when the original system is reformed.
The general results of this view are in qualitative agreement with those which follow from the assumption of a retarding force proportional to x. They show the same gradual change in phase in passing through an absorption band, and indicate a maximum absorption and a refractive index equal to unity, when the period of the light is approximately coincident with the natural period of the substance.
In view of the
mode
facts that absorption does occur,
of occurrence is
still
doubtful,
we
and that
its
shall for the
present content ourselves with the assumption of a term, in the expression for the force, proportional to - x. This term is to be regarded, not as the expression of a fundamental truth, but as a simple
and
convenient mathematical approximation whose consequences simulate the observed effects.
In general we shall have to take account of the effect of the presence of a magnetic field. We have seen (Chap, vi, p. 68) that a magnetic field of intensity acting on a moving electron, (3)
H
charge eS) gives rise to a force whose components are
all
The total force acting on. the electron will be compounded of the forces mentioned, and, in the most general case, its equations
of motion will be
x
&z
*
Zs
^ Zs
Tt
a J? j. P /Q j. - ( P t + *,Pt =e,(E )---f3,^ + I
\
6s
...... (1)-
DISPERSION, ABSORPTION
146
AINJLI
There will be three equations similar to these, but with different n classes of electrons. The solution coefficients, for each of the most general conditions leads to the under of these
equations and does not lead to very complicated expressions,
any
results of
which cannot be obtained more simply otherphysical importance We shall therefore consider the operation of the different wise. forces separately, in so far as
and thus find out the kind of In particular the the presence chapter,
deals
We
phenomena.
the last two terms, which depend on magnetic field, will be deferred to a later
effect of
of a
which
they can be separated physically, which arises from each cause.
effect
with
magneto-optical
and
shall also confine ourselves to
spectroscopic the case in which
in the plane of yOz, are propagated along plane waves, polarized the axis of z, since all other cases may be made to depend on this, if necessary.
No It
Friction.
seems desirable to treat this case separately, although of we include friction and
course the same results are obtained if
put the coefficients /5 equal to zero. The chief advantage of thus lengthening the treatment is that .effects due to resonance are then sharply marked off from those then in the
final expressions
due to absorption and are brought out the two effects are considered together.
more simply than when In the case under con-
sideration the equation of motion of the sih type of electron
m*^ = or,
e
(E +a
P)--
8
dropping the subscripts for the present,
0=
if
is
p and D = -. maximum electric
the frequency of the electric waves
constants which represent the electric polarization in the wave. differential equation of the
function
Ex + aPx
is
E and
is
1>
are
intensity and is a linear
Our equation
second order and the
complementary
A
are arbitrary constants and p Q = (\m) * is the frequency of the natural vibrations of this electron. The particular
where
integral
:
arid
A
2
is
\e
m The complete solution is the sum of the expressions (3) and The arbitrary constants A and A* are to be determined, in (4). If so and x any particular case, by the given initial conditions. are zero when the waves begin to fall on the electron the natural vibrations A I 6 ip + A 2 e tpot have the same initial amplitude as the l
t
forced vibrations of frequency p, and in any event the amplitudes of the vibrations of frequencies p Q and p are initially comparable. There is however a very marked difference in the behaviour of
the two vibrations as time progresses.
The energy
of the vibra-
tions of frequency p Q will gradually disappear owing to radiation and after a time their amplitude will become negligible. On the
other hand any loss of energy from the vibrations of frequency p continually made good by the action of the waves which have
is
the same period and the same or opposite phase. These conclusions can be established quite strictly if a small coefficient @ of frictional type is introduced into the equations. It is then found that the complementary function contains a factor e~~ at where a is positive constant, whereas there is no such term in the particular This shows that after the lapse of a sufficient interval integral.
a
the vibrations will be represented by this particular integral only. The student is recommended to work out in detail the case in
which there
is
are, let us say,
a small frictional term and the initial conditions
x
=
It follows that
and x = 0.
when
sufficient
time has elapsed for the system x8 is given by
to have got into a steady condition, the value of
m
8
m
o
Let
z/ s
volume.
the number of particles of type 5 per unit represent Then the polarization
Therefore
and
From the symmetry
Now
of equations (1) it
is
clear that in general
the equations which determine the propagation of the
waves
are rot
= ~-^r, H=--^c ot G ot
and o ou
The 7??
o
velocity of propagation is c/Vyu/c
= V/-6/C.
media,
Since we may put
/-t
=l
ou
and the refractive index
for
practically all dispersive
we have n
-,,
s,
2
2-
m?=l+ 1
or
o-l
a
- i^(?V^-) a2 A^
^
-
VA
^
For substances which are sufficiently isotropic (sec ]>. 72, Chap, iv) the value of a is one-third. Formula (5) then becomes
DISPERSION, ABSORPTION
AND SELECTIVE REFLEXION
149
medium is made to change, by compressing the example, only quantities on the right-hand side which are altered are the z/s, each of which is proportional to the density. The right-hand side may therefore be written in the form Cp, As the density
of the
it for
the density and C is a quantity which is constant for any given substance and wave-length, but which varies from one substance and wave-length to another. The formula
where p
is
was first deduced empirically by L. Lorenz* and was subsequently shown to be a consequence of the electromagnetic theory by H. A. Lorentzf. The most exacting test to which the formula has been put is that of calculating the refractive index of vapours from that of the corresponding liquids. The following figures, the extent of the show agreement which may be given by Lorentz, obtained in some cases :
It is to be borne in
mind that comparisons of the kind here made with light of the same wave-
discussed are always to be length.
be observed that in the deduction of formula (5) nothing which compels us to consider the medium as a chemically simple substance. Provided a has the same value It will
there
*
is
Ann. Pliys. Chem. vol. u. p. 70 (1880). A rather similar formula (see p. 155) had been derived from theoretical considerations much earlier by Maxwell, Cambridge Calendar (1869). Cf. Lord Rayleigh, Phil. Mag. vol. XLVIII. p. 151 520 (1872). (1899), and Sellmeier, Ann. der Phys. vol. CXLV. pp. 399,
AND SELECTIVE REFLEXION
DISPERSION, ABSORPTION
150
for the electrons in all
the different kinds of molecules present
the value of
m -! 2
m will consist of the
sum
of the product of
two
2
a-1 -which consists
of a series of terms each of
The
factors.
first
factor is the
number
of
of given type per unit volume and the second is a particles function of the frequency which has a definite value for each
Thus the value of
type of particle.
m -! 2
for a
may be
mixture
fraction
which
calculated
by multiplying the value of this
characteristic of each of the substances present,
is
by the proportion of the substance present in the mixture. a
=
J this result
m -1 2
m
may
m-f-l
1
Putting
be written
m -1 2
,
1
m*?
2
m
and p for the 5th constituent p s denote the values of is the mass of it in unit mass of the mixture. This s present and/ formula has been found to be fairly satisfactory for mixtures of where
s
,
different liquids.
It is found that this additive
law of refractivity
is
not confined
merely physical mixtures. Something of the same kind holds for the individual atoms of which different bodies are
to
composed. Supposing that a has the same value (-J) for every electron no matter in what atom it may happen to be, expression (5) may be written
_.
ii m +2 2
where a S)
bs
of electrons
summations n
a
e
3
*** ,
LT m, (Pi -?)
i
m
+ s
*
(p*-p*)
etc. represent, for each atom A, B, etc., the number of given type per unit volume, and the separate are for each distinct kind of atom Thus
,
present,
2
>-
is
t0
be taken over
a11
the electr ns in an atom of
AND SELECTIVE REFLEXION
DISPERSION, ABSORPTION
A (supposed liquid) then, if the refractive power the state of chemical combination, -
a
=
ma + 2
is
151
independent of
,
2
3
m
!
(p?-p*)>
8
where a/ is now the number of electrons of type s present in unit volume of the liquid element. If is the molecular weight of the
M
compound, A, B, C if
M
a molecule of
are the atomic weights of the elements, and contains q atoms of A, r of B, s of and so on, ...
M=
then
If the density of the compound elements are p a pi p c etc., then a^
O bo that j
1
TT
o
z i
_ pm A 8
'C
i
is
p m and the densities of the
,
,
@s"
T
-,
in s
(p s -p*)
bs
= a*
Pwi &-
~f M
pm B
__
*
x
^
6
pa
^ Z i
C^S &s"
m
-,
s
i
;
^r
(pi
-p-)
etc.
,
~
The 4 quantity J
_ m + 2p Bl /^|
s
xi ^^ and the quantities compound -^ H i
the molecular refractivity J of the
is called
i
ma -
ma + 2
1
= 2
m
--4 -
, '
/} fl
l
B
m + ^2
p6
-
b
2
&
,
etc.,
are
called the atomic refractivities of the elements.
The
fact that the molecular refractivity of different substances
can be calculated from the atomic refractivities
is
quite important.
shows that the mechanism which gives rise to refraction is an atomic property and is so deep-seated in the atom that it is almost It
uninfluenced by the changes of configuration of the atomic conThis stituents which take place during chemical combination. of cases to in a number be true certainly seems fairly large of chemical combination. It is not, however, universally true.
Thus is
all
in the case of the
compounds of carbon, the carbon atom
found to have a sufficiently constant atomic refractivity in the so-called saturated compounds. If, however, two of the
carbon atoms are united by a
them
"
double bond
" it
is
necessary to
a quite different atomic refractivity. Again, the value of the atomic refractivity of oxygen is quite different for
assign to
molecular refract! vity magnitude of the aid in determining the constitution of is regarded as a valuable On the whole, the general constancy of the organic compounds. definiteness of the changes when atomic refractivities, and the are so distinctive that the
take place, supply us with striking evidence that changes do not very profoundly disturbed by the structure of the atoms is
chemical combination. Dispersion.
way in which, according to our index depends upon the frequency or waveFormula (5) may be written the incident vibrations. length of form in the somewhat more simply,
We
shall
now
consider the
formulae, the refractive'
where a
Now
.
since the incident vibrations
vary as &&,
OL
and the natural vibrations as
&***, e***, etc.,
they are respectively
If X, X 1? X 2 etc., denote etc. periodic in times 2ir/p, 2ir/pi, 2tr/p^ the corresponding wave-lengths measured in the free aether (not ,
by the waves) X2 =
in the substance traversed
X=
Xj = QTTC/PI,
27rc/p,
m
2
1
=
T>
+
47r2 c (l 2
^?~+~a
=
S
j.
But
r-
:
-\
o
\g-
+
infinite
2 i
summation
p 2^ 4 -~4 2 -1
'
*>'
----
i
.
;
m
Now when X is
^ 7 wv(X^T7)
a)
so that n
etc.
27rc/pa
v s ef\ 2 \*
1
s
-ij
m
s
(X
X,
nx 2
=0,
since every J term in the
)
But when the wave-length is infinitely briginfinitely slow, the case under consideration
vanishes.
and the period
is
approaches continuously to that of an electrostatic Held.
The
quantity K, which enters into the equations of propagation, must therefore become identical with the dielectric coefficient as measured
by
electrostatic
methods.
Hence
for infinitely
the dielectric coefficient of the medium.
ic
+a
47f2 c 2 (l
+ ai
We
m
s
long waves m-
therefore have
=
K,
1 Y
m +a and m are 2
vs
,
es
\s
,
/c
_a
47rV2
(
+
1
Cl}
......
T m, (X - V) 2
a)
*
constants characteristic of the material under
s
may be
written in the form
_, -X
X2
where
_
.
4-
consideration, so that (8)
substance.
JtlJl<.c LiJCj A.JLL'IN
JtLi
n
"* v
.
X2
2
2
-Xn
2
Jj
C^,...Cn are constant quantities characteristic of the There will be one of the terms Gl9 (72 etc., correspond,
ing to each of the natural periods.
In the case of transparent colourless substances the natural periods must be either in the infra-red or in the ultra-violet part of If they are in the infra-red the synchronous wavebe large compared with X, and if they are in the ultra-violet the X-/S will be small compared with X. Thus, as an
the spectrum. lengths \ r
will
approximation, we
may
write (9) in the form
Qr Xr2
m'-l^K-l
m
2 -f-
a
K
+a
^Q
v
~"
C = Constant + 2 r~
X2
, 1
approx ............. (9
a).
/Yp*
~\
~
"X
"
In these formulae we have neglected the fractions -~- and X Xr ~ changes compared with unity. Since the refractive index
m
roughly j as 8 j in the same way
m
2
1
m +a 2
it
,
follows from (9 a)' that for v
these substances the refractive index will increase continuously X diminishes, in this region. Since the transparent colourless
as
first to have their dispersion investigated, As we shall see, it this type of dispersion is said to be normal. can only be said to be normal provided we are a long way from
substances were the
the natural frequencies of the substances.
The behaviour whose frequency is substance is most X=X
6%)
of formulae (8) and (9) in the case of light close to that of the natural vibrations of the interesting.
When p
and the corresponding term in then
(9)
ps
for
becomes
2
example, then infinite.
If
X
is slightly less X,, X/) has a large negative C\./(X 2 Xs2 ) has a value arid if X is slightly greater than X s then (7S /(X As X approaches X s from smaller values of large positive value.
than
;
,
X
m """ m
9
AND SELECTIVE REFLEXION
DISPERSION, ABSORPTION
154
2
4-
approaches rr
a
X ^
values of
m
2
4-
o>
and as X approaches \8 from larger
,
+
approaches a
change 5 in the value of synchronous frequencies.
m
oc
from 2
-
+
to
oo
a
-f
we
If
There
.
is
thus a sudden
oo
at each of the
plot -~TT"a as ordinates against
X
have two branches, one extending to and the other returning from plus infinity at each
as abscissae the curve will
minus
infinity,
of the values
= \ X = X2
\
1}
,
Thus
etc.
in crossing
an absorption
in the direction of increasing wave-length, the refractive
band
index will suddenly increase, and the rays close to the band on the red side will be more deviated than those on the violet. It was disThis kind of dispersion is said to be anomalous. covered by Leroux* in experimenting with a prism filled with iodine vapour. The later investigations of Kundtf showed that it
was related
to the presence of absorption
bands in the way
We
indicated by the type of theory now under consideration. also observe that if X is less than every one of the values
Xl3 X 2
Xw
...
,
,
of the terms
all
Moreover in
sign.
all
the
^/(X
2
cases
same
Xf), etc., are of the
known
at
present
1
+a
is
the other quantities which make up the C"s are positive, 2 2 Xj ), etc. essentially positive, so that each of the terms (7i/(X is when X of is less than each the values X etc. 1; X2 negative
and
all
,
X
greater than each of the quantities X^ XL>, ... \ n all of the quantities Ci/(X 2 - V), Cn/(\:2 - \ n2 ) are positive. It 2 follows that for is sufficiently short waves always less than K,
Also
if
is
,
.
. .
m
and
for
sufficiently long
waves
the case of the latter assertion tion, since infinite.
as
m
2
On
X becomes
unity.
This
m
3
it is
always greater than K. In necessary to make one reservais
clearly approaches the value
the other hand, zero is
m* =
tc
when X becomes
it
follows
from equation (5) that
and p becomes
infinite
m
in accordance
2
approaches the value
with
experimental results, since the Roentgen Rays, which, as we shall see, may be looked upon as electromagnetic waves of very high frequency, are not deviated in
passing through a prism.
A *
great
many
dispersion formulae, that
C. M. vol. LV. p. 126 (1862).
is
to say, formulae
AND SELECTIVE REFLEXION
DISPERSION, ABSORPTION
which connect the
index with the
refractive
frequency of the transmitted to time. Cauchy's formula
light,
m=A +A l
2
155
wave-length or have been in vogue from time
\~2 -f AsXr* -h
. . .
was the
first to have a theoretical justification. Cauchy developed from general considerations of wave theory on the hypothesis that the distance between the vibrating particles of the medium it
could not be regarded as completely negligible when compared with the wave-length. The formula thus obtained gives a fairly satisfactory representation, when three terms are used, of the dispersion of a number of substances. As it makes the refractive
index increase continuously with diminishing wave-lengths (the constants are all positive), it fails absolutely to account for
anomalous dispersion. The idea that time rather than length was the determining common factor of the light and the matter which gave rise to Maxwell dispersion, appears to have occurred first to Maxwell*. supposed that when the atoms (we should now say electrons) were -displaced, forces of restitution were called into play, and that there was also a resistance to their motion. similar idea
A
occurred somewhat later to Sellmeier, after
which
this theory gives rise
single
mode
of vibration
is
and in the absence of
(p.
73)
the wave-length of the light, and
= 0,
friction,
the case
is
to the natural vibration of the substance.
a
the formula to In the case of a
usually named.
is
contemplated by Sellmeier, the formula
where X
whom
\ s that corresponding In the case in which
formula (5) becomes
m =1+S 2
n
Y y & 47r 2 c
_ "" 2
m,
2
-_
t 47rV w,2 (X - X/) 2
so that Sellmeier's formula can be regarded as the particular case when we put a = and n ~ 1.
of formula (5) which arises It is
still
an open question whether the best dispersion formula
100 the Lorentz formula (5), or a generalized Sellmeier formula It is however pretty clear that formula (10) is (see p. 176).
is
other things unsatisfactory, since, as the density of the medium,
vary
from the truth than
^f a
7iif
-
1 being equal, it makes m* and this is a great deal further
Both
p.
(5)
and (10) give the same
~i
when any one of the zeros X = Xj 2 X = X 2 etc., is crossed. The infinities of w are however in different Thus in a formula of type (10), viz. in the two cases. positions of behaviour of general kind
m
2
,
,
*-
in the neighbourhood
of, let
* 2
1
us say
X = X S we may put ,
A.
the rate of variation of the terms not involving \ s being comobserve that m* is negative from X = \ s
paratively negligible.
We
1
1
-
A \ ~-} approx.
Thus there
is
" X/<2s/
of wave-length S\ s = s /2q s \ s for case of a formula of type (5) we of an absorption band
A
m
2
which
may
m
is
imaginary.
a range
In the
write in the neighbourhood
A
1
or
where It is clear
crosses
from
m
from (11) that
\>\
'
8
to
2
- oo as \ changes from + oc to 2 then becomes continuously
X<X/. m
smaller numerically, but remains negative as reaches the value zero, which is determined
to
by
or
\--\-
Aa/(l
+ aq).
Thus
-)*
to
m is
imaginary from
\**\*-
sign until
it
Except in the immediate neighbourhood of an absorption band is practically the same whether we assume a dissipation term or not. Moreover the general the value of the refractive index
character of the behaviour in the neighbourhood of an absorption band, as outlined above, is the same in both cases. It is important to bear in mind that the quantitative results are not the same, and many of the foregoing formulae are therefore to be taken as
than representative. The same caution is to be observed in regard to the following treatment of residual rays, which neglects absorption. The more complete theory will be
illustrative rather
considered
later.
The Residual Rays. It
is
which, as If
now necessary to interpret the imaginary value of m we have seen, arises in a certain range of wave-lengths.
m is the refractive index of
wave propagated along the
27T t
A
If
m is imaginary,
a real quantity.
the material, the vectors in a plane
may be
axis of z
represented by
- mz) (ct
be equal to The expression now becomes let
us suppose
e
it to
2rr~z A e
-f
/3
where
/3 is
.27T ^ -7- ct
A
.
positive this represents a disturbance periodic in time which falls off indefinitely in amplitude as z increases. If ft is
If
/5
is
negative
it
increases
indefinitely.
require infinite energy to maintain
As
sideration.
/3
As the it,
it
varies from oo to zero
latter
case
would
may be left out of conand \ is small, /3/\ will
be large over most of the range, so that the obvious interpretation of our result is that light of the wave-lengths for which the value It is not is imaginary is incapable of entering the medium. of
m
is absorbed by the medium when it gets the case for example in most coloured liquids which show body colour. It is almost unable to enter the surface. In fact all the light of the wave-length under consideration is
merely that the light inside, as
is
from the surface of the substance, even when the For light of this particular range of wavenormal.
totally reflected
incidence
is
length the body behaves like a perfect reflector.
Many
of the aniline dyes exhibit
phenomena
of this character
In fact something similar arises whenin the visible spectrum. ever the medium possesses intense absorption so that it will be more convenient to consider the case of the aniline dyes when the ;
has been discussed. It is clear from theory of absorbing media the considerations which have been urged that it is not necessary to have absorption of the type which is accompanied by degradation of energy for the rays to be unable to traverse the
medium.
It is only necessary that the period of the light vibrations should of the medium. agree with one of the natural periods
The best examples of this type of phenomenon have been found in the behaviour, in the infra-red region of the spectrum, of a number of insulators which are quite transparent to light in the visible spectrum. rock-salt, sylvite
example,
is
and
The most conspicuous examples are quartz, If the reflecting power of quartz, for
fluorite. it
examined,
dence, for all
is
found to be small, for normal inci-
wave-lengths from the visible spectrum
up
to about
^ (I/* = 10- cm.). It then begins to increase rapidly as X increases, until at a wavelength in the neighbourhood of 81 ft quartz is almost as good a 4
7-6
reflector of radiation as
a metal.
This state of things continues
up to about 9 /A, when the reflecting power begins to diminish. The transparency varies in the opposite way to the reflecting power. In fact between 8*1 and 9/x quartz is so opaque that Nichols* was unable to detect any radiation through a layer of it 2'5 in thickness. The relation between the only wave-lengths reflecting power and the transparency of quartz is exhibited in the accompanying diagram (Fig.
25),
which represents the results of
Nichols's experiments.
Since the rays which correspond to the natural periods of substances are incapable of entering them, and so are always almost totally reflected, we are furnished with a new means of investigating the optical periods of substances. This
method, due
to
Rubens, consists in submitting a beam of radiation from sonic source, such as a Nernst glower, to a series of successive reflexions, at incidences as nearly normal as possible, from surfaces of the' material under The rays which are obtained after investigation. *
Ann. der Phys.
vol. LX. p.
401 (1897).
only of those which correspond to the natural periods of the substance. These rays have been named by Eubens, Reststrahlen or Residual Rays.
a sufficient number of reflexions
The wave-lengths
of the residual rays for a
are given in the following table
20-75
8-l_9,
Quartz (Si0 2 )
24-4
Fluorite (CaF 2 )
62,
Sylvite (KC1) Kock-salt (NaCl)
(PbMo0 4 (CaW0 4 ) (A1 2
70
45_48 5056 10-814 1 0-813-2 10-615 ?
Wulfenite
Corundum
number
)
3)
Potassium Bromide (KBr)
of substances
:
Wave-length of Eesidual Rays in 10~ 4 cm.
Substance
Scheelite
will consist
76,
87
Authority
[Nichols*
\Rubens and Nichols t Rubens and Nichols t Rubens and Hollnagel J Rubens and Hollnagel { Coblentz
Coblentz Coblentz
Rubens and Hollnagel {
We have seen that according to the theories of dispersion with which we are concerned, the refractive index between certain *
Loc.
cit.
Ann. der Phys. vol. LX. p. 418 (1897). I Mtz. Preuss. Akad. Wits. p. 26 (1910). f
We
shall conan imaginary quantity. of reflexion under these circumstances a
limits of wave-length is
sider the
case
phenomenon more fully, limiting ourselves of normal incidence. Referring
that
the electric intensities
in
refracted waves are respectively
X
little
if
2 sin
Y ~~
to the comparatively simple to
2
cos Q l
sin 9 2 cos
- cos Y ~
and
3
where
#1 si*1
I
+ sin Q
l
cos # 2
we
see
#2-- sin 0i CQS #2
cos #! sin # 2
+ sin Q
l
'
cos #2
and #2 are the angles of incidence and
^
132,
-^' '
'
2
p.
X% and X>, then
l9
<9
Chapter Vn,
the incident, reflected and
refraction,
and
In the the magnetic permeabilities /iq, /A 2 are taken as unity. all the are incidence case of normal angles vanishingly small, so that
we may put sin0 1 =
1
= msinfl
2
=:W^^2*
Hence
2 ~t
**!
.(12).
X
and
971
When reflected
m
X
=
3
+
I}
when
1
real the intensities of the light in the incident
is
and refracted waves are cX ?, cX/ and
nd we observe that me X} + oXf intensities
2
3
4m (m +
/!
'Let us consider the
incident light as is
real
m
and
than II
.
As
,
and
J
3
/m-l\
m
2
-^
change in the ratio of the reflected to the
travels over the range of the order unity, as in
tion in the laboratory, less
is
Il} J and 7 we have I,
m
mcX/ respectively,
required by the Calling the corresponding
cX^, as
principle of the conservation of energy.
When
=
I
3
is
of the
we
same order
increases to the value
are considering.
most cases of as
+
I but
oo
I
,
/.,//!
is
refrac-
always
becomes
m
When becomes zero, 73 is still equal to /,. equal to unity. consider what happens when has the imaginary values = iff which it takes between oo and zero. may put
m
Now
We
where
/3 is
real.
Then
m-I
l-i3
l-2-
m
Let us put this into the form p
+ i sin ff) = pe
iQ
then p* and 6 will measure the difference of phase of the two beams under comWe have parison. (cos
,
will represent the ratio of the corresponding intensities,
2
1
yS
Thus the intensity of the incident light for
between
+
oo
and
equal to that of the which intervene the imaginary values of It is clear that the intensity of the transreflected light
is
m
all 0.
mitted light must be zero throughout the same range of waveoo and length. This is at once evident for the limiting values H-
;
from the expression
4m
J2 The width
of these regions appears to
be quite considerable
and is different according to the type of dispersion formula used*. For a formula of the Sellmeier type, we have seen (p. 156) that m is
= imaginary from X X S to
X=A/X V
AJ2qs \ s rays range from \8 to \s Lorentz type of formula (p. 156),
m
v
A/
2
S
-.
Thus the
qs approximately. is
residual
Using the
imaginary from
==
1
Thus the width
of the residual rays
8X=
is
+ aq
approximately
9X
We
shall illustrate these results by considering the example furnished by rock-salt f. Paschent has shown that the refractive index of rock-salt can be represented over the range from '18/-6 to 22 p by the following dispersion formula of the Sellmeier type :
m* =
b*
+ ~~-~ -\f
'
\a
-V
X*-V
*
Havelock, Roy. Soc. Proc. A. vol. LXXXIV. p. 515 (1911). The numerical values are taken from the paper by Havelock J Ann. der Phys. vol. xxvi. p. 130 (1908). t
(loc. cit.).
= 5-680137, M, = -01278685, M, = -005343924, M = 12059-95,
where
6
2
V= X2
X 32
9
2
"0148500//,
,
= '02547414^, = 3600/t
2
2
.
[In the expressions above, and in the remaining formulae of which are given are such
this section, the values of the constants
that the values of the wave-lengths
= 1 x 10~4 ft
X
in
terms of the unit
cm. have to be substituted.]
This gives two natural periods in the extreme ultra-violet and In the immediate neighbour60/-&.
one in the infra-red at about X 3 =
m=
-
5*680142 + 12059'95/(X hood of the latter we find 3600). This gives for the lower limit of X which corresponds to an imaginary value of m, the wave-length 38*4^. Thus the residual rays from rock-salt should extend from 60//, to 38-4/4; since the 2
2
value 60//, corresponds to the natural vibrations of that substance, according to the dispersion formula given by Paschen.
A dispersion formula of the Lorentz type has been given by Maclaurin*, which shows an even better agreement with the This is experimental values of the dispersion from '48^ to 22/x.
m -! 2
K-I
G
l
Xs a = 5*51,
where
*r
V
a X2 -X 2
2
'
= 5*9,
= 00191605, X1== 012652^, = C 683-816, X = 51-3/*. 2
2
In the neighbourhood of \, #'=0-169652, and the upper X which gives an imaginary refractive index is
limit of
-
The lower
limit is V{V oCi/(l + aq')} = 01027^. Thus in this the residual region rays would cover a range of '0226^. In the infra-red the value of ?/ is 0*429449 and the limit is
upper
The lower
limit
is
In this case total reflexion should occur over a range of *
Roy. Soc. Proc. A.
vol.
LXXXI. p. 367 (1908).
The experimental results on the distribution of the energy among the different wave-lengths in the residual rays are not in accordance with the theoretical conclusions which we have just deduced.
There
no evidence of the existence of
is
finite stretches
of spectrum in which the reflecting power is equal or nearly equal to unity. In every case the curves show a very sharp rise to
a definite maximum, which is sometimes followed by a rather similar second maximum. The reflecting power at the maximum
There is no evidence of the generally comparable with 75%. existence of an extended region showing a reflecting power near to unity. It is probable, as we shall see, that this difference arises
is
fact that we have omitted to consider the resistance term in the equations of motion, which corresponds to absorption.
from the
Absorbing Media.
We
144) that absorption can be accounted for in a general way by the introduction of a fictitious retarding force If the proportional to the velocity of the vibrating particles.
have seen
(p.
component, parallel to the axis of which acts on the 5th particle is
x,
of the force of this type the equations of motion
ft s d)s
become (13),
with similar equations for the displacements parallel to the axes of y and z. We shall suppose as before that the impressed electric intensity
E
is
parallel to the axis of x.
We
shall leave out of
account the displacements along the y and z axes, since in an and z are equal to zero, the mean values, y isotropic medium, if
E
E
averaged in space, of y s zs and their time derivatives will also be Dropping the suffixes for the moment the equation of motion may be written ,
zero.
mD +0D + \}x = e (Ex Ay 2
f \
If
Ex^Ee** and
will
Px ^P^
k
the forced vibrations (see
be given by
x
m D* + D +
= \
e
Ay
(E + aP)
p.
147)"
is
Thus the displacement of the sth electron
............... (14).
The
polarization
Pe^
is
equal to
S
V8 e8 x89
where
v8
is
the
summation electrons of type s per unit volume, and the in extended over all the electrons in the molecule. Proceeding
number of is
exactly the
as before (p. 148),
same way
of the refractive index is
we
find that the square
now given by
H i
,
*
m'-l or
m
2 small compared with s (Pi?-p ) So that except in the except when p is very nearly equal to ps immediate neighbourhood of the natural frequencies of the
In
all
the important cases ft^)
is
.
substance
we
shall
have
-
-
1
-
-
r+ a (m - 1 ) 2
-
N
=2 !
r-V- ^ m (p? p) 2
very J nearly. y
a
we obtained previously on the assumption that there was no damping. Thus the introduction of a dissipative force has practically no influence on the phenomena This
is
the same formula as
save in the immediate neighbourhood of an absorption band.
In tracing the phenomena in the immediate neighbourhood of the natural frequencies, it is desirable to effect a simplification, otherwise the formulae
become
too
cumbersome
to
manage.
We
suppose that in the immediate neighbourhood of p^ we can treat the contribution to the right-hand side of (15), which arises from electrons other than the 5th one, as a constant quantity q independent of the frequency p. This will be legitimate if p varies over only a small It would range in the neighbourhood of p 8 lead to erroneous results if it were probably to a very applied broad absorption band. This difficulty, however, might be avoided if q were replaced by a few terms of an expansion in of
shall
.
powers
AtfSU.KFTJ.UJN
AJN
S ttJL UlU 1 1 V JU
L>
.tUi;*
1OO
.LttAUJJN
normal dispersion caused by
p, sufficient to take account of the
the more distant natural frequencies.
With hood of
p=p
m
putting unnecessary suffixes ,
for
and
is
m =
shall
have in the neighbour-
the refractive index and dropping
m
**
2
!4-a(m -l)' V Thus
we
this restriction then,
-
'
2
m = n (1
Let us put
complex.
Both n and K are
The
real.
IK),
so that
vectors which specify a plane
wave
.2rr
x propagated along the axis of z will be proportional to e where \ is the wave-length measured in vacuo, c is the velocity of and %7rc[\ = p. Thus the vectors will be light in the free aether,
proportional to e
As
-2^z ^
in the previous cases in
e
^(ct-nz] A
which
it
therefore equal to
nx,
has occurred, the real exextinction coefficient
The
a decay factor. ponential represents is
/i HA (17).
and the medium exhibits absorption.
shows that the phenomena are periodic in a distance equal to \/n so that the medium behaves otherwise as to n. It is clearly of conthough its refractive index were equal siderable importance to deduce the values of n and ntc from
The second
factor
y
equation (16).
After rationalizing the denominator in the expression for
and then equating
real
and imaginary parts we
n2 (l-K 2 )
=
4+-^^
(18),
3
.-^ where
A
1
-f-
-**
-&
:.(..).
ve 1
D
aq ,
l
7=
q
-^
(1
- a<"
(20). j-
'
mf
find
100
2
Thus
2
= - / A: + 1
r-^-r--
-
+^+
Feeble Absorption.
There
is
one particular case, which
of importance in nature, of these formulae may be
is
in which a considerable simplification of substances exhibit absorption in effected. large number their volume, and are said to possess throughout
A
varying degree This colour.
body
minerals,
and
is
in fact
the case with most coloured solutions and with most coloured substances which are
of the visible spectrum. transparent to a considerable proportion a distance comparable in In these cases the absorption is small
with the wave-length of light, although it a distance comparable with 1 centimetre.
may be
Consider the behaviour of waves for which as to
make 7 =
We
0.
*
=
it
has a value such
p
see from p. 165 that the absorption causes
the intensity of the light to z
considerable in
fall off'
as e
A
In a distance
.
llK to unity. will therefore diminish in the proportion of e~
Since the absorption in distances of this order is very small it so that n/c follows that e~ nic must be very nearly equal to unity must be a very small quantity. Turning to formula (22) we see ;
that
nrc will
quantity.
and
when 7 = 0, provided
only be small
It follows that 8 is a large quantity
that
/$ is a small
compared with
J3,
compared with 7 within a reasonable distance of the value of p corresponding to 7 = 0. In this region we may use, as an approximation, the formula also that it is large
(23).
At the frequency ~- (UK)
and therefore
op
which the absorption
for
(
dp vy-
i
1
- OLO
_
+ b"J
]
vanishes.
_
is
Hence
a
maximum
QHiJU.DU.LJL V
By
putting p=p<> on the right-hand side as an approximation
maximum
see that the absorption
*
m*
ml-aq
Near the natural and put -4 = 1, a =
W
271*
If
(24
b),
we we
^^
2
_l^
m1
awP__
aq
frequencies of a gaseous substance we can = 0. In this case it may be shown* that q
n and UK are
satisfactory approximations for
and
we
lies close to
=1+ =
................. < 24
2WF)
^
7?
-^
........................... (246).
neglect the variation of 8 compared with that of see that the maximum absorption is given by 7
=
p=p
7 in or,
These relations have to this order of approximation, by been utilized in interpreting certain spectroscopic and magnetooptical experiments.
(Of.
.
Chap. XX.)
The Residual Rays from Absorbing Media. If
we
are to discuss the
phenomena which
characterize the
evidently necessary to consider the behaviour of n and K, when S/S is not small. Solving for tc the formulae for
residual rays
we
it is
find
m and, treating S as constant
a,
- - B/2A and when 7 = oo Treating we find that the maximum and minimum
This vanishes when 7 2
n and
2
ri
*
2
similarly
2 values of both n and
nV are 4-
*
Cf.
.
SBrf
roots of the cubic equation
-
AS 7 2
4*
S 2 = 0.
H. A. Lorentz, Theory of Electrons, pp. 154, 310.
Let us now turn our attention to the intensity of the light reflected
at
medium under
the
of
surface
the
consideration.
to the case of normal incidence and following Confining ourselves the method of Chapter vn, p. 138, we see that the ratio of the reflected to the incident electric intensity is
X Tor
all
W here
,
1-fe
l
the reflecting media that
and since ax
=
c/V/vei and a 2
e
,
e
-Z* -w-
,
so that
A
=
= we have
= c/V//,
>
^ = /^ =
1,
we have
= A/ - = m,
m 1r = m 4- 1
=
l
2 tf 2
to deal with
n n
i#n
1 =-
;
.
&# ft -f 1
ie Putting this into the form pe we get
n2
.
p
c^
2
so that
p
.
= {(tt
+ I) 2 -f O..9.
and
tan 6
As p
is
The
before, 6
.
2
ft
2
*2
}
.
=ft
2
1
2?z
,
sm = +
,
and
n2 * 2 -
-f-
+7l2 /C2
~l
measures the change of phase on reflexion, and
the ratio of the intensity of the reflected and incident rays. expressions which result on substituting from equations (21)
and (22) are very complicated, and there does not appear to be any suitable approximation of a general character. The value of 2 p can only be obtained satisfactorily by numerical computation after the constants in the formulae have been determined. The 2 character of the graph of p as a function of p will be discussed after the theory of dispersion has been considered from a rather
more general standpoint. a
number
of the
In the same place we shall also review preceding results for absorbing media, using a
rather simpler dispersion formula.
(See
p.
178.)
Generalized Theory of Dispersion.
In order properly to realize the limitations and approximations to which the foregoing theory of dispersion is subject, it is desirable From to consider it briefly from a somewhat wider standpoint.
what has been
clear* that dispersion is essentially a dynamical problem in which the machinery is determined by the fundamental structure of the atom. Unfortunately we know very said it
is
little about this structure, so that the necessary data cannot be stated very explicitly. Fortunately the methods of generalized dynamics enable us to find out a good deal about the behaviour of
such a system even when we do not know
much about
its
exact
constitution.
In order to determine the motion it is necessary that certain functions of the state of the system should be known or, at least, be capable of definite specification. These are the Kinetic Energy, the Potential Energy, the Dissipation Function and the function which is equal to the work of the external forces. Let us consider these briefly in order.
Regarded as a dynamical system the optical medium consists of a system of electrons which may be treated as point charges
when undisturbed, to unknown conditions of equilibrium. The equilibrium is not necessarily a static one but may involve
subject,
motion in
orbits.
In any event the expression for the kinetic energy is quite simple, since it is equal to the sum of the energies of the individual electrons. If there are n electrons in any sufficiently large element of volume of the this
element
medium
the kinetic energy
BT which belongs
to
is
(28),
where
?^,s
.
is
the mass and
&' iS.,
ys and
z8
denote the components of
the velocity of the 5th electron.
To obtain an expression complicated.
for the potential
energy
is
much more
If the equilibrium state involves steady motion, the
This is only true provided dynamics is adequate completely to account for the behaviour of atoms in this respect. This point is now doubtful, but, at least, it is of interest to
examine the
results to
which generalized dynamics
leads.
will
potential energy
involve the velocity as well as the space
This makes the analysis considerably more
coordinates.
com-
intended to be purely As the present discussion plicated. to the extent of supposing illustrative we shall forgo generality to be a static one or at least one the undisturbed is
equilibrium
which can be treated as though it were static on the average. The effect of orbital motions has been considered to some extent
by Larmor*.
Under
for
expression
(suffix s) will
the most
conditions
these
important
term in the
energy of
any particular electron form of the be undoubtedly the potential
a <>/
+ yf + s
3 ),
y s and z8 are the components of the from the equilibrium position. This displacement of the electron term represents the work done by the force of restitution conIf all the electrons always templated by the former theory. moved similarly and preserved similar geometrical relations in all
where a
a constant and
is
as89
the motions contemplated, the whole of the potential energy could term of this kind. probably be represented by a single constitutive
But even when forced
oscillations alone are considered,
the relative
displacement of the different kinds of electrons is affected by the frequency of the vibrations, so that it is necessary to consider the influence electron
of the lies
in
separate electrons on each other. the field of force of the doublets
Now
each
which are
equivalent to the displacements of the other electrons, so that the
complete expression for the potential energy of any particular electron will contain terms depending on the displacements of all the other electrons which
lie
within some considerable distance of
Consider the part of the potential energy of the 5th electron which arises from the displacement of some other specified electron it.
which we
may
denote by the
suffix a.
the undisplaced position of e s be r'
When
e g is
displaced an
*
Let the
line joining e a to
where
r,
=
amount
oc 8
,
r will
become
Phil. Trans. A. vol. cxc. p. 236 (1897).
X,
The
AJtJSU-KFTlUJN
potential energy of
doublet arising from
ea
es
the
AJNJD
SELECTIVE KEFLEX1ON
in its undisplaced position, moment of which is ea xa is
1/1
due to the
,
a
The corresponding quantity
There
for the displaced position of e8 is
be a term in the potential energy of egj due which is of the form e 8 ea a!s^af(^ #) where /(r,
will therefore
any other
electron,
to 8)
depends upon the undisplaced positions of the electrons in space. Similar remarks apply to the y and z components of the displacements of the electrons, so that the complete expression for the potential energy of the element of volume may be written r=ti
S
W= 2
s=n
2 [A rs xr xs + Br8 yr ys + Grs z.r zK
r=ls=l
TS
where the
yr
...
28
H
of this expression for the potential energy calls for The cross coefficients rs etc. are
A
fuller consideration.
little
.(29),
.
The nature a
. .
involve the fundamental A, B, (7, F, Gr, medium, but are independent of the displacements
coefficients
structure of the #?r ,
y r zs ]
proportional to the inverse cube of the mutual distance of the electrons involved, so that they are small except for pairs of
which are quite near one another. On the other hand of electrons at a distance between r and r + dr varies 2 as r dr but, on account of the periodic character of the phenomena, the distant electrons are in layers which exert opposite and electrons
number
the
:
approximately equal effects. Thus the contribution to the potential energy of a particular electron, which arises from the cross terms, will
come almost
entirely from other electrons in its immediate in other words the whole of this potential energy
neighbourhood may be considered to ;
legitimate to express if
this
is
taken
it
arise
as a
fairly
from local causes.
summation
large.
It
is
therefore
over the element of volume
Unfortunately the
size
of
the
appropriate element will depend to some extent on the period of the vibrations, so this process can only be regarded as an approxi-
mation after
all.
The summation in (29) can be split into two parts, one involvingelectrons in the same molecule and the other representing the influence of the electrons in the external molecules.
represented by the term
The
effect of
aP
in equations the last named part (1). will with This part of the summation clearly vary changes in the The effect of the part of the summation density of the medium. on the other electrons in the same molecule is not which is
depends
considered in equations (1). It is not likely to be influenced if at all, by changes in the density of the medium.
The
discussion on p.
143 shows that we do not know
much,
much about
the nature of the forces leading to dissipation of energy in systems It is therefore desirable to make some fairly of this kind.
general assumption about it and we shall suppose that the dissipation function SF is a quadratic function of the velocities of the electrons. The consequences of such a supposition have not, as yet, been
shown to be incompatible with the results of experiments.
An
expression for the work function SJ7 may be found by considering the energy of a dielectric medium in which an electric field resides.
volume
If the electric intensity
\$%DE = %E(E + P).
is
The term
E
the energy per unit %E can be interpreted 2
energy per unit volume of the space occupied by the dielectric, leaving %EP as the work done by the field on the as
the
electrons.
The work function
2 If
we
er
therefore
is
{Ex
r
+ Ey yr + E zr z
consider only plane polarized waves in is to the axis of cc we can put parallel
}
............
which the
(30).
electric
intensity
(31).
The equations of motion become greatly simplified when the and F (dropping the S's) become sums of squares. is well known that by means of a linear transformation of
functions T, It
W
W
F
coordinates any two of the functions and can be transT, formed into sums of of the new variables each squares
by an appropriate simultaneously for
multiplied
But we are unable to do thi.s three even when two of them are already
coefficient. all
sums of squares, as is the case now with T and possibly also F. For instance suppose we replace the 3n variables xl9 y^ ... zn by 3n new variables, ql} q.2 ... qsn which are linear functions of x y^ ... zn l
.
,
Let
xr = ar q 1
where the
2
(3?i)
l
4- a,?q.2 4-
coefficients Oj 1
a rs q s +
...
cw8
'1
. .
.
+
371
a,.
are constant quantities.
The
condition that this transformation should reduce the expression for
T
to a
the
9>i
2
sum
of squares
is
.
quantities a^, a?
,
etc.,
new
which
T
arise
expression for
In a similar manner
~--
will lead to
-
2*
the cross terms in the should be satisfied.
^-
that the
-
,
equations each.
equations between
when the
coefficients of
T
are equated to zero, and the functions
W
As soon
as
F
we have more T
2t
than two functions to reduce, the number of equations exceeds the number of variables, and the transformations will only be reducible if a number of the equations happen, on account of relations In the inherent in the original coefficients, to become identical. present case where
we have
three
functions,
9n 2
9n
-^-
^-
of
the
equations would have to be alike. There is no reason to expect that this condition can be accurately satisfied with the kind of systems under discussion, so that we really ought to consider the general case in which 7 functions.
We
shall
1 ,
W and F are any homogeneous quadratic
do this very briefly later
Nevertheless there
is
(p.
177).
one case of simultaneous reduction which
well worth considering, that in which the kinetic and potential are converted into sums of squares by an appropnuto energies linear transformation and is assumed to be to a sum of
is
F
equal
There ;ire two squares of the velocity coordinates so obtained. reasons why this treatment maybe considered plausible. In the first place we are quite in the dark about the real nature of F, und 1
,
assuming it to be expressible as a sum of squares of the coordinates which enter into the normal forms of T and we are the about it. making really simplest possible assumption Secondly there are a large number of cases in which the dissipation is small, in
W
in these cases the
and close
approximation
will probably give a assumption outlined
to the truth.
variables g>,, q t ...q* which satisfy equations Introducing the becomes like (32) the kinetic energy
T = 'T0,q,* .............. ........ -(33), =
where
r=l
that for all values of
provided I
m between 1 and 3n, excepting
and
Z
= m, r=l
There are
(3n- 1) such
2
Turning
to
relations
between the 9n*
coefficients.
W we see that since
and
etc.,
we
shall
have
7W/... F=|1 *s=l
(34).,
where ^ 7i =
I 2
{Arsarta
provided that for
all
values of
^
and
u,
except
t
=
,
between
1
and
3?i,
1 I {4 r, (a/ a/ + of of)
-4-
5
tt rir
(6 r '6,
+ bfbf) + Crs (cfc +
cfcr")
r=l s=l
+ Fr8 (or b u + afb ru ) + Grs (c^r t
8
There
We
will
also
t
cs
u
+ as*c r
+ Hn (6 r*C +
6/c r
w )}
= 0.
1) of these equations. again be ~-(3?i^
have n
U=
2 r=l
where
)
s-^u JSTe r av
= 2 s=l
Ss
^
(35),
^= o
and we assume
^
where the
s
s=l
*>
^
are constant coefficients.
e's
of motion are given for each q
The equations
by the extended
Lagrange's equation*
dt\dqs j
They
if
dqs
are therefore /O
and
dqs
dqs
dq s
X varies as e
ipt
""
/O Q\
*
S\
i
i
the forced vibrations are given by B.
The natural
vibrations
be obtained when the external
will
X
Those corresponding to the electric intensity is equal to zero* be determined will therefore by the equation displacement qs
&? + VsQs + e They
be proportional to
will
Evidently
TTS is
e
complex and
i7rst
s
=
qs
(40).
where
if ive
put
TTH is
irs
a root of
= p + iks s p s
,
will
the frequency of the corresponding principal period and k8 decay factor. Since
z
we have
>
/3 S
k^)
(p s
p/ =
Hence and
-^
6 g /^
-I-
7^
= 0,
j~^
(41),
'
ft
=
ry
*
r
Lord Bayleigh, Theory of Sound,
s
vol.
i.
chap.
v.
be its
in wie ^^^ of volume. in any sufficiently large element electrons n the over all the volume unit in polarization of such elements If v is the number
The summations
will
be
=vS
e,
S
n
= fCo
where
Thus
a
for
medium
of unit
2*
the complex magnetic permeability
m is given by
refractive index
m = 1+ 2
-
where n ^_
~- 2
It
is
m r (a/
V
-f
evident that the refractive index
must be independent of
The
n and v except in the combination nv. particular values of n and v are arbitrary 'except that n has to be a sufficiently large number. The product nv is equal to the number of electrons in unit volume of the substance and is therefore a characteristic
The
constant.
requisite independence
is
secured by the fact that
on large the constants which enter into (44) keep summation s. Thus values of the for different themselves repeating in (44) is really a summation over the different principal modes of
when
n
is
mode is multiplied by the number of times The total number of terms, coincident equal to three times the number of electrons
vibration in which each it
or
occurs in unit volume. is
otherwise,
present in unit volume.
Formula (44) is of the Sellmeier type except for the inclusion of the dissipation term. With energy functions of the type now under consideration, the relation between the refractive index and the density of the substance is not an obvious one, since the constants A i/r g 2 and 8 will involve the density in virtue of the relations on pp. 173 175. A formula of the Lorentz type would, however, arise if we assume that the only part of the force on an electron which depends on the density of the medium is = aP, where a is a constant and P is the polarization. Formula (44) then becomes t9 ,
<j>
~
2
1
-a
'
w -1 a
(
)
,
-1
V/
2
where the constants are now somewhat portional to the density of the substance independent of it.
A s/
different.
and
\fr 8
and
is proare nearly
(j> 8
W
and F are not In the general case in which the functions T, of the values of x8 etc. sums reducible to squares simultaneously Sn simultaneous
are the solutions of
written
down
linear equations
in the form of determinants.
and can be
Consequently these
determinants enter into the expression for the refractive index and
make it difficult to handle except by approximate methods. In general the symmetrical coefficients which lie along the axes of the determinants are large compared with the remaining imsyminetrical coefficients
as a series of
magnitude.
sums
;
so that the determinants can be
expanded which decrease progressively in can be shown that the Lorentz and
of products
In this way
it
Sellmeier types of formulae result in virtue of approximations which are equivalent to the physical assumptions which have already been
We
shall
made
in
deducing them.
now return
to the behaviour in the
neighbourhood of
an absorption band and the residual rays, using the simpler formula (44) instead of (15). .,
m
19
near the Critical Frequencies. Absorption and Reflexion natural periods, In the neighborhood of one of the close a as approximation, sufficiently we shall have,
= Putting 7?i n on as find, p. 165,
(1
-
IK)
where n and
are real
/c
-n
and
so,yp=p89
positive
we
N
(48))
A=q
where
t
,
5=A
g
g
(50)
When
the absorption
wave-length we
is
small in a distance
get, as before, to a first
compared with one
approximation
Substituting the values of A, B, y and 8
we
find
^-
vanishes
if
we may put ty s2 =p, as a sufficient approximavalue of p for which the absorption is a maximum
In the fraction tion. is
Thus the
given by
Thus the corresponding true natural frequency is the frequency which the absorption is a maximum. It is somewhat less than the constant tyf which enters into the dispersion formula. for
We
shall
now turn
radiation reflected from
to the
problem of the intensity of the a surface of the substance under con-
,
ASSUMPTION AND SELECTIVE REFLEXION
sideration at normal incidence.
2 ?i
Substituting the values of
/c
2
?i
it is
_n
this
by given by
Denoting
already brought forward show that
~
(1
and
+
-?i/c
A:
2
)
-
/r,
IV
hJ
considerations
2?i 4- 1
found previously, we
get
2
\^-iY
_
l
1
f
/^7& + 2ABy
.
By
,
where
We menon
have seen that
for
substances which exhibit the pheno-
body colour U-KT is small compared with unity, and for it is necessary this to be the case for the particular value 7 = that 8 should be large compared with B. There is no guarantee of
that this will be the case with substances which give rise to the residual rays, since Nichols has shown that in the case of quartz the amount of the residual rays which are transmitted through
a slab of the substance only two to three wave-lengths thick is The value of the extinction incapable of experimental detection. coefficient, ntc, for
such substances,
unity or greater,
and
may
therefore be of the order
this corresponds to a value
comparable with that of
It does not
of
B
at least
seem
likely that there is any approximation of general application in the case of the residual rays which leads to any very marked simplification of
the formulae.
It is therefore necessary to evaluate the formulae
troublesome process. The involved are obtainable from the correspond-
in each particular case
constants
A
S.
and
B
and
this is a
ing constants in the usual Sellmeier dispersion formulae and so 2 2 also is (=7-f_p ). Determinations of 8 from the experimental
^
results
do not seem to have been carried out as
yet,
but the value
the most important factor in determining the proportion of the incident energy reflected. of 8
is
maximum
\
180
AND
DISPERSION, ABSORPTION
The of 7 or
of the curves precise nature
X depends upon
which express p 2 as a function
the values of the various
constants.
Nevertheless they always possess certain common features the example in the accompanying figure. are exhibited
by
which
The
ordinates represent the percentage of the incident energy which 2 and the abscissae are the values of 100 p is reflected, i.e.
they
6-5
;
7
8
7-5
95
8-5
10-5
4 Wave-lengths in 10~ cm.
Fig. 26.
wave-lengths (X = 2?rc/p) of the incident radiation. Alt the constants except s have the same value in each of the graphs = 2'05, A,. = 2-563 x 10~H 1, 2 and 3. The common constants are:
are the
^
and f/=4'53xl0 28 and in graph 3,
.
^=
indicates
the value
In graph 1, 1-42 x 10 13
.
X. of
,=0: in
The
X which
graph
2,
^=l'42xlO
vertical line at
corresnonds tn
la :
X=8'855
t,l^
^Mfi^i
AND SELECTIVE REFLEXION
DISPERSION, ABSORPTION
frequency tys
The points marked thus
.
mental measurements of 100 p 2
:
x, are Nichols's experi-
we
If
for quartz.
call
\ s the
=
^
and to the value 7 0, \ s will wave-length corresponding to be equal to the constant wave-length whose square enters into the denominator in the usual Sellmeier dispersion formula. It is usually assumed that \ s is identical with, or, at any rate, very close to the wave-length for which the energy is a maximum in the residual rays.
assumption
may
be
It is evident far
from being
from the figure that this justified.
Starting with values of X which are less than \ s (7 negative) a small value which gradually decreases to a very small has p* minimum. From this it rises very sharply to a maximum beyond it again diminishes, but more slowly than it rose. Thus the curves are far from being symmetrical about the position for which p 2 is a maximum. The maximum value corresponds to a value of \ which is distinctly less than \ s The positions of the
which
.
maximum and minimum may
of course be obtained
2 tiating the expression for p with respect to which results is of too high an order to be of
7,
by
differen-
but the equation
much practical use. minimum is very
It happens, however, that the position of the easily obtained
with sufficient approximation, since
we multiply the maining
it is
practically
minimum
If value of the numerator in p\ 2 2 top and bottom by 7 -f S the numerator re-
coincident with the
is
ID
The minimum value
of this
is
at 7
=
-
spending
minimum
2 provided S
is
value
of
p-
is
-.
jA.
The
corre-
JL
given very approximately,
rather small, by
Thus the position of this minimum and the corresponding value of p 2 should give an important check on the constants in the dispersion formulae.
CHAPTER IX THE FUNDAMENTAL EQUATIONS the electron theory may be equations of from the results of abstraction or regarded as a generalization assumes that matter is The electron theory Chaps, in, v and vi. but a distribution of electrified elements of volume in
THE fundamental
nothing space.
There are thus no magnetic charges in the sense in which
there are ultimate electric charges or electrons. fields which occur in nature arise entirely from the electrons.
The magnetic
motion of the The simplest assumption which we can make as to the
nature of the universal equations of the field is that they are identical with those which we have derived for the free aether containing electric charges.
an assumption, as it What we can be sure of
is
is
is
It is important to realize that this sometimes regarded as self-evident. that the fundamental equations must
degenerate into those for the free aether at points not in the immediate neighbourhood of material particles; but this is a very different thing from being sure that they are valid in the an atom or an electron. The assumption of their
interior of
universality
is
a hypothesis which will only be justified
conclusions to which
it
leads are in
if
the
agreement with deductions
from experiments.
We
therefore assume for the universal equations
p 0....,
"**--;
:
(1), (2),
W
THE FUNDAMENTAL EQUATIONS where the mechanical is
V
relative to the
force
on an
charge whose velocity
electric
measuring system
per unit charge,
is,
(5).
c
show that these equations, which are assoof Lorentz, are not inconsistent with any of our previous results. Looked at superficially they do appear to be inconsistent since, by simply writing average values in equations (1) to (5) we do not arrive at equations which are obviously identical with those which we found to comprise the behaviour of dielectric and magnetic media in Chaps, in, v and vi. It is necessary to
name
ciated with the
;
to be
remembered, however, that the vectors defined as the and magnetic intensities and inductions respectively, in those chapters, were all average values of the true electric and magnetic intensities but formed in different ways. When this difference is taken into account the discrepancy will be seen to It
is
electric
We
disappear.
shall
now
consider the equations in order from this
point of view.
Equation (1) is supposed to apply to any element of volume however small. The corresponding equation div D = p is an equation between average values, and only applies to an element In of volume which contains a very large number of electrons. order to compare large volume.
them
We
let
us integrate (1) over any sufficiently
have
1
7
pdr r where
=
dE,, dE,} -^ + JL+=l.
[f[\dE., \
.
7
7
dxdydz J
E
=
E
at any point of n represents the normal component of the surface. But we have seen that the induction n is the
D
D
n average value of the force in the flat cavity perpendicular to so that the normal induction is nothing else than the average value of the intensity taken over a surface perpendicular to it. ,
E
only strictly true provided the surface is so the that of polarization charges of a given sign excess large In other words, of the doublets whose inside of it is negligible.
This identification
is
axes are cut in two by the surface the difference between the number which leave their positive and those which leave their
negative ends inside must be negligible.
On
the other hand,
if
THE to
is not big enough satisfy indefinite. becomes induction of the
the surface
this condition, the
meaning
E=p
In any event we saw at the end of Chap, ill, that div that p represents the total average density of the always, provided arises from conduction or polarization electrons it charge whether = p is only The apparently inconsistent equation div or both. of the volume electrification the true provided the part of density which arises from the polarization electrons is left out of account.
D
= 0,
viz. div 5 Equation (15) of Chap, v, with div average values, and is consistent
is
H=
an equation over
for precisely the the same reasons as those which establish consistency of (1) and In this case we do not need to consider the possibility div D = p.
of an excess of magnetic poles of a given sign being situated For as the elementary magnets consist of inside the surface.
charges in motion,
electric
such a
way
it is
impossible to cut
them in two in The detailed
as to separate the equivalent charges.
formulation of the magnetic properties of bodies from this point of view will be left to a later chapter.
The equations obtained in Chapter VI also refer to average The equation which is equivalues of the dependent variables. valent to (3) is
In order to show that these equations are consistent to consider their geometrical interpretation.
analytical expression of the
fact that the
Each
it is
of
necessary
them
is
an
of the component, parallel to the contour, of the vector on the left, round any contour, is equal to the integral of the normal component of the vector on the right over any surface bounded the same contour.
Thus the
E
line integral
by
of (3 a)
is
the average value of the tan-
component of the
electric intensity taken round the to the of evidently equal It_is average value of the when derived from (3) because as in Chapter iv is equal to the average value of in a filamentous have
gential contour.
E
E
V
E
We
cavity. already seen that the average value of Sn over any surface is equal to the n over the same surface. average value of The equations (3)
H
and
(3 a) are therefore consistent with one another.
A<J>
THE FUNDAMENTAL EQUATIONS
of Quite similar considerations apply to (4) and equation (17) write we which may Chap. VI, .................. (4a) -
H
is
round a contour The average of the tangential component of the over corresponding evidently H, and the average of En
surface
There
is is
clearly
Dn
,
so that these equations are also consistent. In is worth remarking in this case.
one point which
surface bounded interpreting (4 a) it is desirable so to choose the cross it during the of electrons the none that contour the by
under consideration. Otherwise there is a contribution pV term owing to the motion of the electron. (4 a) then
interval to the
becomes inconsistent with contour.
This difficulty
the surface integral surfaces terminated by the
itself since
not have the same value over
may
all
does
same
be overcome either by choosing the
surface so that the polarization electrons do not cross it, or by taking the element of time large enough to include the average
value of effects arising from such translation.
This
is
the motion of an electron across the bounded surface to the creation of a separate
zero because is
equivalent charge in the unlike charge in the old
doublet with
new
its like
position of the electron, and its The creation of this doublet introduces a local term position.
which just wipes out the effect of the motion of the charge across the boundary. These remarks are pertinent to equation (4 a) only. Equation (4) is always consistent with itself, and is consistent with (4) when the latter is self-consistent. in the force
In comparing equation (5) with the corresponding equation
the agreement of the first term on the right is clear enough, hut the second requires fuller consideration. Here we have, to deal with the average value of taken a line to which // is
H
normal.
It
is
difficult to see
along
how
may be done directly, but, an indirect method may be employed. Considering the ease where (at least so far as average values are concerned) lei, this
E=Q
us apply the universal equation
F=-[V.H]
to find
i,he
force
material
the resultant force on the circuit
for
round the circuit p.
If the strength of the current is i
medium.
83
;
it
s.
\\Hn dS
is,
it
follows that (5)
and
that the average value of direction of
But, as
H
.
H]
taken
B.
is jjuH
En = ~\ En ds = K,E = D, s J
same
contour.
normal component of the
we have
seen,
(5 a) are consistent and, incidentally, taken along a line normal to the
H
From
this the
and magnetic vectors would
electric
[ids
similar to that in Chap, v, equal to i/c times the rate of
of course, the
universal magnetic intensity.
Thus
I
(5) gives
./
over any surface having the
Hn in this integral
The
o
t
By an argument
follows that the force is
change of
the value -
lead
analogy between the us to expect that
the suffix n denoting that the vector
is
perpendicular to the direction of integration.
By
dividing the space up by means of tubes of induction, it that the average values of the universal expressions %E* 2 for the electric and magnetic energy densities respec-
is clear
and
^H
2 2 This is only true provided \icE and |yu,J? neglect constant terms which may be regarded as representing the intrinsic energy of the electrons and of the molecular magnets.
tively are equal to
.
we
The Differential Equations
satisfied by the Vectors when Charges are present.
In Chapter
vn we
were concerned with the solution of equaand the extensions of them, which have just been considered, in the cases in which the density p of the charges was
tions (1) to (4),
zero. The results thus obtained naturally applied to the propagation of electromagnetic effects in insulators, including the free aether as a shall now consider the particular case.
everywhere
We
nature of the solutions in the more general case, when electric charges are present and contribute to the resulting phenomena by their motions and the forces they exert.
THE FUNDAMENTAL EQUATIONS The new satisfied
equations,
E
by
analogous to
is
true
A
are
A
if
if
we
first
K
1
-^
,
rot
dA z
dA y
dy
dz
dA x
dA z
dz
dec
easily
than
A,
The components
any vector point function.
is
which are
prove the general theorem
V*A = rot
A
grad div
of rot
^2
V 2 E=
and H, may be obtained rather more
would otherwise be the case
which
0.01
'
'
dAy
JdA a
dx
So that the x component, rotation of
A dy
example, of the rotation of the
for
is
_
dy \dx
dy
dz\ dz
da \dy
dx
dA y --r ~^
dA z \
fdA x
d = ^~~
"^
ox \tix
-r -^
dz J
oy
dz
X (&A ~-~_\
-}
ox-
Since a similar result follows for the other two components
grad div
A - V 2A = rot
rot
A
we have
............... ((>).
In order to obtain the differential equation satisfied by t and obtain
E we
differentiate equation (4) with respect to 2
#
^+
8'
Substituting the value of
a
T~ (pF) ,
,a//-. = crot-
H from (3) we get =~
whence, from (6) and
g
'
C
J
rot
(1),
In a similar manner, starting with (3)
we
find
(H).
Each of these equations is a vector equation and is equivalent each of the three components. to three separate equations between V are x ,Ey ,EZ) x y ,Hz and of E, Thus if the components x the Vz respectively, components are given by the fa, V
H H
E
H
,
,
,
Cartesian equations
,_
VH
,
'
and
I
_
--^~dF~~~c\W
......
~**
I
There are four other similar equations for the other components.
The nature
of the solutions of equations (9)
and (10) may be
discovered by considering the equation
**-where
o> is
which we
a function of
x, y, z
and
t.
.....................
In the
electrical
<>
problems
a given function of these variables. The solutions of (11) have a certain degree of resemblance to the potential in the theory of attractions. The potential
V
satisfies
shall
have to consider
the equation
o>
is
V 3 F=p, where
p
is
attracting matter, measured in suitable units. the integral of this equation is
the density of the
As
is
well
known
Thus the potential at any point P is obtained if we take the element, pdr, of mass at any point, divide by ^irr where r is the distance from P, and integrate throughout space. We shall see that a precisely similar result holds for the functions ty which are the solutions of (11). The only difference lies in the fact that in we replace p in (12), not by the calculating the values of
^
instantaneous value of
the function on the right-hand side of (11), but by the value which this function had at the point of integration at an instant r/c previously, where r is the distance co,
from the point at which If in (11)
we introduce
equation becomes
E a
or
H
is
to
be calculated.
new independent
variable u
ict
the
THE FUNDAMENTAL EQUATIONS
The
left-hand side would be the value of
V
2
in rectangular i/r coordinates in a four-dimensional space so that the problem of finding the solutions of (11) can be looked upon as the problem of ;
finding the potential in a four-space.
(11) is
an example of a
number of electromagnetic equations whose symmetry is improved when the time t is replaced by the imaginary variable u = ict.
Kirclihoff 's Solution.
in
A very complete discussion of the solution of (11) was given 1883 by Kirchhoff* in connection with the theory of the pro-
pagation of
light.
As a preliminary
an auxiliary function % which
to solving (11) let us introduce
satisfies
the equation
This is the equation to which (11) reduces when the righthand side is put equal to zero and \jr is a function only of t and the distance r from a fixed point. If we put ^> = r^ (13) becomes 5
The most general
where
F
is
solution of this equation is (see p. 117)
any function whatever.
The two terms correspond
physically to disturbances velocity
c.
x=-F(t + -)
giving
where
We
propagated in opposite directions with shall only consider one of them and take
F is
Next
(14),
a perfectly arbitrary function.
consider the integral '
y^"^)
C&T
(15)
taken throughout a closed volume limited by an internal surface <j and an external surface 8. By Green's Theorem
Fig. 27.
the normals being directed into the enclosed volume.
have from
(15), (11)
We
also
and (13) 1
From
(16)
and (17)
m*s '
This
is
true for
all
to dt between limits r^a
r r /
values of
^ and
3^ on
3^\ _
j
dy _/jk 3t
it
with respect
We
tz .
y ^ onj
r
Let us integrate then get
t.
dn
^g.
A/ /C
3^\
f>
-
:
.(18).
Let us now return
may be any
This
suppose that
it
is
consideration of the function F.
to the
We
shall function of the argument t H- r/c. such a function that it takes the value zero
values of the argument except those in the immediate shall then have neighbourhood of the particular value zero. for all
We
F(&)=Q unless x lies between, let us say, shall also suppose that small quantity.
where
e
e is
a very
We
\ ................. ...... (19). .
Since
F
(so)
zero unless x
is
between
is
e
we
evidently also
have 6
^
J -oo
If the value of r
F(x)dx =
r F( -e
./
is
fixed rt+r/c
provided
to
+ r/c > e
finitesimal but
we
still
shall have, if
and ^
-f
suppose
co is
rfc
<
Moreover,
e.
F to have the property
any function of r and
if
I
since,
a>'
the value of
is
except when
t
infinitesimal interval
Now
let
t*
(o
at the instant
(oo)
dx
=1
t,
t
=
r/c.
......... (20),
This follows
between + e, F 0, and throughout this may be considered constant.
lies co
have a definite positive value and ^ a very large
^ being
definite negative value,
enclosed volume ^
4-
<
r/c
e.
so large that for all points in the
Then the
values of
So
in the integrated part of (18)
all
since the derivatives of -^, ot
are also zero except
We may
F
e in-
J -e
= G> where
we make
r+e
%
vanish.
write the term containing
co
also
% which occur
do the values of
between +
in (18) in the
6.
form
fi
This in (20).
is
to equal i
1
1
jjj r
In a similar way
dr where
a)'
has the same meaning* & as
where
f
denotes the value of -~- at the point of integration un
)
\dn
I
(distant r from
P)
at the instant
to da integral with respect
We
may
=
t
The corresponding
r/c.
be similarly treated.
also have
dn
dn
(re
Thus
= where
1
ff (I) f J]dn\rJ^
1 ^ + Jjrcdn dS f
f
^
the value of
is
\|^'
'
f
J
V r
t}
f
\
*
+ -) * c)
at the point of integration at the
rut
instant
t
=-
c
.
The right-hand
integral
may be
integrated by
parts, giving
since
F(t + -} t~\
value of
vanishes at the limits.
Here
(-^-)
denotes the
i
-
Vv
at -the point of integration at the instant
t
=
,
in
C
accordance with our former notation.
The
left-hand side of (18)
may be
treated similarly, giving
rise to
>Y
Now
let
the surface
$
cr
,
become coincident with a sphere of ^
infinitesimal radius p about the point P.
left-hand side becomes
cp
=
Then on
^ ~~~
dp
and the
LVO
THE FUNDAMENTAL EQUATIONS
When
p
infinitesimal
made
is
to
become very small the terms in - become
compared with
-^r
So that the left-hand side
.
p
becomes identical with point P at the
instant
t
4^^ =
where
"SP
is
the value of ty at the
Hence, we have
0.
__3/i\ fa,(rjv
+ !&;/&
crdn(dt (21).
Now
let
the
surface
S
recede to an infinite distance and
suppose that at infinitely distant points the functions
^, ot
all
^ ^-,
and
have the value zero until a definite time T, then when
r becomes infinite the time
t=
,
to
c
which tlr', Y
{-},
and
\dnj
-(
\dt
]
J
T
so that every in the surface integral refer, is always less than } element of the integral is zero. This supposition is legitimate
physically since we always presuppose that physical phenomena are independent of past or present occurrences at an infinite at the point at the thus see that the value distance.
W
We
time
t is
P
equal to
r-
[({d T
(22), ^ '
47TJJJ r
where the integral is taken throughout space, and the value of each element of volume is that which it possessed at co for T
the instant
t
.
c
The Propagated Potentials.
we have just obtained physical interpretation of the result of the electric and the values means that It very simple. at any instant are at intensities any particular point The
is
P
magnetic
not, in general,
determined by
the state of the rest of the field at
The effects that particular instant, but by its previous history. at P, in so far as they are due to a particular element of volume distant r from P,
depend upon the
state of that
element of volume
than the instant considered. This time r/c is the time which would be required for light to travel from equal to The nature of the field distant element to the point P. at a time r/c earlier
the is
therefore such as
would
arise if
each portion of
which were propagated emitting disturbances tions
it
were constantly
from
it
in all direc-
with the velocity of light
When we come
to the actual calculation of
H
the values of
E
found that equations (7) and (8) in particular and of to, given by the right-hand values the to are unsuitable owing cases
it is
them, being somewhat complicated. The calculations may be simplified by the introduction of two new functions, the scalar sides of
and the vector potential U, from which
potential afterwards be derived
by
appropriate operations.
prove that JET =*
and if
JS;
and
U satisfy
rot
U
shall
..................... (24)
........................
v
prove first of all that a function This function is in fact
H = rot U.
r
.
For
if
(27)
is
true
dr
<**
.................. < 26 >-
U always
exists
such
.................. (27). v '
we have r
Now
may now
the equations
i>
We
shall
........................ (23),
= ----grad(
-
that
E and H
We
4}7rJJ]
r
H
the values of
in the integral refer to the different is measured. points of integration and not to the point at which Let x, y, z be the coordinates of the at which is
and
a, 6, c
point the coordinates of the element of
^= a (-a) + (y-6) a
2
+(2f-.c)
a
U U
volume
required
Then
dr.
and the equation above
may be
written more clearly as
Ux = JL fff fij** 4-rr JJ J
\db
_ 9#_A dc ) V(a
_
dadbdc
- a ja~+ y~_ (~
&)-
+ (Z - CY
'
THE FUNDAMENTAL EQUATIONS
195
,
, (rot t
and
_ _L II ~ 4nr 1
=
f [f
/^ _ <^A 86 7 - dHa \
(fffidHt
-
_ r
(dyUJ \da
d
- f
l\
8a
V 3c
3
dH
fY/aH.
*,
/
r \ 3 ./1
(r 8
al
= Surface
6a after
-^-
Integrals
an integration by
dcdc
Jr
db
The
parts.
da
r
surface can always be chosen
so that the surface integrals vanish, and thus rot
i
7r
^=7
H\ M -----
rrrfrot rot
[
,
dr
d
r
Now
the divergence of
H
is
always zero so that
v (28). -
r
By comparing
this with the equation for the potential
47r
we
see that the right-hand side of (28) is equal to J?. the theorem, which is seen to be true for any vector
proves
divergence
is
always zero.
Having proved that a .always be found,
and obtain
This
whose
we
vector
substitute the
rot
(E +
U
such that
new value
= 0.
-^-J
of
H = rot U
can
H in the equation
THE FUNDAMENTAL EQUATIONS
196
1
This equation scalar quantity
U
-
r)77
E H--^~ c ot
is
the gradient of some
So that we may put
.
the same as equation (24). It will be observed, however, and ^> are not completely determined by the considerations
This that
is satisfied if
is
which have been brought forward. The only condition we have imposed on U except (24) is that it should satisfy the equation = rot U. Also $ may be any scalar quantity. If U and
H
<
are particular values of
U
and
consideration, they will also
be
which
satisfy the equations under
satisfied
and
<
by
=
+ C ut
where
ty is
it satisfy
It
is
satisfied.
There
some
scalar function.
We
shall
determine ty by making
the condition
necessary to show that this condition can always be Substituting the values (29) in equation (30) we get
is
so that (30)
which will satisfy this equation, always some value can always be satisfied.
^
We have p = div
E= - -
^ (div U)
whence making use of (30)
We
......... (29),
also
have
Substituting from (23) and (24) rot rot
U= grad div U-
- div grad <,
197
THE FUNDAMENTAL EQUATIONS 1 r)fh
and
div
U= --c
-
,
ot
V^-^-i.F.
sothat
Thus (25) and (26) are the equations satisfied by the scalar and vector potentials respectively. satisfy (25) Conversely if (f> and will be given by (23) and (24). and and (26) the vectors
U
E
H
In the light of our discussion of equations
(7)
and
(8) it is clear
that the values of the potentials are
where the dashes denote that in carrying out the integrations the values of p and p V, respectively, at the instant t = r/c, previous which the integrals are being evaluated, have to be
to that for
substituted.
Electron at Rest and in
As an
we
illustration of the results
Uniform Motion. which we have just obtained
shall consider the case of a single electron.
has always been at rest then
U
V
is
always zero
;
If the electron
so that the vector
Moreover p becomes identical with p for since the every point, position at any previous instant is the same
potential
vanishes.
as the instantaneous position. Thus the scalar potential with the ordinary static potential, the electric
intensity
is
identical
is
identical
with the usual value of electrostatics and the magnetic force The solution in this case is identical with the results vanishes. of the usual electrostatic theory.
Next consider an electron which is moving and has always moved with a uniform velocity w in a straight line parallel to the axis of
P!
at
z.
Consider the values of the two potentials at any point ^. They will not be determined by the instan-
an instant
taneous state of the electron, but by instant
/.
/ will in fact
its
state at
be given by the equation
some previous
THE FUNDAMENTAL
198.
be called the retarded radius. It is of course a r the velocity w and function of the instantaneous radius l} It is not, however, necessary for our present c. of
where n' definite
may
the velocity purpose that
light
we should
the potentials at a point of *, and at a time t, }
Let us consider evaluate r/ explicitly. is parallel to the axis that such Z P,P 2
P
such that
P
P-.P^w^-t^
Both the
have moved forward in space a distance electron and the point and notning else in the Pro1:)lem nas changed. equal to w (t,2 at the instant t 2 as will have the same values at The
P
potentials the instant l at they had at
^ since they must be determined of P and the moving charge and cannot by the relative positions Thus the field due their absolute positions in space. on depend carried is electron a to along with it as though uniformly moving
P
it
were fixed to
it
by a
;
rigid framework.
Fig. 28.
QiO represents the direction of motion it is clear from symmetry that the potentials will have identical values at all If
points such as a.
point
P
which
lie
on a
circle
on the line of motion.
of given
Let Q
1
=
>
^
radius about
and QiP
= r1}
being the instantaneous position of the electron at the time t for which the potentials at are being calculated. The potentials will be functions of z, r and t only. Moreover the resultant
Q
l
P }
velocity lies
along the axis of
of the vector potential vanish.
z,
so that the
We may
*-/(*, n, 77
*)
x and y components
therefore put
199
THE FUNDAMENTAL EQUATIONS
/
where
and
F
are functions which
it is
not necessary for us to
evaluate more explicitly at present.
We .,
1 ciTT
.&=---grad C Ou
have
,
.
so that
cy
"c ~~
_
_ dz
c dt
Thus the
electric intensity
components, the
regarded as made up of two the axis of z (the direction of
may be
first parallel to
o
rr
of the electron)
and equal to
Oj?
-~r
~
c ot
QZ
-j
motion
and the second /i
directed along the instantaneous radius
The components
#
z
oz
.
_? =0
= 05?
Thus the magnetic lie
.
_dU - 2 dUv dy
centres
/
~-
of the magnetic intensity are
y>
and
7\
and equal to
.
9^
intensity
is
tangential to circles whose
on the axis of motion and whose planes are perpendi-
cular to that direction.
The
distribution of magnetic force is to
this extent similar to that arising from a straight current lying the axis of motion.
on
Accelerated Electron.
We motion
shall
next take an illustration in which the nature of the
during the interval under consideration. Let the be at rest at the point Qx until the instant ^ let it then particle alters
;
THE FUNDAMENTAL EQUATIONS
200
then
infinitesimal interval of time;
instant 2 complete at the rest in the position at ever for remains .
The
P& = n and PQ = r 2
P
field at
taneous position of
at
Q
to
/
move
is
Q
2.
let
Consider the
field at
.
= - r'/c,
the retarded radius.
/
a
;
the instanany time t is not determined by time the at but by its position if
where
;
the stoppage be After the instant t2 the particle
stopped
P and let
move with uniform Q2 at Q 2 it is
let it
reaches the point
a straight line until velocity in as suddenly as it was started at Q, it
t
Up
to the instant
when Q began
was fixed and equal to r1} so that provided
satisfies
t
the inequality
t
or
the instant that
nM
P will
field at
the
< ^ -+
2
After be that due to a static charge at Q x the radius r becomes permanently equal to r2 so .
if
or
the field at
P
between t~t three stages acceleration, velocity,
and
:
1
is
+
that due to a static charge at Q.2 = 2 4- r2/c the field at rjc and t
(3)
P passes
through
moving with a positive that due to a particle moving with uniform that due to a particle moving with a negative
(1) that (2)
In the interval
.
due
to a particle
acceleration.
We electric
shall see in the sequel that the field
due
to
an accelerated
charge possesses novel features of great interest.
CHAPTER X THE ACTIVITY OF THE FORCES
WE
shall
next consider the rate at which work
we
is
done by the
mode
of expression, by the in volume r. The enclosed on the aether, any given charges forces in the field, or, if
prefer this
mechanical force on a unit charge
is
E + c- [VH],
where
V
is
the
velocity of the charge relative to the instrument used in measuring H. The force acting on the electric charge in an dement/ of
volume dr
is
therefore
and the rate at which work instant
is
equal to the scalar
being done by this force at any product of the resultant velocity of
is
the element by the resultant force The rate of acting on it. working of the forces in the field on all the charges present in it, is therefore
Now
the part of the electromotive intensil.y
perpendicular to the plane containing Thus (V always perpendicular to F.
V
and
[VII
that the activity of the forces reduces to
But
pV=c rot H -
c
.^
ct
,
so that
|)
[
// is
Vll
and
is \
is
always
always
therefore xero, s>
THE ACTIVITY OF THE FORCES
202 Eeplacing rot
If
we
If
Cartesian equivalents this becomes
terms containing =-
,
~
and
re-
and integrate each term by parts this becomes
\ldydz (Ey H,
of the
its
collect together the
spectively c
H by
- Ez Hy ) + dzdx (EZ HX - EX HS )
m, n are the direction cosines of an element of surface dS = IdS, dzdx = mdS, boundary of the volume T we have dydz
I,
H
H -EX H Ex H -E Hx
dxdy = nd8, and Ev H,-Et v EZ X the at, y and .0 components respectively the integrated part
is
,
Z
,
of the vector
v
y
are
[EH]. Thus
equal to
where [EH] n denotes the resolved part of the vector [EH] along the normal to the element dS. The volume integral, after rearrangement, becomes
-I
and, since rot 2?
o rr
= --^-
and
H*= Hx*+ Hy + H*, *
this
may be
written
Thus
^ = c (([EH^dS 1
f f fa
^^ + 1^) dr
...... (1).
THE ACTIVITY
OBY
THE FORCES
the magnetic energy per unit volume of the It is aether and \E* is the electrostatic energy per unit volume. of this arid to be borne in mind that the investigation are
Now %H
2
is
H
E
the universal values of the forces introduced in the last chapter and are not the same inside material media as either of the defined as the intensities average values which we have previously and inductions in such media. The mean value, over a region of
the appropriate dimensions, of the present ^H* is identical with 2 2 and the mean value of the or or the former
^H
present
\E*
When E
is
and
%HB
B /^
identical with the former \tcE~ or
H are the universal functions
\ED
or
represents the rate at which energy is lost by the aether, or space, within the limited region r. Thus the whole of the work done by
the forces of the field on the electric charges is not covered by the energy lost by the aether in the immediate neighbourhood.
In general we have also to consider the quantity represented by the surface integral. Since the left-hand side of the activity equation is the rate at which work is done on the electric charges, and the volume integral represents the rate of loss of energy by
the electromagnetic field in the
enclosed volume, the
surface
integral which is equal to their difference must represent the rate at which energy flows into the region r from outside. Any possible alternative to this conclusion
would involve a denial of
the principle of the Conservation of Energy.
Poy ntiny 's Th eorem. The occurrence of the surface integral cff[HH] H dS in the equation of activity of the forces was first remarked by Poyntirig*, who gave to it a very definite physical interpretation. He pointed out that the behaviour of the field could be explained by the supposition that at every point there was a stream of energy
e<jual
per unit area to c[!.H], the direction of the stream being coincident with that of this vector, and therefore normal to the plane and H. The Poynting Flow of containing Energy thus vanishes when and // are coincident in direction, arid has a
E
E
THE ACTIVITY OF THE FORCES
204
value, other things
maximum
being equal,
when they
are at right of course, consistent
This interpretation is, another. angles to one with the equation of activity, and is, in fact, the most obvious It is, however, not the only interpretation. of it.
interpretation
For we may evidently add to
c
[E H] any vector .
R which satisfies
the condition that the surface integral of its normal component closed surface vanishes and the equation of activity will over :
any be satisfied.
still
We
know from Gauss's Theorem that this con-
we have div R = everywhere so that there number of vectors in addition to Poynting's which
dition will hold if
are an infinite
;
of activity. satisfy the equation It is interesting to consider to
what picture of the flow
of
energy we are led in typical instances on the supposition that it In the case of a straight wire coincides with Poynting's vector. carrying a current, for example, the electric force is parallel to the length of the wire, and the magnetic force is in circles about its
Thus the electric and magnetic forces are at right angles to one another, and the flow of energy is at right angles to both. That is to say, it flows perpendicularly into the wire from the
axis.
insulating
medium which surrounds
it.
Probably the most convincing case of the flow of energy in accordance with Poynting's vector is that furnished by the propaConsider a parallel beam of gation of electromagnetic waves.
We have seen that in such a beam the and magnetic vectors may be represented by
plane polarized light. electric
E = H = A cos (pt - x). It is
important to notice that the two vectors are always in
phase, and that they are equal in magnitude when expressed in the units used in this book. They are also at right angles to one .another. Thus the resultant flow of energy is perpendicular to
both
E and H
;
that
to say, it is along the direction of propaIt is equal at any instant to
is
gation of the light.
per unit area.
Its average value over a single period, or over
very large interval of time, o
l
T
TT
J
o
is
any
THE ACTIVITY OF THE FORCKS
Now \A*
mean energy present in unit volume of the this energy as flowing Poynting's Theorem represents is
the
wave, so that Since this is in accordance with the* c. along with the velocity of Poynting's Theorem to results of observation the application solid to radiation evidently rests on very questions relating grounds.
Forces exerted on the Charges.
In the last section but one we have considered the rate of working or "activity" of the forces acting on an enclosed electrical system. We shall now consider the value of the resultant force acting on On the electron theory of matter the results a similar system.
be applicable to any material system since, on this theory, the force acting on a material system is the aggregate; effect of the electric and magnetic forces which act on the electrons whieh will
constitute
it.
The
force exerted
on a unit charge-
is
K+
t
j
V II
j,
where V is the velocity of the charge relative to the system of instruments used to measure the forces. The reason for this
V
will be clearer later (see Chaps, xni particular specification of It will be observed that it is not inconsistent with
and xiv).
the deduction from the magnetic properties of electric currents
which led us to include the term c
force
on a charged body
has been considered well have taken
V
up
(p. 114).
[VH] So
to the present
in the- expression for the
far as is
any evidence which
concerned,
we
might, as
be the velocity of the charge' relative to the which we aether, might suppose to be absolutely fixed in space. to
When we come to consider the electrical and optical properties of bodies in very rapid motion, we shall see that the assumption that refers to the velocity relative to the measuring system
V
effects
very important simplifications.
The charge present
in the
element of volume dr he ing pdr,
the force exerted on this element of volume will
dF=p(jK+
l
\Vni\dr,
he
THE ACTIVITY OF THE FOKCES
206
and the equation
'
= divE.
since
whole -volume will be given by the vector
force on the
But
hence
V= c rot H
p
E.
and
^-//Jjdiv
-^-
,
ot
E + * [crot#. fl] -
.
If
BH-l =~ [^. rot #]
= - [rot .#.#],
and hence
F= -
~ [EH]
[[[c ot JJJ
.
dr +
4-
1
1
This
is
/
f ffdiv l
JJJ
.
F
.
L
J>
+ [rot H. H]} dr
/{divjfif. fl"
the total force on the volume
ponent of
E E + [rot E.E}\dr
Consider the x com-
r.
due to the third term of
Call
(2).
-H,"
1
...(2).
z
it
XH
.
Then
*--* <*-?-*
ty
dz
.,
dr
THE ACTIVITY OF THE FORCES where
in,
Z,
normal
to
n are the direction cosines of the outward drawn dS of the enclosing surface.
the element of area
The third term
of (2) will give rise to similar expressions for the
If we treat the to the y and z axes. components of force parallel shall we same the in in get similar exsecond term way (2) that the components of the magnetic intensity pressions except are replaced by the corresponding components of the electric
If
intensity.
we put
Pa = the components of the last two terms of (2) may be written, in the form selecting the x component as an example,
The
last
two terms therefore reduce
to surface integrals over the
boundary and are, in fact, identical with the forces due to the Maxwell stresses discussed in Chap, n, except for the addition of the magnetic terms which were not then being considered.
We now
notice
a very important difference between our
present problem and the
static case considered in
Chap.
The
n.
the forces acting on the volume from without, as calculated from the Maxwell stresses across the boundary, is no resultant
of
longer equal to the force tending to accelerate the charges enclosed by the boundary, the former being the greater
Thus,
in the absence of electric charges,
by
the resultant, force
due to the stresses over the boundary does not vanish unless the value of this volume The most natural interintegral is zero. In static cases, conpretation would seem to be the following. sidering the action external to the
between the charges enclosed and the region bounding surface, the resultant force on the external region, given by the Maxwell stresses, is equal and opposite to the resultant force on the charges enclosed. In general, however, the action and reaction between the *
charges
THE ACTIVITY OF THE FOKCES
208
the reaction to the force exerted on the external region falls on the aether enclosed by the surface, the amount of this reaction 1
o
- rper unit volume being C ut [EH].
There are two other
possibilities
which are deserving of consideration. We may either deny the of the Newtonian law of action and reaction to
applicability
the physical electrodynamic systems or we may deny reality Of these two alternatives the latter is of the Maxwell stresses. to be preferred, at
theory of matter.
any rate from the standpoint of the electron For if we are to regard material systems as
electrodynamic systems (which is essential that the
shell) it
for
them
That
the electron theory in a nut-
is
Newtonian laws should be true
to the extent that they are true for material systems. they must be exceedingly close approximations to
to say,
is
the truth in the case of systems the parts of which have velocities which are small compared with that of light. It
happens that we can retain the law of action and reaction and
stresses if we interpret unit volume of the medium. per In that case equation (2) shows that the stresses acting across the boundary of the region are equal to the force tending to move the
also the physical existence of the
the vector \EH~\jc as
Maxwell
momentum
charges within the region plus the rate at which momentum is communicated to the enclosed medium. The idea of electro-
momentum was first suggested by J. J. Thomson*. The momentum per unit volume is very closely related to Poyn ting's magnetic
[EH], being, in fact, equal to the latter divided by the square of the velocity of light.
vector, c
We
electromagnetic momentum is extremely useful in enabling us to make calculations about the mechanical effects of light and of moving electrons. At the same time it suffers from a serious disadvantage in so far as we have no shall see that this idea of
any mode of motion of the aether to "Thomson has suggested f that it represents
satisfactory conception of
which
it
corresponds. the inertia of tubes of electric force.
These are supposed by him have a definite concrete Each tube is physical existence. to be anchored at one end to the electron to which it supposed
to
*
Recent Researches in Electricity and Magnetism, p. 13 (189&).
t Loe.
cit.
THE ACTIVITY OF THE FORCES
209
belongs, whilst the other end extends to an indefinite distance. In a recent paper Thomson has examined the consequences which
would follow if the number of such tubes attached to each electron were quite small*. It will be observed that this theory attaches the inertia in reality to the electrons rather than to the medium in which they move.
The most important
momentum
is
to the
application of the idea of electromagnetic
dynamics of a moving
electric charge.
We
however, defer that question until the next chapter and occupy ourselves for the present with the question of the pressure exerted by light and other electromagnetic waves. This subject
shall,
excellent illustrations of the application of the ideas both momentum and of the aethereal stresses.
.affords
of electromagnetic
The Pressure of Radiation.
We
shall consider the pressure of radiation first of all from the view of the Maxwell stresses. Let us apply equation (2) of point to any closed surface containing matter or electrons and consider
the average value of the quantities occurring in the equation, taken over a considerable interval of time T. The average value of the left-hand side will be the average force exerted on the
There are important cases in which the Maxwell stresses can be very easily calculated and in which the volume integral vanishes. Considering the lastnamed term first, its average value over an interval of time T is
matter or electrons. .average effect of the
" cT This expression conditions holds
is
equal to zero
when
either of the following
:
(1)
time
The volume
integral has the same value at both the be the case when the electromagnetic
This
will
actions are periodic
and
limits.
T
is
an integral multiple of the periodic
time. (2)
The volume
integral
fJJ[EH]dr 9A1 /1OAH\
is
finite
throughout
A.V--
JL J.
the time considered and
V A
T
A
is
JL
an interval of time so great that
it
be regarded as infinitely large.
may
It is evident that the electromagnetic
unimportant except when
momentum
is
relatively
rapid changes are taking place in the
state of the electromagnetic field.
Let us now apply our results to the case of a plane polarized
wave of monochromatic
light
incident normally on a surface which absorbs it completely. Let represent the ab-
A BCD
sorbing substance, the light being incident on it in the direction of the arrow which Fig. 29.
E
Q cos pt. along the axis of x and be equal to components of the electric and magnetic intensities are
wave
is
Let the parallel to the z axis. electric intensity in the light
lie
Then the
Applying equation (2) to the cylinder whose cross section by the plane of the paper is and integrating over a complete period, we see that the part coming from the volume integral on
EFGH
the right-hand side vanishes. The left-hand side is equal to the average force exerted by the light on the matter in by which it is absorbed. The components of the Maxwell stresses
ABCD
are
:
Pxy
=Pyx
= Ex Ey + Hx Hy = 0.
Similarly
j>y
= jp=0.
EH
Thus there
is no stress on the cylindrical surface of which are sections, since this surface is everywhere parallel to the axis of z. The stress vanishes over since there is no
and
FG
GH
light there.
The only part
stresses are effective is the
which the Maxwell end EF, and thev are here
of the surface over
THE ACTIVITY OF THE FORCES
211
Thus the force exerted by the to a pressure E'2 per unit area. light on the opaque surface is a pressure which is equal per unit area to
The energy per unit volume
of the
so that the intensity of the light,
amount of energy transported
beam
which
is
of light
is
equal to the average
across unit area in unit time, is
If rn
i
Thus in the case of light incident normally on a perfectly absorbing surface the radiation pressure is equal to the intensity of the light divided by its velocity of propagation. It is clear that in the case of a perfectly reflecting surface
the value of the radiation pressure will be doubled since the will be twice as great as with a intensity of the light at when the incident intensity is the surface, absorbing perfectly
EF
same.
Since these results are independent of the plane of polari-
zation of the light they will also be true, at normal incidence, when the light is unpolarized. It follows, on similar grounds, that
they are also true
for light of
mixed
frequencies.
Isotropic Radiation.
Next consider a perfectly reflecting enclosure filled with isotropic radiation. By isotropic radiation we mean radiation which is being propagated in all directions in such a manner that the probability of the direction of propagation of any ray, selected at random, being found within a given solid angle is proportional to that solid angle. The thermal radiation which would fill the enclosure in the final state of equilibrium
which ensues when there are material bodies
of this character (see Chap. xv). Consider an infinitesimal area dS of the reflecting enclosure and apply equation (2) to a cylinder, similarly situated to that, in Fig. 29, but whose height
within
is
it
is
infinitesimal
before, let
compared with the dimensions of
dS be
perpendicular to the axis of
z.
its
ends.
As
We may now
THE ACTIVITY OF THE FORCES
212
the smallness of their dimensions.
The components
of the force
the Maxwell stresses therefore vanish everywhere arising from front face of the cylinder, and are there equal except on the respectively to
Since by hypothesis the trains of waves which constitute the radiation are as likely to be travelling in any one direction as in
any
other, the electric
and magnetic intensities are as likely to
are as any one direction as in any other and, incidentally, have the We therefore as be to following negative. positive likely relations among the average values of these quantities, taken over lie in
a long interval of time, for the 'radiation which crosses the end of the cylinder :
EX E = Ey E = H x tt = Hy H = 0. Z
z
Hence the mean values of the
z
p^dS = pyz dS^O, j^dS = -E~* + H~*dS = -
and
Here E* (= if 2 )
is
the
z
tractions are
~E* dS.
mean square of the electric Thus in the case
the radiation in the enclosure.
intensity in of isotropic
radiation the pressure is equal to one-third of the energy in unit volume of the .radiation. This result has an important application to the
Thermodynamics of radiation. Radiation Pressure and Momentum.
Let us now consider the pressure of radiation from the point electromagnetic momentum. Confining our atten-
of view of the
tion first of
to a plane wave of plane-polarized monochromatic observe that this consists of an alternating electric and magnetic field which travels forward with the uniform velocity of light. As we have seen, the electric and intensities
radiation,
all
we
are always at right angles to each other
magnetic and to the direction
THE ACTIVITY OF THE FORCES magnitude at any particular point at any instant, and for the same If E and point in space they are harmonic functions of the time. are the values of the two intensities at any instant it follows
H
that the electromagnetic
c
momentum
per unit volume
= ~ E? cos*pt. [EH] = -EH~-E* c c c
Thus a plane wave (whether polarized
may be regarded as through the medium with
p/2-TT
in the
This
light, since
or not) of light of frequency
a stream of
momentum which moves At a fixed point
the velocity of light.
medium the momentum per
momentum
is
unit volume
is -
c
2 E
always in the direction of propagation of the
is
E and H always have the same sign.
Now
suppose that a train of plane waves, such as we have been All considering, falls normally on a perfectly absorbing surface. of the radiation disappears so that there is a constant stream momentum flowing into the absorbing surface. But interchange
of
momentum implies the existence of a stress. The radiation will therefore exert a pressure on the surface which will be equal to the The rate of change of momentum per unit area per unit time. of
average pressure
in
is
therefore
agreement with the value found previously.
In the case where
perfectly reflecting the forward momentum will not merely be destroyed but an equal and opposite momentum will be given bo the reflected wave. Thus in this case there is twice as great a rate of change of momentum so that the pressure will be twice as great as with a perfectly absorbing surface, for the same
the surface
is
;
intensity in the incident light.
In the case when a plane wave making an angle 6 with the normal to a it
convenient to resolve the
is
is
incident in a direction
perfectly reflecting surface, in the incident and
momentum
waves into two parts, one normal to the reflecting surface, and the other parallel to it. The momentum parallel to the normal which falls on unit area in unit time is (1/c) S cos 2 0, where S reflected
is
Poynting's vector T,'
I,
!.,
c
[EH].
The momentum
in the
same
direction,
THE ACTIVITY U*
214
- (l/c)Scos
2
The
0.
TJtu&i
enters twice because, in the
factor cos
first
the resultant momentum is along the direction of propagaplace, tion of the wave and we are dealing with the component at an angle 6 with this direction and, in the second place, on account of the radiation falling obliquely on the surface, unit area of the
wave
front will
be spread over an area I/cos 6 of the surface. The momentum normal to the surface is evidently
rate of change of (2/c)
8 cos2 6 per
average value
unit area.
is (2/c)
cos2
Thus there
6S = E
2
is
a normal pressure whose
6.
The momentum parallel to the surface lies in the plane of incidence. The amount of it which is incident on unit area in unit time is clearly (1/c) S sin 6 cos 9. The amount leaving unit area in unit time is also equal to this, since the direction of this part of The rate of change of the momentum is unchanged on reflexion.
the tangential component of the momentum is therefore zero, so that there is no tangential stress, even when the radiation is inciIt is evident dent obliquely, at the surface of a perfect reflector. of radiation is the true when no is that this conclusion part longer
There
absorbed at the reflecting surface. stress
which
will
then be a tangential
between the
proportional to the difference and the reflected waves.
is
in-
tensities of the incident
When light is incident at a transparent surface, we have to deal with a refracted as well as a reflected beam. Let l denote the angles of incidence and reflexion and Z the angle of refraction. In this case we are no longer dealing with the free aether, so that
we have
to use the expression for the
appropriate to a material substance.
momentum
per unit volume the making changes in the
By
argument on p. 205 which are necessary when dealing with media whose dielectric constant and magnetic permeability are not equal to unity, we see that the general expression for the momentum per unit volume is [DB]fc, where is the electric and the magnetic induction. for the incident, Denoting the values of this
D
B
quantity
refracted
and
waves by
reflected
G
19
and
Gr 2
clear that the instantaneous value of the
V, (G,
where
F
a
and
7
2
+
cos 2 0,
- VG 2
3
respectively,
2
-
ff,)
of the tangential stress
sin 6, cos e l
- F (? 2
2
it is
is
cos2 0,,
are the velocities of radiation in the
The instantaneous value "Pi (
G
normal pressure
two media.
is
sin 0. cos 0,.
THE ACTIVITY OF THE FORCES Here
sin ^/sin
2
=
T^/Fo and the values of
215
G G and l3
z
6r3
portional to the respective instantaneous intensities of the
are pro-
beam.
The ratios of the average values of these can therefore be obtained from the expressions for the reflected and refracted intensities found in Chap, vir, and will depend on the plane of polarization of the light.
add that the pressure due to electromagnetic radiation was predicted by Maxwell*, as a consequence of his theory of stresses in the medium, in 1873. It was first demonIt remains to
by experiment by Lebedewf in 1899, and later independently by Nichols and Hull} in 1901. Several of the more strated
complex cases of the effect of examined by Poynting.
light pressure
have recently been
Isolated System.
An interesting application of the
theorem expressed in equation
Take any surface isolated system. of the dimensions the the system, enclosing surface surrounding field of that the at it so may be considered any point being great (2) arises in the case of
negligible.
an
Then the Maxwell
stresses vanish over the
boundary
and we have
But the left-hand side is the force acting on the charged bodies in the system and is, therefore, equal to the rate of increase of (supposedly) material momentum of this part of the system. Calling this
momentum
M we therefore have
or the
momentum
system
is
gained by the material part of the charged
equal to the
momentum lost by the electric field. We
see in the next chapter that if matter
is
made up
shall
solely of electrons,
*
Treatise on Electricity and Magnetism, 792 (1873). t Arch, des Sciences Phys. et Nat. (4), vol. vra. p. 184 (1899)
433 (1903). Phys. Rev. vol. xnr.
vol. vi. p.
p.
293 (1901).
;
Ann. der Phys.
THE ACTIVITY OF THE FORCES
21$
M consists of a distribution of momentum in the From
this point of
field of
view the
the same,
effect of the
as (?. type in reality be regarded as giving rise forces on the parts of the system may to a rearrangement of the electromagnetic momentum in the field,
amount remaining constant throughout the changes which take place. This result is only true provided the motions otherwise may be regarded as quasi-stationary (see p. 262) the over radiation lost be momentum will boundary. by
the total
;
At present we can only be certain that matter is made up in of electrons. If this should turn out to be ultimately true we
part should have to say that during quasi-stationary dynamical actions the electromagnetic momentum of the field is converted partly into the electromagnetic momentum of the individual electrons and partly into
unaltered.
material
momentum, the
total
momentum
being
CHAPTEE XI CHARGED SYSTEM IN UNIFORM MOTION
WE
have seen that a charged system in uniform motion carries its field along with it as though it were rigidly attached to it and that the potentials and forces are given by the equations
We
shall
suppose the charged system to be moving along the
w
axis of z with the uniform and that it is symmetrical velocity about the axis of motion. The equations above refer to axes y
whom the system is moving with V ~w. If we consider axes which move uniformly velocity with the we can take along system advantage of the fact that the fixed relatively to the observer, past
field
If jr-
is
invariable relative to points measured along these axes.
denotes the rate of change of any quantity with respect to
time at a point
where
d/dt
x, y, z fixed
D ~
d
JJt
dt
with respect to the moving axes, then d
--
8
8
dx
denotes differentiation at a point fixed with respect to w, v, w are the velocity components of the
the fixed axes, and
so that moving with the moving axes, -^
for a point
also
have ^ = # =
0.
= 0. We
So that a
-a
g^a?"
and
along by the moving system to it enables us to eliminate attached were rigidly though On substituting for the time from the equations for
Thus the as
r)
it
2
~,
the equations -for
where
/3
= w/c.
these equations
$ and
By changing
may be
U
become
the variables to
a? x
,
y ly z^ where
written
These equations are of the same form as Poisson's equation. scalar and vector potentials due to a moving charged system may be obtained from the scalar potential for a slightly
Thus the
system at rest. The transformed system is obtained from the moving system by stretching all the lengths in the to the direction of motion in the ratio moving system parallel
different
1 to J. J.
Vl /32 This type of transformation was first obtained by Thomson* by a different method and also, independently, by .
Heavisidef. It will *
Phil.
be noticed that the scalar and vector potentials are
Mag. April 1881, also Recent Researches
p. 19.
t Phil Mag. April 1889.
in Electricity
and Magnetism,
219
CHARGED SYSTEM IN UNIFORM MOTION
the constant factor ft. proportional to one another, differing only by It will therefore be necessary for us to calculate only one of them.
Let us now denote our moving system by S and suppose that corresponding to S there is a fixed system $x which is obtained by ratio enlarging the dimensions of the moving system S in the Vi /3s in the direction of motion. Every point CD, y, z in the 1 moving system will correspond to a point #1? y l9 zl in the fixed :
system, where the relation between
x, y,
#19 ylt
z and
^
that
is
already given. Also, corresponding elements of volume will contain equal charges, since straining the dimensions does not alter the quantity of electricity present. Thus, if p is the volume .
moving system and p : the corresponding quantity in the corresponding element of volume in the fixed system, we have
density in the
the equation
= p dx
p dx dy dz
But
doo
= dx
l ,
hence
l
l
dy dz
dy = dyl} and dz = dz- Vl doc dy dz
dasl dy 1 dz^
^ = p A/1
and
- /3
.
2
/3
V1
2
l
;
2
/3
,
.........
.
.............. (3).
The
potential fa of the distribution in the fixed system satisfy Poisson's equation
But the
scalar potential
which we are seeking,
must
satisfies
the
equation 32
d 2
<
2
a^T
d 2
2+ a^2=:=
9y7
^ = Vl^/3
whence
~^
2 .
........................ (5).
Thus the scalar potential in the moving system is equal to 2 1 times the electrostatic in the /3 1/V potential corresponding fixed system. It is desirable to emphasize at this point that the fixed system which we have imagined is simply a mathematical device to
We are not supposing that the moving transformed physically in any way into the corresponding fixed SVSf.Pm. Whn.t W<^ IICIVA rrr\\rorl IQ fVof fK/-> o^ol^v ^^A facilitate
system
is
the calculations.
CHARGED SYSTEM
220
UJN1FUKM MOTION
IJN
vector potentials of the moving system are related in a certain way to the electrostatic potential of the corresponding fixed to calculate the value of the system, and it is comparatively easy I have mentioned this because I find that students who latter.
have a little knowledge of the principle of relativity are apt to become confused as to the point at issue.
We
now
in a position to express the energy and the in the moving system 8 in terms of electromagnetic the potential 0! and the coordinates x l9 y ly z-^ of the corresponding are
momentum The
fixed system Si.
We
energy per unit volume
is
notice that 1
30
30i
30i
'
3# 3
1
30
30
T
J
2
VlT
3flq
OO
We
electric
J
-1
00?
-L
'
3#i
/3
-L
Vi
dy O
2
/3
3yx
f
vQ.)!
have seen that the resultant vector potential We thus have
is
parallel to the
direction of motion.
Us TT-
Hence
The
A^ = j-j
electric
~=
= Uy = 1
0(p
.
.
and 001 ^ -^-
U = /30. s
Y-. ,
-&
energy of the moving system
=
8
-
is
W= 2
.(6V
CHARGED SYSTEM
The components
Z
2
==
UNIFORM MOTION
of the magnetic intensity are
y --o
,
~T:
9#
jjj{H/
The components
_ v.
3y
The magnetic energy
T=\
IN
of
S
is
therefore
+ H/ + H\ dxdyde
of Poynting's vector
S=
c
[##]
are
v x (10).
We
have seen that the
momentum
per unit volume
the components of the total electromagnetic
system
8
are
is
~
S, so that
C"
momentum
of the
CHARGED SYSTEM
222
The
results
UiNJ.ru.KM
IJN
are true for any electrical system apply them to two simple cases
we have obtained
in uniform motion.
We
shall
on the ground of their simplicity, will give us but also because we may reasonably expect that they The case electron. that we the an insight into the behaviour of of thin spherical shell uniform shall consider first is that of a rigid which we
shall select partly
electrification.
The Rigid Electron.
We
shall
now suppose
that the
R
consists of a
moving system
The uniformly to be be supposed negligible compared with its radius. It follows from the results that have already been established, that the energy, momentum, etc. of the field due to such an electrified shell when in motion may be obtained of radius spherical shell thickness of the shell will
if
which
electrified.
is
calculate the static potential
we can
system which
direction of motion are lengths parallel to the 2 whilst lengths at right angles increased in the ratio 1 Vl strained so that
all
:
,
remain unchanged. The moving sphere will evidently strain into an ellipsoid of revolution in the corresponding fixed of the ellipsoid coincides with system. The axis of revolution to this
the direction of motion of the sphere.
The major
axis of
any of
2
whilst the the principal elliptic sections is equal to jR/Vl /3 of the moving sphere. is equal to the radius ,
minor axis
R
be observed that these results will only be true provided the electrified sphere is rigid. Certain experiments, which will be discussed later, have led physicists to suspect that It
is
to
the lengths of bodies depend on their velocities relative to that of the observer engaged in measuring them. If this kind of change affects the electron itself, as well as the aggregate of electrons
which we suppose constitutes the material substance, the figure in the fixed system which corresponds to the moving sphere will no longer be the ellipsoid which we have described. This follows because the moving system which
we suppose
when
is
to
be spherical
no longer spherical when in motion. We shall see later that if the shape of the charged sphere does change, and if the changes are such as would naturally at rest
becomes distorted, and
be suggested by the results of the experiments, the
CHARGED SYSTEM
IN
UNIFOEM MOTION"
223
become much simpler than those which we are now carrying out the rigid spherical
for
shell.
In order to make our notation agree better with that usual geometry of ellipsoids we shall suppose the motion to be
in the
x instead of that of z. The appropriate parallel to the axis of changes in the formulae of the last section may easily be made on
The equation of the ellipsoid into inspection. sphere distorts is
where a = R/Vl minor respectively.
2
/3
are confocal with (14)
if
2 2 p =a
of
which
and
=R
6
The equation
may be
which the moving
the semi-axes
are
major and which
of the family of ellipsoids
written
the square of half the distance between the foci. b2 The spheres which bound the rigid electrified shell will transform into two infinitely near, similar and similarly placed ellipsoids, one is
given by the equation (14). The space between the is filled with a distribution of electrification of
similar ellipsoids
uniform density. The potential due to such a distribution is constant within the region bounded by the ellipsoidal shell, the distribution being equivalent to that on an ellipsoidal conductor maintained at a constant potential. Outside the shell the equipotential surfaces are the confocal ellipsoids given by equation (15)*".
The
difference of potential
between two confocal
ellipsoids
whose equations are given by \ a /V3
,
where
and *
e is
7=
A a A-3
f A3
Jc
O
a
cd\3 2
V-C'
,
the charge on the inner ellipsoid.
Applying
this result
Cf. Webster, Electricity and Magnetism, Chap. v. + Maxwell. Electricity and Maanetism. 2nd ed. vol. i. D. 237. eanation f28L
CHARGED SYSTEM
224
to the ellipsoids (14) to (15), \s
=a
UJNJLtfOKM
1JN
MOTION
= Vp + X corresponding c = p = Va* - 6 2 and
and (15) we have X 3
corresponding to (14),
2
outer ellipsoid to an infinite distance and on account of the difference between our introducing a factor 1/4-Tr units of electric charge and those used by Maxwell, we find that
By removing
the
the potential at any point in any ellipsoid confocal with (14) and is
given by equation (15)
Since the potential soidal shell, the total
is
the same at every point of the
energy of
ellip-
its electrostatic field is
2 .
22
^ log 6
a
2 b + Va_____ -
2
,
.
This must also be equal to
We have seen that the energy and system depend upon the integrals
momentum
of the
moving
'
We
therefore have one simple relation
2
It is necessary to obtain one of
between them, namely , 2
log
.
them separately by
..(19).
direct inte-
gration.
To obtain the value of J^ we use confocal coordinates.
The
level surfaces are (1)
the system of ellipsoids
x
2
Vi
'x""
1 ' rotated
trough
TT
......... (20),
CHARGED SYSTEM IN UNIFORM MOTION
225
the system of hyperboloids
(2)
/y.
yZ ^l =
2
l
?
P
rotated through
TT
(21),
and a series of planes passing through the axis of as which an angle (f> with the standard plane. The coordinates are
(3)
make then
p and
X,
As the problem
.
is
one of revolution we only
by the plane of xy. Let dS be an element of area in this plane included between the two ellipses characterized by X and A, -f dX and the hyperbolas characterized by p and -h dp. Solving the ellipse and hyperbola equations simultaneously we find that the Cartesian coordinates of the points of need
to consider a section
JUL
intersection are
Let
A BCD
Let
A=oc ,y l
represent l
,
dS
magnified.
then
Let the direction cosines of spectively.
so that
AD and AB
Then
dS = AB.
AD (Im - I'm)
be
J,
?n
and
V,
m
re-
From the
expressions
for
^S = d\ dp
Whence
xlt y l at intersection
4
v^X (p
The element of volume dr
, 2
X) (p
2
obtained by rotating
is
It is therefore given
the axis of motion.
- /t) .
4-
dS about
by
fJL
To
any of the confocal
for
From we
^
calculate the value of
ellipsoids
the equation ^
=
find
-
p
x z2
+
we remember
OX 1
that fa
is
constant
and therefore depends only on
v
X.
2
+ X = X ^-
1,
-
^
2X
The and
limits of
=_p we 2
/u,
X
are
X=&
2
to
X=
oo
.
With the
limits
p=
cover half the space, so that
By changing
the variable to
into partial fractions
we
x,
where
x-p*
p,
and turning
find
+x-p
CHARGED SYSTEM IN UNIFORM MOTION cr Since
,
<& Yl
+
J-
and
t
3
=se
r \ log&
167rp J
^
(
,
log &
2 Vp + X - jp
Vp
2
4-
X
By expanding
= log ,
1
log
-f 2
V^ + X-p
,
1
+
- X-
7
~
V_p + it can be shown that this vanishes when X = oo and a = J? -r Vl - /S we find p = \/a -6 & = V
2
2
2
On
.
substituting
2
;
From
(19)
we have
whence
We
have seen that the potential
W = H(! -
s 4 /8 )
^i
and the kinetic (magnetic) energy ,;.
The components
and
(electric)
+ (1 - &)
~
energy -
is
J
*}>
is
T = ^(l-^)~-J
2.
of the electromagnetic. momentum are
For a system which is symmetrical about the axis of motion For, corresponding to the last two integrals are equal to zero. o
i
p\
for any point in the field
there
is
another point, which
^
xOz, for which plane r
-^
.
have certain
^p
the reflexion of the
has the same value but
o%i
values,
in the
first
an equal
-^ oyi
Similar considerations apply to the integral
and opposite value. containing
is
^ and l
which
If
we
substitute the values of J^
and
J"2 in
the
Gx we
find
&*i
other formulae
we
get the values of W,
T and
Grx
.
For
The electromagnetic momentum in the field of the rigid The present calculaelectron was first calculated by Abraham*. which is practically identical with Abraham's, is taken from Lorentz's Theory of Electrons.
tion,
Electromagnetic Mass.
By expanding
the logarithm in the last formula
Gx
shown that
for small values of /3 the value of
where u
the velocity of the moving charge.
is
sponding expression for small.
T
reduces to
^p
may be
it
reduces to
Again, the corre-
u 2 when
/3
= u-
is
It will be noticed that these expressions are respectively
of the form
m u and Q
2 ^7w w
,
where
a constant quantity. The nature of these expressions leads us to a very important result. For they show that the electrified
is
sphere,
which we have supposed to be Devoid of mass in the it had a mass m Thus it
ordinary sense, behaves as though
folkws from *
"Prinzipien der
(1903).
.
the principles of electromagnetism alone that
Dynamik
such a
des Elektrons," Ann. der Physik, IV. vol. x. p. 105
CHARGED SYSTEM
IN"
UNIFORM MOTION
229
charged body, when it moves with the velocity u, carries along with it an amount of electromagnetic momentum m Q u and of electrokinetic
^m %
energy
2 .
Moreover
momentum and
this
energy
remain unaltered so long as the velocity of the body is unaltered. But this behaviour is precisely what characterizes the motion of a so-called material particle whose mass is would have to be modified for
results
m
Q
.
It is true that our
particles
moving with
comparable with the velocity of light. But they would nevertheless be exact, within the limits of accuracy of measurevelocities
ment, for such relative velocities as have been imparted to any On the other hand considerable material masses in the universe. the differences should be perceptible in the case of the very rapidly moving charged particles emitted by the radioactive substances, and, as we shall see, it is precisely the properties of these particles
which have confirmed the results of the electromagnetic theory.
The
idea of electromagnetic inertia, which is due to J. J. Thomson*, is fundamental to the electron theory of matter. For
opens up the possibility that the mass of all matter is nothing than the electromagnetic mass of the electrons which certainly form part, and perhaps form the whole, of its structure. It it
else
obviously opens
dynamics.
Our
up the
possibility of
This will be considered
an
electrical foundation for
later.
calculations so far have presupposed
that the moving and has body possesses always possessed a constant
charged
A
fuller discussion of electromagnetic velocity in a straight line. mass involves the consideration of bodies undergoing acceleration
and
such cases the results which we have obtained are not
for
This difficulty is one which is peculiar to the electromagnetic theory and arises from the fact that when a charged body is accelerated part of its energy travels off to infinity in the form strictly true.
of electromagnetic radiation.
Longitudinal and Transverse Mass.
We
shall,
however, see in the next chapter that, provided the body is sufficiently small compared with its
acceleration of the
velocity, the values for the * is
Phil.
Mag.
vol. xi. p.
of this character
energy in the
229 (1881).
The
field
it
for the electro-
mass of ordinary matter contains appears to have been
idea that the
on account of the electrons
and
CHARGED SYSTEM IN UNIFOBM MOTION
230
which we have calculated above, will only magnetic momentum, values true the from differ by amounts which are exceedingly small.
Under these circumstances, which will be stated more later, the formulae which we have obtained for constant
precisely
systems moving with varying velocities. The state of systems which satisfy this condition has been called
velocities
may be applied to
by Abraham* quasi-stationary.
Assuming that the quasi-stationary condition is satisfied we now consider the behaviour of the moving electrical system under the influence of an accelerating force. shall
There are two cases to consider: (1) when the force motion and (2) when it is perpendicular to
direction of
is
in the
it.
Any
We shall suppose be compounded from these two. the moving charge to be placed in an external field whose action If we consider gives rise to the accelerations under consideration.
other case
any
may
the whole electrical system, will vanish, so that the force exerted
infinitely distant surface enclosing
the Maxwell stresses over
by the external
field
it
on the moving charge will be equal to the
rate of diminution of the electromagnetic ternal field. But the total momentum
momentum
of the ex-
whole system must be equal to the
of the
remains constant, so that the external force
momentum of the moving charge. Since massless it follows that the force exerted on is, by hypothesis, the moving charge by the external field is equal to the rate of
rate of increase of the this
increase of the electromagnetic momentum of the This charge. equality is clearly a vectorial one and is therefore true for the different
components of the force and the momentum independently
of one another.
V----/v^
.
In the case of a
force acting in the direction of
motion we
evidently have
since
Gx
only contains
with the equation accelerations
is
Fx =
t
implicity through
m^~ we
see that the
x.
mass
Comparing
this
for longitudinal
When
CHARGED SYSTEM
IN
transverse
to
the force
is
UNIFORM MOTION the direction of motion the
begin to describe a circular orbit with constant speed. body If r is the instantaneous radius of curvature of the path, the rate will
of change of the momentum Q is always directed along r and, by the principle of the hodograph, is equal per unit time to Gu/r, where u is the instantaneous velocity. Hence
But the instantaneous acceleration is also along r and equal to u-/r, whence it follows that the electromagnetic mass for transverse accelerations
is
T^-^..,, '
The mass
equal to
^^
= C=l
..................
for transverse accelerations is therefore different
from
the mass for longitudinal accelerations. This difference was first pointed out by Abraham*. It happens, as may easily be verified, that the difference between the longitudinal and the transverse electromagnetic mass becomes vanishingly small for small velocities. properties of an electrically charged body which we have been considering have a close analogy in hydrodynamics. Any
The
geometrical figure moving in a fluid sets the surrounding fluid in motion. In the steady state when the figure moves uniformly in
a straight line the fluid motion
is
carried along
by the moving
were rigidly attached to it when the state of though motion changes, waves are set up and part of the energy of the system is radiated away to great distances. We shall see in the next
figure as
it
;
chapter that this also has its counterpart in the electrical case. Confining ourselves to the case of uniform motion, in the steady state it is found f that if the moving figure is intrinsically massless nevertheless possesses inertia and behaves as though it had a certain mass coefficient which is a function of the mass of the fluid it
In the case of a massless sphere moving in a displaced by it. perfect fluid this apparent mass is one-half of that of the fluid displaced by the sphere. When a circular cylinder moves at right angles to its length the apparent mass is equal to that of the fluid displaced by the cylinder. *
Loc. dt.
In the case of unsymmetrical figures
CHARGED SYSTEM IN UNIFORM MOTION
232
the .apparent mass
no longer the same
is
The hydrodynamical
directions.
for
cases differ
motions in different from the electrical
mass ones in one important respect the apparent the of of the velocity moving figure.
is
always in-
;
dependent
This analogy was very clearly seen
by
J. J.
Thomson
in the
memoir* in which he developed the idea of electromagoriginal We shall take the liberty of quoting his exact words: netic mass. "The charged sphere will produce an electric displacement throughand as the sphere moves, the magnitude of this out the field ;
will vary. Now, according to Maxwell's displacement at any point electric the in displacement produces the same theory, a variation a field in which electric currents and current ; effect as an electric
exist is a seat of
energy
;
hence the motion of the charged sphere
has developed energy, and consequently the charged sphere must as it moves through the dielectric. But as experience a resistance the theory of the variation of the electric displacement does not take into account anything corresponding to resistance in conductors, there can be no dissipation of energy through the
medium hence ;
the resistance cannot be analogous to an ordinary frictional resistance, but must correspond to the resistance theoretically experienced
by a solid in moving through a perfect fluid. In other words, it must be equivalent to an increase in the mass of the charged moving sphere." The Contractile Electron.
We have
seen that the determination of the field due to a rigid spherical shell of electrification in motion can be reduced to the determination of the electrostatic potential due to a certain ellipsoid.
ellipsoid
We have pointed out already that the particular which we have been led to consider as the equivalent upon our supposition that the spherical shell the negative results which have been obtained in
fixed system depends is
rigid.
a
number
Now
of optical experiments on
moving systems,
instituted
largely in order to try to detect relative motion between the system and the luminiferous medium, seem incapable of explanation except on the hypothesis, suggested by FitzGeraldf, that, on
account of the motion, the matter of the testing system undergoes *
Phil Mag. V.
vol.
n. D. 230 (1881).
+ Nntiir*. Inn*
Ifi
CHARGED SYSTEM IN UNIFORM MOTION The negative
contraction in the direction of motion.
233 results are
immediately accounted for if this contraction is in the ratio of 1 to Vl - J3r, where /3 is the ratio of the velocity of the system to the velocity of light. The dimensions transverse to the direction of motion are supposed to be unchanged (see Chap. xin).
Let us suppose that this contraction affects the electrons as The resolution of the problem is well as the material as a whole.
due
much
It is in reality
to Lorentz*.
simpler than that of the
For the shell which was spherical when at rigid spherical shell. rest becomes an oblate spheroid when in motion, the polar axis coinciding with the direction of
R Vl
2
/3
,
where R is
unchanged and equal
torial radius is
arises as to
be obtained
what
is
we
if
motion and being equal to
The equaThe question now
the original radius of the to R.
shell.
the corresponding fixed system /Si. This will multiply all lengths parallel to the direction of 2
motion by 1/Vl
leaving the perpendicular directions unchanged. Thus the corresponding fixed system is simply a sphere of radius R. The potential fa is symmetrical in the distorted /3
,
space of the fixed system and equal to e/^irr^, where distance from the centre of the sphere in this system. trostatic
is
energy
But everything
is
^Pl
7
also
Gx = -
Vl - /3
Gy = Gz = 0,
the
elec-
symmetrical about the centre of the sphere
III U fVl /%v.dTl= n'(^7) dTi=lll(-) dxj
6-rrcR
is
thus
in the fixed system, so that
and
i\
The
'
2
as before.
\
7
/
/
\
1
dr lt
CHARGED
234
SiiSTJJim UN
u JN JUB UJCUM.
For longitudinal accelerations we have
*
so that the
where
w
longitudinal
a-
mass
the value of the mass for small velocities.
is
For transverse accelerations
(29).
Thus the
ratio of the longitudinal to the transverse
mass
is
^ = (l-^)-i
(30).
m.2
These formulae are
all
simpler than the corresponding ones for
the rigid electron.
The Experimental Evidence.
We saw in Chapter I that the experiments of Thomson, Wiechert and others had established the existence of particles whose charge per unit mass e/m had a value some 1800 times as The smallgreat as that for an atom of hydrogen in electrolysis. ness of the mass of these particles relative the with together largeness of their charge suggested them as a likely field in which to look for experimental evidence of the existence of electromagnetic mass.
A means
for the detection of
the latter
is
furnished
by the fact that electromagnetic mass is a function of the velocity of the moving charge, whereas the ordinary mass of the Newtonian scheme
is
assumed to be independent of the velocity of the
particle.
CHABGED SYSTEM IN UNIFORM MOTION mass is small except when the the with velocity of light but it happens, velocity comparable in the that /3~rays from radioactive substances we fortunately, It is true that the alteration in the is
;
have moving electrons whose velocities vary widely but extend almost up to the velocity of light.
The first successful experiments in this direction were made by Kaufmann*. Working with the /3-rays from radium bromide he made use of a device similar to that of crossed prisms in experiments on dispersion. His apparatus is shown in Fig. 30 a.
A speck
o'f
radium bromide was placed at the point
immediately
below the parallel plate condenser of which PjPs represents a section by the plane of the paper. Immediately in a above t 2 was a minute hole
PP
D
The line thick metal plate. and in the vertical, plane
OD
was
midway At
between the condenser plates. some distance above D was a
hori-
A
zontal photographic plate. suitable difference of electrostatic potential was established between the plates of
the condenser, and the whole system
was placed between coils designed produce a uniform magnetic field and lying in planes at right
to
Thus angles to that of the paper. the particles were acted upon, during their passage between the plates, by a horizontal electric force lying in the plane of the paper. If this field alone were operative the particles passing and would through the points
D
paths when between
pursue parabolic the condenser plates the para2 2 bolas lying in the plane of the paper.
P P
Fi s- 30 a
-
,
After escaping from the condenser the would be rectilinear and along plate subsequent path the tangent to the parabola at the point of Thus under escape. Gott.Nachr. 1901, Heft 1; 1902, Heft 5; 1903, Heft 3; Phys. Zeits.
p.
55,
CHAEGED SYSTEM IN UNIFOBM MOTION
236
the influence of the electrostatic field alone, a group of particles the same velocity would be deflected to the right or to all
having
The spot would be the field. according to the direction of Since the shifted from the centre of the photographic plate. deflection depends upon the velocity, if the group consisted of
the
left
different velocities the spot particles having of the paper. line a into lying in the plane line
would correspond
would be spread out Each point on this
to a particular velocity.
the magnetic field run from left to right in the plane of the paper, so that if it alone were operative it would to describe circles and cause the particles which pass through in a plane perpendicular to the plane of the paper. Owing to the the spot would be displaced in the plane then, field, magnetic of the paper, if all the particles had perpendicular to the plane the same velocity, and would be drawn out into a line in this
The
lines of force of
D
had different velocities. plane, if the particles should expect to get a once we at are operative
When
both
fields
curved line on the
the point of photographic plate, each point of which represents impact of particles having a certain velocity. The position of each deflection point gives us, of course, the magnetic and electrostatic It remains to of a particle with a definite but unknown velocity.
be seen how we
may deduce from
the measured displacements the
value of the mass as a function of the velocity of the
moving
particle.
For the sake of simplicity we shall suppose a uniform electric to D and extend, from left to right, all the way from is uniform and then to cease absolutely. The magnetic field
field to
H
parallel to this electric field all the
way from
to the photographic
plate P.
Taking
D
as the origin, let the coordinates of be 0, 0, z, and be #, y, /. Let the axis of y
those of points on the plate lie in the plane of the paper, Assuming that the deflections so far as the
P Y
denoting the electric intensity. may be treated as small, we have,
motion in the plane of the paper
m || =
and
ot~
m^ =
Ye,
from
ot"
Ci2
and
m -^ = 0, "
from
D to
P.
is
concerned,
to D,
CHARGED SYSTEM IN UNIFORM MOTION Thus
x-
= const. = w
,
01
where
w
is
=
when the
Hence
t
of the velocity of projection,
w t 4- const. = w
z
and if *
OD
the component along
particle leaves 0.
=
mwfy =
Thus Since y
= when
and
B=
also
and
A
when
z
=
;
-
Thus
mw
and where ^ is the angle which the tangent to the parabolic path at D makes with OD. Since the subsequent path is rectilinear the displacement
y' at
the photographic plate
y = (/ - * The projection
i}
tan e
is
=
of the path on the plane of
xOz
the magnetic field and is a circle passing and to the degree of through approxima-
D
tion of this calculation.
of this circle
and
is
controlled P,
P
P'
s
s'
Let r be the radius
S^^ (Fig. 30
b) the tangent to it at the point Sl which is symmetrical with respect to and D. and Pl are the
D
-
1
intersections with horizontal planes through P' (#', 0, z) is the inter-
D and P respectively.
section of the projected trajectory on the
photographic plate. S' is the intersection of a vertical line P'S' and a horizontal line
S
is
the mid-point of OD.
Then O
8r
Fig. 30 b.
by
neglecting
$&
compared with
r,
and
'
2r neglecting
P P' compared with a
Thus
r.
- fii? .~.L> '.
The radius
of curvature of the circular
the balancing of the centrifugal force magnetic field on the moving particle.
when r
is sufficiently large. ,
Comparing
this
path
is
determined by
by the force exerted by the
Thus
Hence
Hez
,
,
with the electrostatic deflection
y' }
we have
x'Yz, and eliminating
W
Q
m_y H
2
e~~Hi~-~~Y
z-(z2i)
,nn\
\^/*
2#
These equations are only to be taken as illustrating the principle of the method. Owing to the fact that the electric field does to the point D and that not extend all the way from the point the electric and magnetic fields are not quite uniform, the actual treatment of the experimental data is somewhat different. For these refinements the reader
The experiments
We
T.
must consult the
original paper.
are carried out with certain values of
see from equation (31) that if
we take the
H
and
undeflected
spot as origin and the axes of magnetic and electric displacement respectively as the oc and y axes, then the ratio of the coordinates
of any point on the curve on the plates determines a certain value of the velocity W Q of the particles. Substituting the same values r
we obtain the value of m/e possessed the whose initial by particle velocity had the particular value found. .We can thus find the value of m/e for all the previously of x and y' in equation (32),
velocities present
among the
/3-ray particles.
Since
it
is
highly
CHARGED SYSTEM IN UNIFORM probable from other considerations that as a function of w way able to find
m
From
e is
constant
we
are in this
.
the results of Kaufmann's experiments
it
was immedi-
ately clear that the mass of these particles increased with increasing velocity and that the variation was in good agreement with the
formulae which had been developed on the supposition that the origin. A careful re-exami-
whole of the mass was of electromagnetic
nation of his plates led Kaufmann to the conclusion that his results agreed better with the formula of Abraham for the rigid electron than with that of Lorentz for the contractile one.
The
graphs of the two formulae are however not very different for the range of velocities embraced by the /9-rays used, and it seemed at the time very questionable whether the experiments did not agree with the results predicted by both formulae within the limits of error.
experimental
ment mass
It is to be borne in
tests only the transverse
mass and
mind that tells
this experi-
nothing about the
for longitudinal accelerations.
Bitcherers Experiment
A
very ingenious experiment to test the different theories of the constitution of the electron has been carried out byBucherer*.
Fig. 31.
A speck of radium fluoride, which contains more radium per gramme than any other available compound of radium, is placed at the point R (Fig. 31), which lies at the centre of the lower plate of a circular The distance between A and B is parallel plate condenser AB. very small compared with the dimensions of the plates. and B are maintained at a suitable difference of
A
The
plates
potential and are placed, so that their planes are horizontal, in the centre of a vertical cylinder. CD is a section of the walls of the cylinder by *
Ann. dcr Phys. IV.
vol.
xxvm.
n.
513
A
the plane of the paper. photographic film extends all the way round the inside of the
The whole apparatus
cylinder.
is
exhausted
placed in a uniform magnetic field of the paper. perpendicular to the plane
and
is
H
To simplify the discussion of the experiment we shall suppose the distance "between A
B
and
be so small that
to
it is
quite negligible.
In the absence of the electric and magnetic the /3-rays travel in straight lines all the and their trace on the photographic from
fields
way
R
a circle in the plane of the condenser. This circle becomes a straight line, of course,
film
is
when the
When
film is unrolled.
are applied the paths
the
fields
become more complex.
Between the
plates they are still horizontal after escaping they describe but straight lines, their axes along the with circular spirals
The rays will direction of the magnetic force. only be able to escape from the plates provided the downward pull of the magnetic force is equal to the upward pull of the electric field. we consider a particle starting out with velocity w in a direction making an angle a If
with that of the magnetic for
compensation
field,
Xe = Heiv sin where plates.
X
is
the condition
is a,
the electric intensity between the for the particles to escape from
Thus
the condenser
__ c
He sin a
In Bucherer's experiments this formula was
H
by taking X and Under these circumstances
tested
Fig. 32.
so that
After leaving the plates the particles follow a spiral path of which ST may be regarded as
CHAEGED SYSTEM IN UNIFORM MOTION
241
Consider the position the projection on the plane of the figure. T as a function of the angle a. When w is equal
of the point
to the velocity of light, /? has the
maximum
value unity.
For
particles moving with this speed the theoretical magnetic deflection is zero on account of the electromagnetic mass being
This follows from the formula for the radius of curvature
infinite.
= mw/He. We
should therefore expect the trace on the film to coincide with the undeflected trace (i.e. the rectilinear trace r
obtained in the absence of the
sina
=
fields)
or a
when
=30
150.
or
This was immediately verified. The maximum deflection occurs when a = ?r/2 or /3 = | with the electric and magnetic fields used. of the experimental trace is shown in Fig. 32, the very dark line being the undeflected position. None of the rays escape
The form
from the plate except when the value of a and 150.
lies
between 30
=
The values of the maximum deflection when a 7r/2 enable us to distinguish between the different formulae for the mass of an Since these particular particles always move at right electron. the to magnetic field, their paths are circles. Let r be the angles a be the difference between the radii of the cylinder CD and the plates of the condenser AB and let z be the perpendicular radius, let
Then
deflection from the undeviated trace.
z
= a~l(%r
z\
z H
(ct~\
.
J
f = He m = Her ~~- z-\ w
Thus If
zw
m
is
the mass of the
velocity
(w = 0),
Lorentz,
we have
then
\
moving
a2 z
,
electric
charge
for the contractile electron
/,,= 1(1 -#)-*, and substituting the value of e
2cz
/3
m given
above we find 2c
for
zero
suggested by
CHABGED SYSTEM
242
With Abraham's
IN
rigid electron,
on the other hand,
3 28
2cz
e
UNIFORM MOTION
-tanh
25]
|
m,~~ H(a*
where
j8
The
+ z*)\!S
tanh2S
.(34), j
= tanh 8. relative constancy of the values of
e/m calculated from the of equations (33) and (34) means by experimental enable us to distinguish between the two respectively should The data yielded by the experiments are given in the formulae. Q
deflections
accompanying table
:
In the case of the Lorentz electron the value of e/m Q is constant within the limits of experimental error, whereas in the case of the rigid electron the deviation is much greater than can be accounted for in this way.
These experiments appear to dispose effectually of the rigid and they may be regarded as making it reasonably certain that Thomson's corpuscles are devoid of mass such as is due electron
except
to the charge that they carry. For this reason refer to them in the sequel as negative electrons.
,
We
shall find later
we
on that the relation between
characteristic of the Lorentz contractile electron is
,
!
i
electrical
shall always
m
and
true of
ra all
systems according to the principle of relativity. Bucherer's may therefore be regarded as evidence in favour of
experiment
that principle.
OHAPTEE
XII
CHARGE MOVING WITH VARIABLE VELOCITY IN the case of a charged body moving with variable velocity the field is no longer carried along as though it were rigidly attached to the moving system. That is a state of things which In order to deteris characteristic of uniform rectilinear motion.
mine the
state of the field in other cases our only recourse is to
evaluate the potentials
We
direct integration. shall see how this may be done in the case of a point charge, that is to say of a charged body all of whose dimensions are negligible compared with the other dimensions
by
entering into the problem.
Let us seek the values of the potentials at the point P It will be remembered that (Fig. 33) at a certain instant, t. the values // and (Vp)' which enter into the integration are not the values at the element dr at the instant
t
for
which we are
P
but at a certain instant, say t seeking the potentials at 0, which differs from t by the time which is required for a disturbance travelling with the velocity of radiation to pass from the element = c6 if r is the distance of volume dr to the point P. Clearly r from dr to and c is the velocity of light. shall the }
P
We
suppose
*
coordinates
of the
moving charged body to be given as a of function the time t - 6 at which the disturbance explicitly leaves it. If S denotes partial differentiation when t and 8 are considered as independent variables, we have Sg SB
f,
=
77,
<%
d(t-8)_
d(t-9)'
W~~ '
S0
d(t-d)'
W
with similar equations involving
S_
dl;
d(t-6)~
___ __ d(t w.
r>. ri
-0)~
and
1. 1.
.
St~~S' St
't.
CHABGE MOVING WITH VARIABLE VELOCITY
244
At the
outset
from
arises
are face to face with a
we
the fact that the
new
difficulty
charge pdr which enters
which
into the
the potentials is not all present in the element of volume at the same instant. Thus the charge pdr which occurs in this element of the integral is not equal to the integrals
expressing
true charge which would occur in this element
Consider the truncated cone is
terminated by the spherical surfaces
if it
were
at rest.
P and which apex AOSC and BTRD whose
TOSR whose
is
at
AC
The sphere represents of the instantaneous position contributing charges at the instant 6 6 and the sphere the instantaneous position at t dd. t radii are cQ
and
c (6 -f
d6) respectively.
BD
Thus the part
P from
of the charge which contributes to the potential at OS of the element of volume dr is
the front-end
present
there at a later time than that which contributes from the backend TR. All the charges are, however, present in the displaced f / element dr' = O S at the one instant provided
RT
=
t-9-d8
Hence the true charge de which is element dr is, if X is the angle between r and F, 00'
Vd0.
de *
Thio
r^otllf io
= pdr = pdr
(1
- /3 cos X)*.
effective in the
CHARGE MOVING WITH VARIABLE VELOCITY For, since
dr and
have the same base,
dr'
_
_
We
CQS
-
*
OT
fc~
cd9
therefore obtain the following expressions for the potentials
:
m
''* k
and
for
;>
a point charge
__ ~~--' *
,
(p
rr
47rr(l--/3cosX)'
The dashes 'denote that the
4WC (1 - ^8 cos X)
...... V
;
*
values are not the instantaneous ones
We
= r/c
have assumed in this previous. in p may be reinvestigation that the effect of any real change time dd. the small as during garded negligibly
but those at a time 6
Having obtained
now
are
to
<j>
and U, the
electric
and magnetic vectors
be obtained from the equations
H = rot U.
and
The
differentiations
which enter into these equations are at
P
The fixed with respect to the axes of reference. t. The z and now variables are x, potentials independent y, involve #, y, z partly through r and partly through d, which is the point
a function of functions of
f
,
77,
f
and
x, y,
(t
z
9).
when
t is
They
also indirectly
cosX=
fixed
also
depend on
through
{(a?-
r(l~/3cosX)=:r-i{(^ 6 Let us seek the value of
0.
77, f which are both directly through have
and involves
We
t
CHAKGE MOVING WITH VAKIABLE VELOCITY
246
We
have
/J
*-(/
fc\ fc _]_
/
\
3#
Similarly
3^7
a o
= ~ V..86) 5-
,
8?
oa;
Hence
and -
or
Thus
The
expressions for
-
and
-
may be
written
down from
CHARGE MOVING WITH VAKIABLE VELOCITY
d(t-ff> ~
Similarly
Hence
J
-
{(#
f?)
f
+
(#
and
_
_ec
_^_
*?) *7
+ (*
247
CHARGE MOVING WITH VARIABLE VELOCITY
248
The
expressions
for
Ey
E
and
may be
g
down by
written
inter-
changing the axes.
be observed that of the three terms in the expression Let us for Ex the first does not involve the acceleration f 77, Its comof the electric intensity separately. consider this It will
,
.
part
to ponents are proportional
(y-
(a-f-0|),
17-017
resultant of this part of the electric intensity where O l is is therefore proportional to and directed along Q^P, would the which occupy if it conthe moving charge
The
respectively.
position
tinued from the instant t-0 to move uniformly during the which it had at that rj interval 6 with the velocity ,
following
We may
t
E
l of the part of the electric intensity which does not depend upon the acceleration in the form
instant.
therefore write the resultant
we work out the value of the magnetic intensity from the also expression H-= rot U we find that there is a part of that If
which
we
is
independent of the acceleration.
Denoting
this
by
jETj
find
=
j?;
where
\
is
the angle between
#! sin X x
OjP and the
tangential to the circle passing and whose centre is dicular to
V
........................ (4),
direction of V.
H
l
is
P
in the plane perpenthrough on the direction of V.
In the case of a particle which moves with a uniform velocity, the expressions just given will represent the whole of the electric
and magnetic
E
l
E
In that case
intensities.
it is
convenient to express ^ and the angle it
in terms of the instantaneous radius
makes with the direction of motion, rather than The change is easily made. We have
in terms of r
X
n/sin
X=
r/sin
00 = VO =
and since
l
rf
X
x
fir,
= r + /3 r - 2/3r cos X = r (1 - /3 cos X) + /3V 2
2
2
=r
2
2
2
2
(1
2
- B cos XV + /3 r, 2
2
2
sin 2
\
sin 2 X,
:
and
CHARGE MOVING WITH VARIABLE VELOLTix r (1
hence
- /3 cos X) = n (1 - /3 47rr1 2
(l-/3
sn These formulae were
first
sin2 A : )*
-^L=2-
!!-% =
and
2
sin 2
2
<-
............... (5),
s
X )2 1
V
given by Heaviside*.
In the case of uniform rectilinear motion the resultant electric from P to intensity at any point P is directed along the radius
O of the moving maximum, .that "is,
the instantaneous position maximum where sinXi is a plane,
and
a
is
l
minimum where sinX = 0, 1
that
It is a charge. in the equatorial
is,
along the polar
true of the magnetic force, which howThis state ever vanishes in the neighbourhood of the polar axis.
axis.
The same thing
is
of affairs has been described
by
J. J.
Thomson
in the statement
that the lines of force due to an electric charge in motion tend to concentrate in the equatorial plane. In the case when the
becomes equal to the velocity of light the concentration Since ft is then equal to unity the force vanishes at complete.
velocity is
every point outside of the equatorial plane. It will be observed that this part of the force varies inversely as the square of the distance from the moving charge and is therefore inappreciable at big distances. Acceleration.
Turning to
we
E^
E which
the part of
see that there are
two terms
in this,
involves the acceleration,
and their components are
proportional respectively to
(*-f-0f), and
f,
.
77,
part which
Thus is
E
(y-i7-0$),
may be
a
directed
_
may be
is
0,P
,
written
eP cos
fji
1-0 cos where p
regarded as being made up of a together with a component If we call the acceleration (f 77, =F f)
along
parallel to the acceleration. this result
lies
e
^"
YJ5 l
=<
the angle between r and
E, obviously
(*-?-0)
f
r
"
'
(<})
'
.
in the plane of the
radius
{
P
and tho
CHAKGE MOVING WITH VARIABLE VELOCITY
250
acceleration. if
It
is
r = OP. For by the corresponding
also at right angles to the radius
we multiply each component
of E% in turn
direction cosine z
or
of
OP, we
find, for
the numerator of the sum,
x
Thus Ez is at right angles and the acceleration. If
r
to
[(*
OP
-
2
)
+ (y -
and in the plane containing
X
P
we work out the value
H
J? 2 of the part of the magnetic which involves the acceleration we find that this is
intensity at right angles both to
OP
and to E* and
is
equal to E% in
magnitude.
Thus the part of the electromagnetic field which depends on the acceleration of the particle is specified by two vectors, the electric
and magnetic
intensities.
These are mutually perpen-
dicular and, in our units, are equal to one another in magnitude. to They are both at right angles to the radius from the point
P
the position
of the particle at the instant at which the state of its motion determined the field at P. The vectors may thus be said to be at right angles to the line of flight of the electromagnetic disturbance.
There the
is
another very important difference between the part of
which depends on the acceleration and that which does We saw that the electric and the magnetic intensity in
field
not.
the latter were both inversely proportional to the square of the distance from the In the former both the moving charge.
So inversely as the first power of this distance. that at great distances from the the of moving charge the part
intensities are
field
which depends on the acceleration will become verv preat
CHARGE MOVINO WITH VARIABLE which does not depend upon it. The compared with the part have considered the of this will be clearer after we
importance
field. distribution of the energy in the
Energy
We know the
field at
in the Field.
volume of that the electromagnetic energy per unit
any point
is
and (^^2) denote the scalar products of the vectors. The energy per unit volume may thus be split up into three
where
(fixfia)
parts,
U,
and
Following Langevin* we energy separately.
U, =
W
= i {HS + E$, = J7,
(H&) + (E.K),
+ -&}.
shall consider these three parts of the
CHABGE MOVING WITH VARIABLE VELOCITY
252
We
field at
have seen that the
the state of the
determined by
when it was at a point The same statement is through P with
at the time
charge at the instant
moving
true of the field at
all
t
t is
-
6,
OP = r=c9.
where
distant r from P,
The
P
the point
points on a sphere
thus be regarded as the moving charge in all directions with the from out spreading 2 on OP the uniform velocity c. If we consider another point as centre.
field
may
P
field at
P
no longer be determined by the state of the
will
2
moving charge when
at 0, because the disturbance initiated at
P
The field at has already passed over P 2 and reached P. instant t will be determined by the state of the particle some point
The
at a
2
time later than
t
6,
let us
say
2
at the
when
at
+ dd.
t
points at which the field at the time t 9 4- d9 the state of the particle at the time t
of the
is
determined by the sphere PZ R^ whose centre
is
locus
At the time
t
is
2
and whose radius
is
the state of the field in the excentric shell bounded
by the two excentric spheres whose radii are
OP = cd
and
2
P =c(0-d9) 2
be determined, at every point, by the state of the moving 6 and t 6 -f- dd. charge at some instant between the times t
will
We U l3
shall U.2
now
and
U
3
consider the energy of each of the three types which is present at the instant t inside the
excentric shell.
The energy
of the first type per unit
f
-f
H?}
32-71* 2
e (1
- /3
32-7T 2
(1 4- /3 sin
^(l-jgcosX) 2 2
)
2
2
is
\)
XO
6
n + /3V sin X r (l-cosX) 1 + 2/3 - 2/3 cos X - ff 2
2
2
8
2
_e (l-/3 ) 327r s r*
by
2
a 3 7^(1 -/3 sin
327T-
__
= 4^
volume
2
2
(l-/3cosX)
8
cos 2 X
B
making use of the various relations on pp. 248 and 249.
The energy
of this type within the excentric shell
is
ff f J7,
dr
CHARGE MOVING WITH
VAIUAJ>.-~
of the shell taken throughout the volume of volume nates r, X, the annular element
27rrsinXxrdX x
Now OP = 00, pendicular
from
P = c (0 - d6\
2
2
2
and
PP
2
Q
if
Using polar eoonli is
.
is
the foot of the per
on OP,
In the limit when d8
is
made
to vanish,
QP, = Oa Pa
Thus
.
^ C 0^ c (e-d0)~ c/3 cos Hence rrr
JJJ
^ dT=
# (l -
~
-
^ Jo "Ife^ )
cd
f-
(wh<;r(!
a:
=
?
\
^ ft
X sin
('(isXK
the value of the part of the energy, wlijch does nni. at th< time / IM*|.\VM*U depend upon the acceleration, which is found It will and are off dff). the excentric spheres whose radii <;(() of the be observed that for a constant value of odd this part, energy
This
is
It is therefore negligible ;il, varies inversely as the square of r. a great distance and is practically all concentrated in the imme. diate neighbourhood of the moving charge.
In the case in which the charge has always IHMIJ in motion with the uniform rectilinear velocity F"=/9c, all the energy is this type. The formula we have just obtained enables u,% to
!'
the total energy of this type which lies outside a sphere of radius 7? described about the moving rlinrge. The smallest of the excontric spheres will have the ijistanhmemi:.
calculate
small
position of the moving charge, that time t as centre.
to say its position
a{
tin-
All the other spheres, see Fig. *$;"">, are external to this our behind its centre () since we are supposing thai
and
is
}
their centres
charge
is
moving with a
(] ,
velocity
which
is
smaller than thai
tini,f
When this is done the as centre. that they have the point same as when volume of the shell between any two spheres is the field is clearly the the in total The energy were excentric. they
was spread uniformly as if the energy of each excentric shell shell. It is concentric over the volume of the corresponding
same
therefore >
3+j8
a
dr
e
2
2
4/3 ...(8).
Fig. 35.
In our units the potential (electrostatic) energy outside the must therefore be sphere of radius R is e^/SirR. The remainder the energy of the magnetic field of the moving charge. Since the electrostatic energy is the same whether the particle is at rest or in motion, the magnetic energy is the same thing as the kinetic thus energy of the moving charge. The kinetic energy is
CHARGE MOVING WITH VARIABLE VELOCITY
255
regarded as the part of the kinetic energy of the moving point charge which lies outside the sphere of radius described about it as centre. In the case of an extended
may be
This
B
distribution of electric charge such as the spherical shell considered in the last chapter, it is necessary to calculate the combined effects of the superposed fields due to the different elements. The calculations are very complicated and can only lead to the results obtained previously, so that we shall not pursue the matter further
in this direction. It is important, however, to emphasize at this stage the two features of the part of the energy of the field of the moving charge which does not involve the acceleration. In the first place
main
the whole of this energy is located in the immediate neighbourhood of the moving charge arid is carried along with it in its motion. In the second place its magnitude is a function of the
geometrical distribution of the charge. Both are clearly estabby the discussion immediately preceding and by the results This part of the field has been called of the last chapter. by lished
Langevin the velocity wave of the moving charge.
We
shall now consider the part of the energy which depends E2 and 2 The amount of this which is present beon only tween the two excentric spheres whose radii are cff and c (6 dd) may be calculated as follows. Since Ez = 2 the energy per unit
H
-
H
volume
is
Now E
2
is
the resultant of two components, one along
(1
and the other
parallel to
- j3 cos X)
;i
P
and
-
-
eFcos/x s
X
l
'
F and />
and
1
<
To
find
(1)
X
the
value
parallel to
of
00
l}
E? (2)
resolve
Y
E
2
into
three
components,
perpendicular to 00^ and parallel
1YHJ V li\ Ur
= 0,
to the plane
=
plane
and
Then
0.
(3)
since
Z perpendicular
00 = /3r we
eY
r_
V
VVJLJLja.
(cos
/z
to
00
(cos
cos 4-7rc
where
e
is
2
also to the
X- ,6) .
sm
fji
""
and
2
2
r(l-/3cosX) j l-/3cosX sin X cos (cos Y'= 1 - ft cos X cos X) 2 ( 47rc 2 r (1 47rc
1
have
r (1 -
cos X)
2
fji
sin
...(10),
X sin
- ft cos X
1
F and 00
the angle between
l
.
We
have now
to
evaluate the integral
dr
where
'
r* (1
- /? cos X)
sin
\d\d$cdd.
__ Fig. 36.
The to
limits of integration are
:
for X,
from
to
TT
and
for
,
from
2?r.
Since
cos
/JL
=
cos X cos e
-f
r%TT
and
sin
X
sin e cos
,
r-2
d
= %7r,
Jo
Jo 27T
and
c
o
after
ft cos X, we changing the other variable from X to x = 1 of this between the excentric energy type spheres,
find, for the
the value I JL
This result r
it
__
is extremely important. Since it does not involve shows that the energy between the spheres remains constant
CHARGE MOVING WITH VARIABLE VELOCITY
257
6 and provided they always correspond to the same instants t t9+dQ Thus as the spheres travel outwards with the velocity c they always enclose the same amount of this energy. The total t
energy of this type which is emitted by the accelerated charged particle travels off, in undiminished amount, in all directions to infinity with the
uniform
velocity
regarded as belonging to the the surrounding space and
c.
This energy can no longer be
moving system. is
It is transferred to
in fact the energy radiated
by
the-
particle.
Since d Us
2 proportional to F we see that there is no radiaThere is no tion except when the moving charge is accelerated. motion. uniform in Moreover a the from radiation radiation particle
is
A
P is positive or negative, retardation acceleration. This is true not only for
has the same value whether
has the same effect as an
small velocities but also for velocities which are comparable with that of light.
Our any
readily be extended from a point charge to geometrical distribution. By considering two spheres
results
finite
may
such that both r and dr(=cd@) are large compared with the dimensions of the charged system, it is clear that the radiation is
independent of the distribution of the charge within the system, provided the acceleration is the same for all parts of the system. This establishes an important difference between the radiant
energy and the part of the energy which is independent of the For we saw that the latter part depended very
acceleration.
directly on the geometrical distribution of the electrification, being in fact a linear function of the electromagnetic mass of the system. On the other hand, the energy radiated by a charged system does
not involve the electromagnetic
When
mass of the system.
the system contains charges of both signs, and also when have different accelerations, the radiation will in
different parts
general depend on the geometrical distribution; but, in any event, the influencing factors are not those which determine the electromagnetic mass. It is clear that so far as the radiation which escapes to a great distance is concerned each electron in any When the material system may be treated as a point charge. velocities are small the
angle
e
and
it
mav be
energy in the shell does not involve the shown, in a manner rather similar to the
CHARGE MOVING WITH VARIABLE VELOCITY
258
that the radiation is proportional to (SeT) 2 foregoing calculations, taken over all the electrons or charged particles in the system. It follows from this result that a material system which is
,
not electrically neutral will accelerated, even is
though
otherwise obvious since
it
made up
clear that
is
it
is
of electrons.
uniformly This result
any system which
external field will not radiate.
have no
arranged so as to
emit radiation when
it is
is
The
2
result that the radiation is proportional to (SeT) enables us to whether a given system will be an efficient radiator see at a
glance
For instance, we
or not.
may
take the case of two equal and
in a circular orbit about the mid-point opposite charges revolving Here both the charges and the of the line joining their centres. have accelerations opposite signs so that the sum of the products ;
a good radiator. In the case of two equal negative charges revolving in the same orbit about a positive charge at its centre the charges have the same sign, and their
is
additive
and the system
is
The value of (SeT) 2 is therefore zero accelerations opposite signs. and there is no radiation from this system. There will of course the negative charges get out of phase that the vanishing of 2 be no will the condition for radiation (2eF) only provided that the summation is taken over a sufficiently small element of volume.
be radiation emitted
The is
if
It is well to observe
with each other.
result that the rate of radiation of
energy from a point charge
equal to (12) V '
for small velocities
was
first
given by Larmor*.
Some of the properties of the radiation waves will be considered in later chapters. The Wave of Reorganization.
The part
dU
of the energy in the shell, which involves the and and of and H^ may be l 2 l geometrical products of obtained in a manner similar to that used in calculating the value of dUB the same Cartesian axes as before, we find the Taking 2)
E
E
H
,
.
X Y
values of the components l} from the expressions on p. 248. *
Phil.
Maa. V.
VOl. TT.TV.
r>_
Z of E and P Q We have already found
lf
503 HftQ7\
l
f!f
l9
l
a.lan
Afh ov
/rn/7
l9
J^ of H^ JST2 ,
Urnttov
r
F ^ 2
,
9.9.7
2
CHAEGE MOVING WITH VARIABLE VELOCITY
E E
P
and and
R
259
readily be obtained 2 are equal to each other and are z The value of also perpendicular both to each other and to OP. this part of the energy per unit volume is
the components of since we know that
s
,
2,
H
Q2
,
2
may
Introducing the element of volume in polar coordinates we integrate this throughout the region between the two shells and find
This energy is a maximum when 6 = and is zero when e = ?r/2. Since Tdd is the change of velocity in the interval d9 we see that Tdd cos e is equal to the resolved part of the change of velocity in the direction of motion.
Calling this c&/3
we have
This part of the energy contains r in the denominator, so that it falls to zero as the two spheres proceed to infinity. a rather simple and important relation between the and the energy in the field when the charge is in energy uniform motion. In considering this, in order to fix our ideas, we shall suppose that the motion is uniform except during the infinitesimal interval between t 9 and t 6+d9. Before t the charge moves in a straight line with uniform velocity Vi and after t 4- dO it again moves uniformly in a straight line but
There
is
dU
2
with a different velocity v 2 Consider the energy in the field outside of the two excentric spheres. Up to the instant t this is that which is proper to the case of a charge moving with the .
uniform velocity v^
At
the instant
t
this begins to
be
altered.
A new field begins to
be established, namely that which is proper to a particle moving with the uniform velocity v2 The new field is shell it as the excentric travels left behind outwards evidently with the velocity c. The relation referred to is that the energy .
dUz is just what is required to change the field outside of the excentric sphere from the state corresponding to the uniform velocity v l to the state corresponding to
v.2
.
For
this reason the
energy dU>2 has, very appropriately, been called by Langevin the
wave
of reorganization.
CHARGE MOVING WITH VAKlA^i,E
260
VJfil.UU.LTY
We have is a very simple matter. a point charge moving with uniform velocity ^ = eft the energy Ul (ft) of the field outside a sphere of radius 6 is r about the position of the particle at t The proof
of this assertion
seen (p. 254) that for
If the speed instead of being value of Ui would be
neglecting higher powers
+ 8) -
t/i (
The
^
were
v2
= c (ft +
of 8j8 than the
us say, the
8/3), let
first.
The
difference
J7i 08) equal to the value of dETa given by (14). extension to a finite change of velocity spread over a finite
is
interval of time
is
obvious.
Acceleration and Force.
Let us consider the work done by an external force on an The law of conservation of energy requires that
electric charge.
be equal to the gain in the energy of the system, and seen that the whole energy of the system can be expressed
this should
we have
as energy of the electromagnetic field. the first consists in fact of two parts :
change the state of the appropriate to the
new
The increment d 172 which
is
of energy
required to
electric field to the condition
state of motion,
permanently and the second dUB which
represents the energy transformed into radiation. force acting on the charge we have therefore
If
Xe
is
the
l-ffsin' 2 3 (1-/3 ) since Sx
=
cyS
cos e dd.
The right-hand
side of this equation
would
represent the whole work of the 'force in the hypothetical case of an electron whose properties were those arising from that part of
CHARGE MOVING WITH VARIABLE VELOCITY
261
the field due to a point charge e, which lay outside a sphere of It is very unlikely that an electron radius r about the charge. this constitution, and where the electrification has an
possesses
extended geometrical configuration the equation has to be modified. We have seen that the second term on the right-hand which from the energy radiated is not affected by the geometrical arrangement of the charge. With the first term it is different. arises
The new value of
this may be found by considering the difference in the fields in the energy steady corresponding to the velocities before and after the action of the force. Proceeding in this way we find, in
that r
the case of any distribution possessing spherical symmetry, is replaced by the radius of the sphere multiplied by a
numerical factor. The precise value of the factor depends on the configuration of the distribution, whether superficial or throughout the volume of the sphere and,
if so,
whether uniform or
the form of the equation is unaltered even charge is not concentrated at a point.
Thus
not.
when the
electric
differs from the corresponding Newtonian Equatwo important respects. In the first place we and in the second have the additional term depending upon is never constant, although it is approximately so when m place
Our equation
tion
F = mT
in
P
/3
is
very small.
considered.
taining
law
is
This aspect of the question has already been the acceleration is small the second term con-
When
F2
will be relatively unimportant, so that the Newtonian an approximation which is true for small accelerations and
small velocities.
For the form of the Newtonian law to be preserved it is necessary that the second term should be negligible compared with the
first.
Thus if a
is
the radius of the electron
it is
necessary
that
SU
2
2(l-/3
2
2
)c /3cose
2 should be small compared with unity. Since 1 /3 may in all cases be taken to be of the order practical unity, the order of magnitude of this fraction is
Fa 2c 2 /3 cos
where T
is
e
2c
a
1
a
IV
c/Scose
2c
v cos 6
2c
the time necessary for the charged sphere to
move
CHARGE MOVING WITH VARIABLE VELOCITY
262 its
own
It is therefore necessary that the change in the the body during this time r should be small compared
radius. of.
velocity
with the velocity of radiation. If this condition
is
satisfied the
motion
will
be the same
as
that calculated on the supposition that radiation does not occur.
Abraham* has suggested that the term
quasi-stationary be applied
motion which satisfy this condition. Since the calculation of the motion when the state is not quasi-stationary is to inquire whether such extremely complicated, it is important
to states of
There is one case are likely to occur in nature or not. which is of very frequent occurrence in which the condition is states
violated,
and that
is
the case in which the velocity of the particle
This state, however, only lasts for an insignificant ,8 know from the interval of time in the case of an electron.
and
are zero.
We
value of the electromagnetic mass of these particles that a is about 10~13 cm. If the particle has moved from rest for a time t the velocity v
=
Tt,
Ta will
ft
2cvcose
and since
c
= 3x!0
10
cms. per sec. the fraction
with 10~ 13 /(6 be comparable r
x!0 x). 10
This
will
be negligible compared with unity for any measurable Even after 10~20 sec. it will have fallen almost interval of time.
clearly
Thus even in the case of a charged particle starting from seems unlikely that serious error will arise if the motion is treated as though it were quasi-stationary.
to
10~4
.
rest it
Another case in which one might expect the quasi-stationary condition to be departed from arises when the acceleration is very great. Probably the greatest acceleration with which we are familiar is that
on an atom.
which occurs during the impact of a /3-ray particle There
is
some evidence that in favourable
cases
a /3-ray particle may be completely stopped within a distance d equal to the diameter of an atom, let us say 10~ 8 cm. Assuming uniform acceleration as an illustration, the equation of energy in this case is 2 ^mv = mTd.
T = v 2/(2 x 10~
Thus and
g= *
i
(1 o-
5
)^
8 ),
=i
Ann. der Phys.
loc. cit.
CHAEGE MOVING WITH VARIABLE VELOCITY when
v
has
its
maximum
possible value
c.
Even
263
in this case the
ratio has a value which is quite small, so that it does not seem the assumption of quasi-stationary motion will lead to likely that
serious error in practical cases.
The Reaction of
the Radiation.
Although the part of the force which is neglected by the assumption of quasi-stationary motion is practically always small compared with the remaining part, there are cases in which it might exert important effects through its persistency. For example in the case of periodic motions, if the reaction of the radiation always tended to stop the motion, it would in time exert an
For this reason it is desirable to calculate the effect. appreciable an accelerated electron without assuming, as we did at on reaction the beginning of this chapter, that the increase in the velocity during the time required by the electron to move over its own radius
is
small compared with the velocity of radiation.
Owing
more general treatment we shall content ourselves with the case in which the velocity, though variable, is rectilinear, and in which the squares and higher powers of the ratios of the velocity of the electron, and the derivatives of the velocity, to the velocity of light may be neglected. to the difficulties
which
arise in a
Let us seek the force acting on an element of volume dr of whose coordinates are the moving charged sphere at the point
P
Let the variable velocity u of the sphere #', y, / at the instant be parallel to the axis of x. Let Q (as, y, z) be the position of some other point of the sphere at the time t. The part of the dxdydz which potentials at P which arises from the element dr is at the point Q at the time t will be determined, not by the instantaneous state and position Q of the element of volume, but = by its state and position Q e x e y, z at a time t #, where t.
,
PQ = [(*- O + (y - yJ + (* - *?}* = eft 2
t
We also
have
PQ = {(* - xj + (y- yj + (e,
and
_ ,_
fl
ox .__
tf
^
= r,
Since cO and r differ only by terms involving u and we can put r = c0 in the equation for
derivatives as a factor
its so e .
^ = a'-
Thus
PQ 6
in Substituting this value of x e
4-
= _
.
7
L
J
terms involving squares of small quantities.
x
u
(u
oo
M
I
|
JL
"\
7Z
I
r
|
+
we obtain
,
I
n
2c 2
\c
u (
_l
n
~T~
.>
/)'
rrl
-
6c 3
higher terms.
We also have ()x fi I
E
=
u
u dr
^
1
H
dx
c
c doc
dr
But
dr
u
dx
2c 3
7"
^
2
=
x
T
dr
n
'
9^
x ,
so that
u x
Thus
u
x
u
/N
,
,
7
to every element of volume dr(=docdydz) of the charge i there will be the corresponding element
at the time
at the time 9 seconds previously, where
u x
The
u
x
,
u
/x
velocity of this charge at the time
u - uO
+ %ud 2
r -
In integrating
..
.,
r2
uO* ,
+ l^___i u
for the scalar
6
t -f
is
... .
r3
_
and vector potentials we must
CHABGE MOVING WITH VARIABLE VELOCITY by
replace
we
/
.
On
substituting the values already obtained,
find to the first order of small quantities that
dr
The
dr
fi
scalar potential ,
1
=
(t>
is
265
ffff }
dr\'
^lll('
evidently
-i/ The vector potential .
w
contains
as a factor, so that the replacement of
by /
-~
will
CC7
of the second order and may be only introduce small quantities It is, however, necessary to substitute in J7X the value omitted. 0. Thus of u at the instant t .
c2
c
=
C/ z
= 0, since
the resultant velocity
The x component of the
The magnetic
is
parallel to the axis of x.
force on a unit charge at
P is
derived by differentiating the It thus contains vector potential with respect to the coordinates. u as a factor, so that the lowest term in [uH] will involve squares of
small
force
quantities.
17) is
r(=rot
This term
be neglected.
therefore
may
Evidently 1
ftrpdr
7
.
u
fff
(
x-afdr n >'-
-*
**
+
w
"r\
/y/
1}
and
Thus
P
the force on a unit charge at arising from the whole on The force an element of volume dr at is therefore sphere,
This
is
P
1
=
p ^
'xJ'J J _ ofo-
fff n
X
-*'j dr -
..
r
(^-jrjjj
j
_L_
-f
o
^
STTC? J ] J
l
-~
\r
r*
7 dr
;
u -
;
p,
DTTC"
since
e=
due to
-
I
/>cZr.
on the whole moving charge
total force
itself is
1 [ff OTTC-JJJ
oo Up 'jdr
The value sphere
The
is zero,
> 7
^
l {{( ( p JJj \r
v + (x-x')*\
r8
-r
dr
+
e
^
[[[
DTTO^JJJ
/
,,
,..
p'dru
+
....
term when integrated over the whole by symmetry. Transforming to polar coordinates
of the first
-
the second term reduces, on integration, to 8
-^
'
6-TTttC
a uniformly charged spherical shell of radius a whilst the third term
is
~
.
07TC3
and
in the case of total charge
e,
Thus
c
The term in u appeared The term in u is new and
3
n -
I
ft
.,
^ ^ )' (18) V
O7TC"
in the case of quasi-stationary motion. is therefore due to the fact that the
The fact that "it is quasi-stationary condition is not satisfied. independent of the geometrical distribution of the electrification at once
suggests that
it is
due to the reaction of the radiation on
the
moving charge, since we have seen that the radiation itself possesses a similar property. This is made clearer if we consider the work done by the
force
Fx
'
during a displacement udt.
CHARGE MOVING WITH VARIABLE VELOCITY This
Fa,'udt
is
The term
6
=
a
-
-*-
^
2
2
uudt
9
(Jw
2
)
electromagnetic energy (Jmv the
moving charged body.
+
e
267
2
uudt
fi
dZ represents the diminution of the 3
of the steady part of the field of
)
The term -
e
2
^
3
^2
is
equal to the
energy radiated.
The work done by the second term from
ti
to
in
Fa
'
in a finite time, say
t2) is
^
r* a
_
zi
6V?
#!
2 <
J*,
J
The integrated part vanishes if either u or u is zero at the limits. In the case of periodic motions this term will vanish when the limits are the recurrent zero values of
limits are not
chosen
u and
u.
Even
if
these
value over a long period of time will be small, being comparable with the maximum value within a The value of ?i2 however, is always positive, when single period. its
,
not zero, so that the integral
[U I
uz dt
will increase indefinitely as
It will evidently tend continuously to diminish the kinetic energy of the vibrating electrified particle, and thus to
t2
increases.
It is, in fact, very similar in effect to the action stop its motion. of a frictional force on a dynamical system.
Planck* has suggested that this force may account for the frictional term which it is necessary to introduce in order to Whilst there
account for absorption in optics.
must be operative
this force for
any
it is,
of the observed effects.
term
frictional
is
smallest
is
is
no doubt that
I think, far too small to
The kind
of action in
account
which the
found in the case of substances which
vm
the residual rays. We have seen in Chapter that even in these cases the coefficient which enters into the give rise
to
resistance
term
the term
-
/r
is
of the order 10 12 in the units employed, whereas
7 -- would lead to only about 10~ of this amount.
DTTC
The
main part of what appears to be a damping force jn optical must evidently be sought elsewhere.
radiators
CHAPTER
XIII
THE AETHER IN our discussion of electromagnetic action so far we have always, explicitly or implicitly, considered the medium, in which the actions take place, to be at rest. It is true that in Chapters XI and XII we have considered in detail the effects produced by
moving charges, and we have seen that changes in the state of the electromagnetic field are propagated through the surrounding medium with finite velocity. Without having specified the matter very definitely we have implicitly assumed that the bodies, whose motional effects were being investigated, were moving relatively to a fixed system. The fixed system embraces the observer and and we have treated the question as though the medium through which the electromagnetic effects are propagated
his instruments,
was
This assumption rigidly attached to the observing system. is clearly, however, an arbitrary one; so far as anything which has yet been brought forward is concerned, we might just as well have considered the medium to be moving along with the moving It is necessary, then, to consider
system.
what
effects
we should
from the motion of the medium, in order to decide It may either, of the foregoing alternatives is true.
expect to arise which,
if
even happen that of electromagnetic
medium
impossible to form a consistent scheme phenomena, without discarding the idea of a
it
is
altogether.
The main question material
medium
at issue relates to the hypothetical non-
the aether
which has long been supposed
to be the seat of optical and electrodynamic actions in space. Although the question of the effects which are peculiar to moving
media is intimately connected with this, nevertheless the two questions are If the first question is essentially different. settled, we shall see (p. 285) that we have n.lrp.a.Hv a.->.nmnlpf,prl all
material
THE AETHER the material which
269
necessary to determine the specific which are caused by the motion of material media. is
It appears to the writer to
perspective
effects
be impossible to acquire a true
of the matter at issue without
considering
it
in its
development. We shall therefore take it up from that of view, at the risk perhaps of lengthening the treatment. point Naturally, optical effects will be treated as a particular case of historical
electrodyriamic actions.
The Aberration of Light.
The study of
this subject arose out of a discovery
by Bradley
made during an
investigation whose object was to detect in certain fixed stars near the zenith. annual parallax Such a It was not, however, directed towards the found. was parallax in 1728,
sun as but
it
it
would have been
if it
were ordinary
was in a direction perpendicular to
The magnitude
earth's orbit.
stellar parallax, this in the plane of the
of the "aberration"
was found
to
be
to the sine of the inclination of the star, but was proportional The results were found to constant for stars of equal inclination.
be capable of complete explanation on the view that the light was propagated in space with a finite velocity in a direction which was fixed relatively to
the star and which was uninfluenced by the
earth's motion.
The problem is one of relative motion and can be made quite by considering an analogous material case. Suppose an observer to be in an open carriage which is moving with uniform
clear
horizontal plane. He wishes to determine the motion of the drops which fall into the carriage in a rain storm. To do this he is provided with a long tube which can rotate about a horizontal axis perpendicular to the length of velocity
in
a
direction of
the
tube,
and the inclination can be road
vertical
The
direction
off on a circle in a then determined by adjusting through without reaching the sides. is
plane. the tube so that the drops fall The direction of the rain as thus determined will evidently depend on the direction of the rain relative to axes fixed in the ground,
on
its
velocity
and on that of the carriage relative
to the
same
THE AETHEB
270
direction of the rain is in a vertical plane containing the direction of motion of the carriage.
OP be
Let
the direction of the tube,
i.e.
the apparent direction
in which the rain reaches the carriage, and let the angle OPP' = a. Then if OP':PP'= V: v, where Fis the velocity of the rain relative to the axes fixed in the earth,
and
v is
the velocity of the carriage
then OP' will represent the position of the path of the rain relative to axes fixed in the earth. For P' is then determined by the condition that the time required by the to P is equal to that required by P to rain to move from referred to the
same
axes,
r
reach P'.
Thus
..
- sinf0jP/
-
F~OP~
sin a
The angle between the true and apparent directions it by /3 we have angle of aberration; denoting /3
= Z POP' = sin-
The analogy between
OP has
1 .
is
called the
sin *
this illustration
and the
optical case
is
by the direction of the telescope complete. of the light passing through it relative to axes and OP' by that F becomes c the velocity of light in free- space, fixed in space. and
v is
to be replaced
the velocity of the earth in
its orbit.
Thus
sin/3= Y^sina,
and
is
therefore proportional to the sine of the apparent inclinais often referred to as the v/
tion of the star, as Bradley found, aberration constant.
V
we make use
of the hypothesis of the aether the most obvious interpretation of Bradley's observations is that the aether If
at rest in space and is entirely uninfluenced by the motion of the earth through it. On this view, as was first pointed out by Boscovich, the angle of aberration will depend on the velocity of
is
light in the
medium
in
which the aberration takes place.
the velocity of light in this
medium
be given by sin
ft
If
V
is
the angle of aberration will
= --, sin a.
THE AETHER
2/1
vary directly as the index of refraction of the experiment to test this conclusion was carried out
It should therefore
medium.
An
who used a telescope filled with water. long afterwards by Airy He was, however, surprised to find that under these circumstances the constant of aberration had the same value as in an ordinary telescope.
experiment based on somewhat similar reasoning had occurred independently to Arago, who argued that since the de-
An
viation of a ray of light
produced by a dense prism depends on
the ratio of the velocity of light in the prism and in the surrounding medium, its magnitude ought to be different according to whether
the passage of the light through the prism is helped or retarded of the earth. With this experiment it is evidently by the motion
unnecessary to use light from extra-terrestrial sources. Arago found that no effect due to the earth's motion could be detected,
although the expected effect was comparatively large.
An
explanation of the experimental results of both Airy and in a general way by Frosnel, who
Arago was given
suggested
was carried along by moving material media, in such a way as to compensate exactly for the difference between the velocities of the light in the medium and in vaeuo. It is that the aether
necessary that the aether should be carried along with a velocity which is only a fraction of that of the medium, lor if it; were carried with the all.
same
velocity there, would bo no alx.irrai.ion
Fresnel's suggestion
was worked out more completely,
We shall now calculate by Maxwell and Stokes. fraction of the velocity of the transparent, medium it that the aether should be carried alon^ in order
required by Arago's experimentthe earth's motion.
that,
refraction
with is
at.
later,
what
necessary the result
t.o
ij'ive
is
independent, of
The path of a ray of light in any medium is determined by the fact that the time required to pass from any one point, of the This extension of path to any other has a stationary value. Fermat's Principle of Least Time follows on the undulatory theory of light from the fact that if A and />' a, re any two points in a, ray the disturbance arising from points in the wave front in the immediate neighbourhood of A must all reach ./> in the same Dhase.
This condition
will
evid^ni.K-
!,.
y.it.idi,.,!
r
K,
k
,,
JK,.
<;
THE AETHER
272 from as
A
to
B
has either a stationary or a maximum value as well Thus if ds is an element of the is a minimum.
when the value
V
path of the ray and
f
the velocity at any point, then .(1).
Consider the case of any optical system, including the observer, which moves through space with the velocity whose components Let I, m, n be the direction cosines of the ray at any are u, v, w.
Suppose that in any refracting medium the velocity of the it is increased by the amount 8 multiplied light passing through medium through the aether. If F is the of this by the velocity
point.
standard value of the velocity of light in this medium when the system is at rest, the velocity relative to axes fixed in space for the moving system will be
F + 6 (lu + mv + nw). But the observer
is
moving
relative to the fixed axes with a
velocity whose components are u, v, w, so that the resolved part of nw and his velocity along the direction of the ray is lu -h mv
+
the velocity of the ray relative to
him
is
F - (1 - 0) (lu + mv + mv). The equation
of the relative ray paths is therefore
ds
F- (1 - 0) (lu -f mv + nw) or to the first order of (u, S
To be
in
I
y+8
I
v,
w)/V
-p^- (udx
+ vdy + wdz) = 0.
agreement with Arago's experimental result
it is
neces-
sary that the relative paths to this order should be independent of This will be the case if the quantity under the second u, v, w. integral sign integral will
not varied.
is
a perfect differential
velocity of light in free aether
and
since
;
the value of the
then depend only on the terminal points which are If m is the refractive index of the medium and c is the
m =c F 2
2
2
/
.
It is therefore necessary
sufficient that
m
2
(1-0)
^
C"
(udx
+ vdy
-f
wdz)
THE AETHER and udx + vdy
is
arbitrary
it
follows that
m
2
+ wdz
satisfies this
-
_a
0=1
-uloa/w 2
(l-6>) .'
1_^2
273 condition as a rule,
constant 4.
c
so that It
is
clear that
so that
A = 1/c
2
.
must vanish for empty space and (9 = 1 - 1/m2
for
which
m = 1,
.
It follows therefore that there will be no effect, due to the motion of a refracting medium through space, of the order of magnitude of the first power of the ratio of the velocity of the
medium to that of light; provided the velocity of light through 2 2 the medium is increased by (m - l)/m times the velocity of the medium. If we regard the optical disturbance as being propagated in an aether capable of flowing we may say that the aether is carried along of the
latter.
2 2 by moving matter with (m l)/m times the velocity This was Fresnel's interpretation of Arago's result.
It can also
be shown that
the hypothesis that the velocity 2 the in medium is increased of light moving by 1 l/7?i of the medium in the space is sufficient to account for the fact velocity of that the aberration constant
it.
medium
independent of the refractive index is used to measure
is
the telescope which For the discussion of the problem which
of the
reader
filling
is
here involved the
may be referred to Campbell, Modern Electrical Theory,
First Edition, p. 293.
Fizeau's Experiment.
The conclusion that the velocity of light in a moving refracting medium is increased by 1 1/m 2 of the velocity of the medium was put to the test of direct experiment by Fizeau. The apparatus he used for the purpose was similar to that shown in Fig. 37. Two parallel tubes c c' were set up so that a stream of water could be made to run continuously through them in the directions
A ray of light / was divided by the thinly silvered The reflected portion travelled along the path f a b c e c' V a g. The transmitted portion on the other hand followed the The two beams thus combined in the path f a b' c e c b a g. indicated.
mirror
a.
direction of
g and gave rise a telescope.
observed with
to It
interference bands which were will
be noticed that when the
water
is
flowing one of the
beams of light is always in the and the other against it. The
direction of flow of the water
the bands is first observed with the water at rest. position of The stream of water is then turned on and the displacement of the bands, caused by the resulting difference of velocity of the
two beams, measured. Fizeau found that the shift of the bands thus caused was in complete accordance with Fresnel's hypothesis. The experiment has since been repeated with improved apparatus
by Michelson and Morley, whose results were also in complete accordance with the view that the velocity of light in a moving material medium is increased by 1 1/m- of the velocity of the medium.
The foregoing experimental results led to two rival views as to the relation between the motion of matter and that of the aether
Fig. 37.
which was championed by was at rest, and matter moved through it without setting it in motion. Thus the aether was supposed to be able to flow freely through matter like water through a sieve. We have seen, however, that it is
in its neighbourhood.
The
first
view,
Fresnel, held that the aether outside material bodies
necessary to suppose that the relative velocity of moving refracting matter and the aether in its interstices depends on the refractive index as well as on the velocity of the matter relative to
This was
in agreement with Fresnel's which the density of the aether in material bodies depended on the refractive index. The relative would therefore have to be different in order to prevelocity serve continuity of the medium. On this view aberration is the simplest possible problem in relative motion, and it is clear from
the stagnant aether.
optical ideas, according to
THE AETHEE 9
that Fresnel's
views are in the foregoing discussion with the experiments of Arago and Airy.
harmony
Stokes was unwilling to accept the view that matter could pass through the aether without setting it in motion, and he
freely
therefore undertook to investigate
whether
it
would not be possible
for the aether in its immediate neighbourhood to be carried along the earth and still give the well-known results for terrestrial
by
The problem
is to find what distribution of velocity aether so as to leave the to the paths of the rays may be imparted If c is the velocity of light in the in space unchanged. stagnant aether in a direction whose cosines relative to fixed axes in space
aberration.
are
I,
m, n and the components of the supposed velocity of the u, v, m at any point, then the velocity of the ray in space
aether are
be c+lu + mv + mv. Applying the principle the equations of the ray path will be detertime of stationary mined by at that point will
or,
to the first order in (u,
*_S C
If
udx
v,
f^
J G"
+ vdy i-wdz
is
w)/c,
(udx
+ vdy + wdz) = 0.
a perfect differential the second integral
depend solely on the values of u, v, w at the terminal points and will therefore be independent of the motion of the medium in will
between.
The condition that udx + vdy + wdz should be a
perfect
medium
should
differential is
be what
the condition that the motion of the
known
hydrodynamics as differentially hrotational*. Provided this It means that there is to be no whirling motion. condition is satisfied the path of the ray passing between two is
in
points whose velocities are given is determined solely by the values of those velocities and does not depend on the motion of the
medium
in between.
As an
of Stokes's result
illustration
we may
consider the
particular case of light propagated from a fixed point &' y 1} z l to a fixed point &'2 yz ,&>, the space between being filled with a 3
,
""
Gf.
,
Lamb's Hydrodynamics.
1st ed., chap. in.
,
uniform stream of aether flowing with uniform velocity having components u, v, w. The path is determined by
xy l
l
z
^'22/2^
or
ds
8
-
f~l
S
-
LC
p
or
2 2/?.z a
cfe
[u
(fl? 2
- x ) + v (y 2 - 2/j) + w (z - ^ ) l
~] }
= 0,
J
= 0,
J Xil/iZi
so that the path s is obviously a straight line as if there were no motion. 23 ^2J just #2,
from alt y lt
z^ to
y
of velocity communicated earth to the aether, by the motion of the through it, should be should behave like differentially irrotational is that the aether
The condition that the distribution
a perfect fluid for slow disturbances, such as the motion of material This seems a natural conbodies through it would give rise to. dition of affairs,
so that thus far
Stokes was able to give a
satisfactory account of aberration and still retain the view that the earth carried the aether in its immediate neighbourhood along
with
it,
in the
manner
of a solid
moving
in
an ordinary material
fluid.
If the aether is
an incompressible
to be at rest relative to
fluid it is
not possible
for it
the surface of the earth and to have
a velocity equal and opposite to that of the earth at distant points motion is continuous and irrotational*. Ways in which this
if its
difficulty could
and Planck|.
be overcome have been pointed out by Lorentzf They seem, however, rather artificial. In order to
explain Fizeau's experiment it is necessary, on Stokes's theory as well as Fresnel's, to suppose that a moving refracting medium
m~ ) of its own velocity to the imparts a velocity equal to (1 aether within it. If the earth carries the aether with it in its 2
immediate neighbourhood, as Stokes supposed, other moving bodies would be expected to have the same effect. An experiment to test this point has been made by Lodge who tried to find ,
*
See Whittaker, History of Theories of the Aether, p. 412. t Arch. Neerl vol. xxi. p. 103 (1896).
J Of. Lorentz, Proc. Amsterdam Acad.
vol.
i.
Phil. Trans. A, vol. CLXXXIV. p. 727 (189S).
p.
443 (1899).
THE AETHER
277
a change in the velocity of light in the region around the periphery of a massive iron flywheel when it was made to rotate rapidly.
A
results were entirely negative. way of escape from most of the objections to Stokes's theory has recently been suggested by
The
H. A. Wilson* (see
p. 283).
The Michelson and Morley Experiment. So far those who held the view of an aether which is undisturbed by the motion of matter through it appeared to have much the better of the argument. The foregoing account represents the state of the question when Michelson and Morley carried out the famous experiment by which they hoped to obtain positive evidence of the relative motion of the earth and the aether.
Before describing
this
experiment we shall
reflection of light at a problem of the
consider the
first
surface.
moving
Fig. 88.
The plane mirror uniform velocity v.
AB
moving towards the right with the plane wave of light bounded by the rays DA, EB falls on it. Let us find the relation between the angle of incidence i and the angle of reflection. When a wave meets the mirror at
A
is
A
on the ray
DA,
it
has reached the point
C
on the
If the mirror were at rest this wave would afterwards ray EB. meet it* at J3, but as the mirror is moving, the wave will meet it at B' where BB'jB'Ovjc, c being the velocity of light. The thus were mirror behaves as though it turned through an angle e }
given by
tan
(L
tan
e) __ L
B'C
BG
-4-
tan
1
c c
v
tan
1
1
i
+ tan*
tan i
tan
e '
e
THE AETHER
278 V c
whence tan 2 6 neglecting tan 2 1
= tan
= The same
effect of
as if it
1 h tan
\tan
Hence
.
(
e
i
L
to the first order in
tan
e
=
/*
-si
the mirror on the reflected wave
were rotated through an angle
reflected light will
e
is
therefore the
= -sin2j.
The
be turned, relatively to a similarly situated
fixed mirror, through twice this angle.
S si/
Fig. 39.
The arrangement of apparatus used in the Michelson and Morley experiment is shown in the accompanying diagram. Light .
S passes through a half-silvered mirror B inclined the direction of propagation. The reflected ray is
from a source at
45
to
back by the plane mirror A so as to pass through the and thus reach the observing telescope E. The transmitted ray is reflected back by the plane mirror D and again by the half-silvered mirror, and so it also reaches the
reflected
half-silvered mirror
The paths of the rays in space will depend on the magnitude and direction of the velocity of the apparatus. The figure has been drawn for the case in which the apparatus is carried along by the earth's motion in the direction of the ray SD. Let the ^~^~i ^ velocity of the earth in its orbit be v. tlmf, nf li^f ^ ^-
telescope E.
THE AETHER
Then the ray BA will not be perpendicular to c. the motion of the mirror, but will be inclined of account on SB, an angle which is equal, to the first order, to at to this direction 1 = ?r/4. Since is the = tan" v/c, by 6 preceding theorem, since i will be inclined to the normal at an to itself, aether being
A
.
moving
AC
parallel
on the other side of it. In an exactly similar way the equal angle the reflection of CD at the moving mirror ray CE produced by The will be inclined at the same angle to GD as the ray AC. will therefore coincide in direction when they reach the two rays
observing telescope. rest, the construction
If
AB=BDl
when
when the apparatus
is at
be as shown, if C has moved into while the light
motion
it is in
will
the position that B and back. BC is evidently equal to 2vl/c. The moves from B to of the interference fringes seen in the telescope will position on the difference of time taken by the light to reach it represents
D
depend
from B. This is equal to the difference of along the two rays time along the two paths from B to (7. If 2\ is the "time along the path
If
T is
BAG
this is given
by
the time required for the other ray to go from
B
to
have
and
if
T
f
is
the time required for
it
to get
D to
back from
= l-vT'.
cT'
r= c ~, v
So that
L
r C
+V
'
c V
I
V"
f, -a + + -+.. + c-
.
c
+-
c
c-
c
Lv-
i
,
,-
i
V1
V !---_
i
nesrlectinsr merrier
powers
v t 01 -
.
G
D, we
THE AETHER
280
The corresponding
If the fringes
difference of path
is
were adjusted when the apparatus was station-
to this path difference should ary a displacement corresponding
be observed velocity v
if
the whole apparatus were moving with uniform If the motion were with
in the direction indicated.
uniform velocity in a direction perpendicular to this the path difference is found to be equal but in the opposite direction thus if the apparatus is adjusted so that SD coincides with the earth's motion relative to the sun, the effect of turning it through a right angle should be to cause the fringes to shift by an amount ;
v*
to 2i corresponding to a path difference equal
.
Michelson and Morley set up their apparatus on a stone slab It could thus be rotated without which floated in mercury.
As the rotation causing strains to be set up in the apparatus. was made to take place the fringes were observed continuously but no displacement of their position could be detected. The that the relative investigators were able to show conclusively motion of their apparatus and the aether could not amount to as
much
as one sixth part of the velocity of the earth in its
orbit.
The
result of the Michelson
and Morley experiment was
to
place the problem of the relative motion of matter and aether in an apparently impossible position. On the one hand the view
maintained by Stokes that the aether was carried along by the earth in its motion appeared to be full of inconsistencies, on the other hand the consequences of the stagnant aether hypothesis An escape from the flatly contradicted by experiment.
were
dilemma was pointed out by Fitzgerald who suggested that the null effect in the Michelson and Morley experiment was due to a change in the dimensions of the apparatus in the direction of the earth's motion, just sufficient to counterbalance the expected
The required change would evidently be produced if the effect. matter of the apparatus contracted in the direction of the earth's 2 2 motion in the ratio 1 -y /2c to 1, the lengths in planes perpendicular to this direction being unaltered.
This change would be
.
THE AETHEK
281
be capable of direct measurement in any case but even if it were not, the operation could not be carried out as any material scale which might be used would also contract, in the too small to
;
;
same
ratio as the material to
be measured.
This hypothesis seems a wild speculation at
first sight, but on further that it is rather what inquiry, appears, might be occur if matter is made to of electrons. In that case up expected
it
the question at once suggests itself as to why a given portion of If the matter were made up of solid matter preserves its shape.
superposable of
elements of positive and negative
infinitesimal
electrification,
the
only state of stable would be in one which excess of charge of one any equilibrium be to would and the dissipated infinity sign remaining equal and capable
subdivision,
The matter opposite charges would be superposed on each other. would thus annihilate itself and disappear. In order that matter should be stable enough to preserve its identity it is necessary that the ultimate elements of electrification should be finite, and it
is
also necessary that the superposition of opposite
should not result in annihilation.
To ensure
elements
this it is necessary
that the ultimate elements of opposite sign should not have conclude, in fact, exactly equal geometrical distributions. that the positive and negative electrons are essentially different.
We
The simplest assumption we can make as to the nature of the which keep them in equilibrium is to suppose that they are under the influence of equal and opposite forces of ultimately
forces
electrical
character, but
it
may
be that
this supposition is in-
adequate and that the electrical forces are balanced by forces of In any event, in static cases the equilibrium of the electrons will be determined by the positions configuration
non-electrical type. in the matter at
We know
which the resultant force
is zero.
from the results of Chapter XI that the
field
due
to
an electric system in motion differs from that due to the same system at rest, in such a way as would result if all the lengths parallel to the direction of (I
v2/c 2 )^ to 1.
Thus
motion were changed in the
for the electrical forces to
ratio of
have the same
value in the moving as in the fixed system it is necessary that all lengths in the former which are parallel to the direction of motion
THE AETHER
282
2 should be reduced in the ratio of (l-v^/c )^ to 1. If the forces are all electrical it follows that the positions of the balancing
points
be nearer together in this ratio and a contraction in length of this amount, in the direction of motion, will have to occur, in the moving matter, if equilibrium where the
is to
force vanishes will all
be maintained.
Since
as far as the fourth order in v/c,
we
see that this contraction
is
of
the right magnitude to account for the absence of an effect in the Michelson and Morley experiment. Another mode of explanation will be developed later.
An attempt has been made by Trouton and Rankine (R. S. Proc. A, vol. LXXX. p. 420, 1908) to detect and measure the Fitzgerald shrinkage by measuring the resistance of a metal strip when (a) parallel, and (6) perpendicular, to the earth's The experiment showed that if any shrinkage occurs it
motion. is
com-
pensated to within 2 per cent, of the expected value by some counterbalancing effect of the motion on the resistance. Trouton
and Rankine were able to show that on the electron theory of metallic conduction the changes in the mass, mean free path and velocity of the electrons would exactly compensate the effect of the expected shrinkage.
It is interesting to note that according
to their calculation the Lorentz
change of electromagnetic mass with velocity gives the desired compensation, whereas the value found by Abraham does not. Besides those which have already been discussed, there are a of other cases, where effects due to motion through the
number
aether might be expected to arise, which have engaged the attention of physicists. As is well known, the phenomenon of ordinary double refraction can be fully accounted for solely by the fact that
the media which exhibit
it
transmit the disturbance which con-
stitutes light with different velocities in different directions.
We
have seen that in order to explain Arago's experimental result it is necessary to suppose that the velocity of light relative to the observer, through a refracting medium which is isotropic when at rest., depends on the motion of the latter through the aether. The
THE AETHER
with negative
but
;
-
..-, -
results.
s*
*
Electricity.
A
,
method
of the of reconciling Stokes', theory tads, the
mtho, <W^V^
moving bodies with
experimental has recently been nd contraction, Ltling the' Fittgerald j ot the ; out that the probkn, bv H A Wilson*. Wilson points the aethe, like the eari, through u ,-jay We the of press motion in such a way that the hat the Un and irrotational, provided everywhere continuous All Ixx ly at the surface ot the relative velocity vanishes perties of
J
Ittf Idy
I^v* .
.
gential
L
conditions
may be *
satisfied
Pkil Man. VI.
flow by an appropriate vol. xix. p.
809 (1010).
ol:
the .u-lhu
THE AETHER
284
All the experiments of the body. along the normal to the surface been made deal only with which have flow to detect relative aether the velocity component parallel to the earth's surface and, owing to strains, etc., caused by the gravitational action, it would be almost impossible to execute delicate tests for motion in the vertical direction.
Relativity of Effects.
In reviewing the outcome of the experiments which have been undertaken with the object of discovering the relation between the motion of matter and aether, the most striking feature appears to be the small number of experiments which have led to a
The only cases where positive effect. to influence optical phenomena are
motion of matter appears
1.
Astronomical aberration
2.
Relative motion of a refracting
;
medium
as in Fizeau's
experiment; and
The Doppler
3.
It can hardly
effect.
be a coincidence that
There
all
these cases entail the
no experimental evidence of of matter relative to from the motion effects any optical arising aether or to space. One is therefore tempted to inquire whether it is really necessary to postulate an aether for the propagation of
motion of matter.
relative
optical
and
electrical effects
;
it
is
may appear
that a
more
consistent
would be obtained solely by the relative motion of shall have more to say aboiit this question later.
set of relations
We
matter.
The Propagation of Light in a Moving Refracting Medium.
The
effect
of the
motion of a refracting medium on the may be calculated on the electron
propagation of light through it theory, as follows :
We which
shall consider the case of a plane polarized
is
propagated along the z
axis.
beam
of light
THE AETHER
P
285
Referring to Chapter vill, p. 148, we see that the polarization related to the electromotive intensity E' by the equation
is
P = (m
2
When
the
medium
is
-!)#'.
in motion, with velocity F,
the magnetic permeability of the medium is unity, in accordance with equation (5), Chapter IX. In the present case, therefore,
if
(2),
x parallel to the electric intensity and the to that of y parallel magnetic intensity and the motion of the medium parallel to the axis of z with velocity w. The current
if
we take the
density
is
axis of
Dx + w ~~
,
the latter term arising from the convection
by the moving medium.
of the polarization
Thus the funda-
mental electromagnetic equations become
_
---
^-*-M/
and
Eliminating
If
F
is
P
=
*
-
and
I f
^ + w -^ L/A
T\
,
x
=*-
\
j
/ I
r>
^3.
+
ty -z
w -~^ -
a;
/ ^ \
\
.
.
.(4).
H from (4) by means of (2) and (3) we have
the velocity of light in the moving dielectric with we shall have x = Ae Lp ^~ zlv \ where
E
respect to the fixed aether
A
rS
P^. -f
and p are constants. 2 w /c' we have
Substituting this value and neglecting
2
or
7=
~+lm \
W in")
..................... (5). ^ J
This is Fresnel's formula (p. 273). This deduction shows that the convection of the polarized dielectric through the fixed aether produces the same change in the velocity of the light as is required by the experiments of Airy and Fizeau and by Fresnel's hypothesis.
Those exDeriments therefore do not Drove that the aether
THE AETHER
286 is
by the moving matter. The substance of this given by Lorentz* and somewhat later by
carried along
calculation
was
first
Larmorf.
Moving Axes and Correlated Systems.
We
have seen that the behaviour of an electromagnetic system at the earth's surface may be represented by the system of differential equations
:
div
E=p
(I),
div#=0
F on
where the force
(II),
rotH=(E + pV)/c
(Ill),
Tot=-H/c
(IV),
a unit charge
is
given by
F=E + [V.H]/c V
(V),
the velocity of an element of charge relative to the If matter has a purely electrical constitution this earth's surface. of equations will also describe the changes which material system
and
is
systems undergo.
A
method,
for
which we are indebted to H. A. Lorentz, of on electrodynamic actions of the motion
investigating the effect
medium, is to transform the equations so that they refer to axes moving relatively to the material system instead of being fixed in it. As an illustration, and to fix our ideas, let us suppose of the
that the above equations are true when the coordinates x, y, z and are measured along axes fixed in the aether. Then (u lt v l} Wi) will be the velocity of an element of charge relative to the fixed
V
t
We
shall now examine how the differential equations are when changed they are made to refer to axes moving with uniform
axes.
velocity u along the positive direction of the x axis, relative to the Let the coordinates referred origin of the fixed axes Ox, Oy, Oz.
moving axes be denoted by x', y z\ we shall also distinguish the time for events described with reference to the moving system by t'. Let the two systems of axes coincide at t = 0. Then to the
',
x'
*
=x
ut,
y'
= y,
z'
z,
t'
Arch. Neerl vol. xxv. p. 525 (1892).
=
t.
THE AETHER and of, y, z, variables we have x, y, z, t
f
t
being respective sets of independent
^
but in virtue of the principle illustrated in dealing with the p. 285 we see that
convection of the polarization charges on
_+
from
.
.,
tVHi
_
x
tt
and
(I),
^+ from
_r . x
x
Thus
( Ul
_
)
p
=
^
S
-
a
-
(III). ,
Similarly
SEy
dEy + 3^
= -^ + V -^ + SB,,
O
-~ + w^ =
and
Also
Thus from (IV)
Similarly
^-/
g^p
=--
V
^ - =-J (CHZ -
(cfly
+ ,
--3
4- -w *
-^~ ox
J
THE AETHEE
288
Thus
if
we put
H' = (ffx
',
H
v ',
HI) =
Hx H
E
+
v
,
--El
z,
c
U
the equations
~ 1
~n7r
^P
"I"
c \ ot
may be ==: )
J
TT
-
written 5~7~
"^~7
ZTTT
;
02
oy
c
rnj '
^r~7
^r~r
oz
oy dy'
I
_ 3EZ ~W~~ "a^ "^"^ '
\3Hy_ r ~~ c
~w
==
ot
cLEV
\
7
dt
'
o
)y
dxf
dt'
'dy' }
The unaccented symbols in the left-hand members of (7) may be eliminated by means of the equations (6). We have, for example, c
thus
and
/3
3
c /
\ 2
if
= (1-
c
u*l&)~\
similarly
(Ha
,
(^,
H 5) = y
J? y
,
,
/
^) = /
'
y
+ - Hz
',
E - -c H,'} '
z
j
Substituting in (7)
we
find
-
a/
...(8).
THE AETHER
289
H
between the values of H' and and E' and E, are in accordance with the by required (6), equations respectively, values of the magnetic force due to a experimental changing electric field and the force on an electric charge moving in a
The
differences
These terms would therefore be expected to field. the arise electromagnetic system under investigation were to the measuring instruments relative in motion employed to magnetic if
it. Equations (8) show, however, that if rapid changes with time are occurring the state of the moving system will be different from that of the fixed system on account of the terms
investigate
^
and of terms of the second order in - depending
2
involving
on/3
/3
^r,
2 .
We
have seen in Chap, xn that the effects due to a moving charge are not established instantaneously but are propagated through space with the constant velocity c which, we have reason to believe,
is
independent of the velocity of the matter at which
It may be that on account of this finite the effects originate. = t') that the time of of velocity propagation c the hypothesis (t an event when referred to the fixed axes is the same as that of
the same event
when
referred to the
moving
axes, is leading to
Just
inconsistent results. as, according to 'Fitzgerald's hypothesis, the lengths of all material bodies change when the system is in
motion,
manner. x, y,
it
may be
that
all
the clocks change in a similarly definite local time tf might be a function of
In such a case the
z as well as
t.
Without pursuing
this discussion further for
the present (we shall return to it later, p. 298), we shall provisionally is different from t, and see to what admit the possibility that '
conclusions
we
are led thereby.
In order to avoid altering the notation in the preceding t and introduce a new variable equations we shall still retain if t" for
the local time.
of local time
is
Following Lorentz, to
whom
the conception
due, let us assume that
O
Then
iust as in changing: from
x
to
x
=xut
we have
-5,
r\
= ^-
THE AETHER
290
and ->== dt
ot
changing to the new variable
in + w ucc 5- we now have,
^
3
d 51'
~"
i
.
O*"
Thus the equations of the
" ^
JL_ ~~ JL 2/v,"
f",
)/*/
become
field
?'-?H dx"
dx" .(9).
dt"
c
3H "
1
M
a<"
X.
C
If
we
W
^
"~
S/V/ X
^/./
ux/
C't/
C^/
overlook the dashes these equations only differ from (III) 2 when it occurs on the /3
and (IV) by the inclusion of the factor
= (1 ft*
2
,
1
^/c )" the two sets of equations are This leads to an identical in form as far as the first order in u/c. Since
left-hand side.
important principle which we have established to this order of accuracy.
Lorentzs Principle of Correlation. If
we can
E Ey E H X9
,
z,
solve (III)
X)
H H y
,
z
,
and (IV)
let
for
any one of the variables
us sa,y'Ex and then express the solu,
tion in the form
E
means that X9 the x component of the electric intensity in the fixed system, is a certain function / of x, y, z and t, the space and time coordinates referred to the fixed axes. It follows from
this
nj
the identity up to the order - of the equations (9) with (III) and
(IV) that
THE AETHER is
precisely
291
the same function of
-< c'
Thus in comparing a moving and a fixed of #, y, z, t. are identical with one another when at relative which system, in the fixed system characterised rest, for any event by a certain set of values of E and H, there is in the moving system a correcharacterised by values of E' and H', which are sponding event are of as, y, z the same functions of #", y', z and t" that E and that
Em
is
H
and
t.
It follows that within a self-contained system, to the order
of U/G, the electromagnetic effects are
of the system, since E' and
H
f
independent of the velocity are the values of the electric and
magnetic forces which would be actually measured by instruments moving along with the system. This is only true, however, provided the time recorded by clocks in the moving system is the. " " true time recorded by clocks in the fixed local time and not the system.
In the first principle of 'correlation is due to Lorentz. instance he only succeeded in establishing it to the first order of
The
u/c after the
manner
of the discussion above.
A little later Larrnor
by including a contraction along the axes of motion, to accord with Fitzgerald's hypothesis, showed that the principle held to In 1903 Lorentz showed that a very similar transthe order i
formation enabled the correlative principle to be established with exactness for all values of the velocity u less than that of light.
We
shall now investigate what is the necessary transformation of the electromagnetic equations in order to establish correspondence up to any order of u/c.
we
Since the differential equations are linear the transformation and the preceding investigation
are seeking will be a linear one
which is most likely to be successful. be the coordinates referred to the first set of axes and #', y' /, t' those referred to the second, which are in motion Assume that both x and t' are linear relatively to the first. functions of both x and t whilst y' and z involve only y and z at once suggests the form
Let
x, y, z, t 9
respectively.
Let us put
THE AETHER
292 where
i,j, k,
I,
Then
m, n are constants to be determined.
3338
'
-
.(ii).
Since the axes are moving in the x direction, k=l by symmetry. We shall now transform the equations (III) and (IV) in which z and t are the independent variables, to new equations in a}, y,
terms of the independent variables
We
have
-zr
+ Uip =
-*/
#',
y,
2'
and
t'.
+ ti ~^J + ^P
Hence
^+ dEv
Similarly
vl p
dEv ..dEy = m-j-X + yij-* + v
= ck
l
p
-^~r
d -%-T - omn
dx'
dt'
QZ
OCC
.
-^~Oil
Thus m^->( Similarly
Also, from (I)
~ dx
Hence
dy %
dz
==
o
P
~
niii
v
~~
dy'
THE AETHER
293
Hence the three equations may be written
^
TT
v
'
m x-, (Ev + cnHz) 4- v^p = ck -TTT 02^
d ^/ ^
t
p= ir v ^/v(E ~ CnHy)+W z
,
1
'
Ci ;r-7
" -r
(
9t
d^?
iJL
I
dx \
-
c
\
-
7/ y
V
(12).
Turning
to the other three equations
o
and from (II) x -5-^
Sx
y, =
=
+ -5-^ + -5 oy
x
.
i
-5-7-
ox
cz
+ mn t
~+ x
.
,
Jc
ot
v -=-f
+k ,
oy
~
z .
oz
Hence z
c
"dy\ \
-m
I (H, + cnEy
)
77^
^
-a
y
)
'dz'\
y
"
LZ
c
^\
(^
4
{ H.)
<**$ ...... (13).
n The problem is to determine the values of the constants i and the functions E ... HZ which will make equations (12) and .
(13) take the is
same forms
that terms such as
as (III)
H
z
+ - Ey
and (IV). One obvious requirement and
H + cnE z
c
Hence n
=j/c".
Making
. .
this substitution
3
y
should be identical.
we have
THE AETHER
294
*-*m
W (15).
Now put -*-n
I
=1 *
>
Then
(16),
Thus
if
we put
the equations for the
pendent variables are
a?'
new dependent
=
- w*), (a?
variables in terms of the inde-
y'^y.z'
^z
and
f
t
= P(t- w/c )
accurately identical with the differential equations (III)
3
and
(IV) connecting the corresponding undashed variables. It follows that the correlation established to the first order is true previously "~" "
THE AETHER given above and the time and space coordinates in the second system are the functions given above of those in the first The space in the second system is contracted in the system. values
direction of
motion by a factor
in the first system,
whilst the time
factor and, in addition, is
We
shall
- ^ /c )i 2
(1
2
as
compared with that
contracted by a similar a linear function of the x distance. is
postpone the further consideration of the physical
consequences of this result, which was first given by Lorentz, until we have considered another method, which we owe to Einstein, of arriving at the
same transformation.
CHAPTER XIV THE PRINCIPLE OF RELATIVITY Newtonian equations of motion retain the space and time coordinates are when form original f = y, z = z and i' = t to x = x z and t from vt, y x, y, changed of the axes with motion to a uniform translatory corresponding this is no longer that We have seen velocity v in the x direction.
As
is
well known, the
their
f
the case with the fundamental equations of electrodynamics. For the original form of the differential equations to be retained it is necessary that the
#'
traction given
of
by
coordinates should undergo a uniform conand that t' should be similarly
= ft (x - vt)
modified and also depend in a linear manner on x, being in fact If therefore the units of space and /3 (t voc/tf). given by if
time have the same values in two systems moving relatively to one another we should expect differences in similar phenomena occurring in them depending, at any rate, on the square of the All the ratio of their relative velocity to the velocity of light. experiments which have been made, however, lead to the conclusion that the actions taking place in any system depend only on the relative velocities of the parts of that system and are in-
dependent of the velocity relative to any other system which the We shall now attempt to see system may have as a whole.
by changing our mode of defining time, we cannot make all phenomena independent of the state of motion of the system as a whole in which they occur. So far as matter consists of electrons and the phenomena in question are electromagnetic the problem has been solved in the last chapter but the following treatment, due mainly to Einstein, is instructive and leads to results which if,
;
are easier of application to
many important problems.
In order to describe any physical phenomenon it is necessary to locate it in time and space. To locate it in space we must have three
THE PRINCIPLE OF RELATIVITY undefendable axes of reference and an equally rigid unit of length. We may then fix the time of the occurrence by having a series of similar clocks distributed at infinitesimal distances apart in space
when brought together they register equal we have not indicated any method by which the clocks may be correlated when they are not in the same place. We shall now assume that this can be done in such a way that the velocity of light in a vacuum measured by means of the and arranged so that
So
times.
far
system of axes and the clocks is equal to a universal constant c, which is independent of the state of rest or of uniform motion
Thus
of the system.
if
A
and
J9
are two material points whose at the instant
distance apart is r, a ray of light emitted from at the instant ts, where tji will reach
A
B
1A
is
A
B
and s are the readings of the clocks at respectively. what is not of course a assume, priori obvious, that this true whatever the state of motion of the axes of reference may and
We be,
t
shall
provided
it is
has been called
not accelerated.
the Velocity of Light. Relativity which
may
It is a particular case of the Principle of in the words The laws of nature
be stated
are independent of the state of
provided this
is
The assumption here involved
by Einstein the Principle of the Constancy of :
motion of the system of reference
unaccelerated.
Now
consider two equivalent systems of axes 8 and S'. By equivalent we mean here such as possess equal units of length and
when the two systems are at rest Suppose that S' is in uniform motion It follows from the fundamental relatively to S. assumption that the velocity of light must have the same value when measured with reference to both S and $'. Consider any event occurring clocks
running at equal each other.
rates,
relatively to
at the point x, y, 2 at the time t when referred to the system S. Referred to the system S' this event will be described by the
corresponding variables
x',
y' z
mine the relations between
x, y,
On
}
and
t\
z and
t
It is required to deter-
and x, y\ z and
t'.
account of the homogeneity of time and space we should
It follows that the coordinate expect these relations to be linear. planes of S' will be in uniform motion when referred to those of
jK.jiiJUAivn
us,
i:
x and #' axes so motion of 8 and 8'.
however, choose the direction of the
coincides with the direction of relative
that
it
Then by symmetry the coordinate planes of S' when referred to S We shall also choose the will remain parallel to one another.
and the directions of the y and z axes so that the two systems 8 and S' coincide at the instant t = 0. Then the two following sets of equations have identical meanings origin of time
:
x - vt = y= 5=
and
of = 0, y = 0, / = 0.
Three of the desired transformations are therefore of the form x'
c^ (x
- vt),
y'
= 6^,
#'
= d#.
same value
Since the velocity of light has the
in reference to
c
=t = f
both systems and since they coincide at equations also have identical meanings
t
the following
:
+ 2* = c*t~, + = c2
,tf-)-y*
and
#
/2
+
/2
2/
'2
/2
Substituting in the last equation
On comparing
with the
and
z.
This
is
we
first
z are independent and there
obvious and
.
we have
see that bf
=
c^,
since
x,
y and
nothing to distinguish between y
is
we might have written
6j
=c
x
originally.
Hence
^
(tf
-
Zvxt
v2 *2 )
-I-
~
- a? =
/2
-c
2 2
*
1
must be an identity. Prom the homogeneity of this equation t must be of the form a (t -f fiw), where a and ft are constants. Hence
for all values of
Hence
-f
^ 2 i2 )
-^ + c
t
and
a?.
2
^"
1
2
^
2
-
-
a
-p (a
a -
3
+ 2a /9^'^ + a /?
^= Q
2
2
2
^2 )
= 3
..................... (1),
THE
=-m
that
PRINCiJrJbu v^
from
*. ----
^
.-**"<'> S= P sothat the
a*, root
3
-
-2
s to
inoo.si^t
..suits,
d
^-*() (-t 18
where
^3
is
Similarly
now written
for I
= () ^ (-)* a/' = ^
y"=^ a" = 4> ()(/) (,! Q" /b So and
But since the systems So that identical with *, etc.
*
-in> -ilwavs .ut. aiw.iyt.
coincident, o
t", etc.
are
THE PRINCIPLE OF BELATIVITY
300
r
or z and Moreover, since the relation between y and y depend on the sign of v,
and therefore
We
(v)
=
1, (v)
=-1
/
cannot
being obviously untrue.
therefore obtain finally
.(4).
It will at once be observed that these values are the
which we obtained in the
last
same
as those
chapter by the direct transformation
of the electromagnetic equations. If these equations are solved for
t
in terms of
if,
etc.,
we
obtain
showing that the system .8 is moving with respect to the system v along the axis of of. S' with velocity It follows
rest
is
from these equations that when a body originally at measured relative to axes in the
set in motion, its dimensions,
original position of rest, are contracted in the direction of motion and unchanged in planes perpendicular to this. Let #/, y/, zf and L yL z% be the coordinates of any two points in the body referred
moving with it (system S'), oc 1) y1 ,^ 1 and #2 3/2, ^2 being the coordinates of the same point referred to the system S relative to which the body is moving parallel to the x axis with velocity v. to axes
,
Then at any time S we have
t
which
is
constant with reference to the system
( 6 )-
y
Thus a length equal / o+.i/-i
i
/I
tf __
to V parallel to the
.1
TTrli/^Y^
axis of x'
is
reduced
301
THE PRINCIPLE OF RELATIVITY
with regard to which the body is moving with velocity v at a constant time t referred to these axes. If instead of
t
constant
moving with the body
tf
the time referred to the axes
then
as constant,
rt-xt'
= (l -&/(?)$ fa- ats) .................. (7).
that a body at rest with respect to
follows
It
we take
S
as
measured
from S' undergoes the same change of dimensions as a body at rest with respect to S' when measured from S. This result is required of course
A
by the symmetry of the motion.
similar set of relations holds
with respect to the time.
Suppose we have a clock moving with uniform velocity v along the axis of x when referred to the system S. Let us take the position of this clock as the origin of coordinates for the system S'. Then always and x
x'
= vt.
tive events as recorded for the
same events
Let /, &>' be the times of two consecuon the moving clock and t1} t* the times
as registered in the fixed system S.
so that
j
Hence
= /3/ and L
We may
take
-^=
t*
Then
= /3/.
(/-$/)
..................... (8).
/ to represent two consecutive strokes of the
/,
then clear from the equation above that the moving clock as observed from the fixed system will appear to have its
clock.
It
is
periodic time increased in the ratio 1 will
be decreased in the inverse This case
/
:
A / 1
tf
The frequency
.
ratio.
may be
realised physically by considering the line a spectrum emitted by moving molecule or ion. Measured with reference to axes at rest with respect to the ion the frequency of the emitted light in the case of many lines is confined within very
narrow limits. The period of the light may thus be taken to represent that of the reference clock and /, /, etc. may be taken as the times at which the emitted vibrations are consecutively in the same phase. The above result shows that in addition to the ixr^il
I
l7-n rv\xr-r
Tlj~vrvYAliT
d^rar^}'. i~r\ci Tv*\rti-i iV/^TT -n /\T l-n^}
1
1
r* r
f-
nci \TCLV\
i~\f\-l-
rvtr
THE PRINCIPLE OF RELATIVITY
302
observation will be less than v that observed with reference to axes at rest relatively to the ion or molecule in accordance with the
equation v
=
Addition of
Let any point move with a Vz
t
relative to the
Velocities.
having components vx vy', then
velocity,
system of axes
S',
',
Substituting for of, etc. in terms of x, etc. from equations (4) we find for the velocity components referred to the system S the values SC
XQ
VX
'
+V
*-7ifc-
.(9).
Z
Z*
CJ
It follows that the parallelogram
a
first
approximation.
If
= Vx + Vy + v^ '*
fl/a
of velocities
is
only true as
we put '*
t
and
let the angle between the tf axis measured with reference to the system S' and #/ be denoted by a,
1
If v
and v/ are
in the
+
cos a
same direction ?;.
=
.
.(10).
303
THE PRINCIPLE OF RELATIVITY each of which of light.
For
than that of light is also less than the velocity X and v/ = c - /*, where X and p, are positive
is less
let v
and smaller than
=c
c,
then the resultant velocity 2c
- X - p,
2c-X-//,+
:
and
this is always less than c. Also the addition of any velocity of the to velocity light gives rise to a velocity which is still equal of to the velocity light.
Another interesting consequence of the foregoing results is that impossible for any signal to be transmitted from one point A
it is
W
another B with a velocity greater than c the velocity of light. Let the point A be taken at the origin of coordinates and the Observers transpoint B lie on the x axis at a distance I from A. to
mitting and receiving the signal are fixed at A and B respectively. Let the signal be transmitted by means of a material strip relatively in the direction A-*B. Now to which it travels with velocity
W
the material strip carrying the signal be itself moving along the x axis with the velocity v, where v < c, the velocity of light.
let
The
velocity of the signal relative to the transmitting
observing system
is
and the
evidently
W-v 1
The time required
-
'
Wvjc-
for the signal to
be transmitted
is
thus
Since v can have any value < c, T can always be made negative > c. This would imply that the effect would be provided before the cause had commenced to act. perceived Although this
W
may to
not necessarily involve any logical contradiction it is opposed whole character of our experience. The truth of the
the
theorem therefore
follows.
Some of the preceding results differ so considerably from those which follow from the generally accepted notions of space and time that many readers will probably regard them as serious If, however, the principle objections to the views here developed. of relativity is accepted they appear to follow with logical certainty.
The a
vriori
argument in favour of the
-principle of relativity will
THE PRINCIPLE OF RELATIVITY
304
to
Application
Optics.
Let the vectors wT hich describe a wave of light when referred to the system 8 be proportional to a>
19
Ix
/
sin
cos
I
4-
wiy
+ nz\
t
,
)
c
v
1
m, n being the direction cosines referred to S of the directions Referred to the system S' let the vectors be
of propagation. proportional to
sin
,
0)
cos
I'af 4-
A, [t V
mV
4- n'z
o f
from the principle of relativity that if If, x, y z' are z drawn from equareplaced by their values in terms of t x, y will be z of functions the two tions (4) t, oc, y, identically equal.
It follows
,
9
}
Thus sin Ct)
cos
,, [t
/ = sin ^ u
cos
-- Ix 4-
myy + nz\
.
,
,
.
-,,
identically.
Hence
z--
So that
c
=
I'
m
in
.(11).
n
The effect.
last
-
formula
If an
is
observer
the complete expression for the Doppler is
moving with velocity
v
relative
to
THE PRINCIPLE OF RELATIVITY
305
to the source) (i.e. the system of coordinates at rest with reference the velocity v makes an angle with the line joining the source and the observer, the frequency v of the light perceived
by the observer is related to the frequency of the light as measured by reference to an observing system at rest with respect to the source by the relation .
cos
1
,
<j>
C
(12).
I
V
hand the angle is given with reference to the with S' the observer, and is denoted by ', we have system moving If on the other
If we regard the source of light as being in motion and the observer as fixed and if the frequency of the light when measured
with reference to axes moving with the source is z; OJ if also the source is moving relative to the observer with velocity v making
an angle referred
to
with the line joining the source and the observer when axes at rest with reference to the observer, then the
frequency v of the light perceived by the observer
is
given by
I7~tf l
V ^0
v ~?
G) <
!
1
--V c
,
COS
(f>
If our reasoning is correct this formula should also be given by v for v and $' for <j> in the first equation for v'jv. For substituting the change from moving observer to moving source is equivalent
axes to which change of sign in the relative velocity and the We thus get was referred have now become the moving axes.
to a
;
Bl-l'vlc'
Aberration.
x axis and if is the angle between the direction of the ray and that of the relative motion when referred to the 8 system of axes, and the same If the relative velocity v is parallel to the
'
quantity
when
referred to the S' system, then cos
cf>
G
'
i
1
- COS c
j.
This formula embraces the whole of the theory of astronomical Its relation to the formulae previously obtained may
aberration.
be
left as
an exercise to the student.
Fizeau's Experiment.
Suppose the light is travelling in a moving medium. Let be at rest with respect to S', the axes of x and of being
this
chosen so as to coincide with the direction of relative motion.
Then the Qc-l
/
T-
co
cos
\t \
system S
sin
vectors will light & ff>
be proportional to r r
/ ce/
cos
(
y^ \ t'
V
^F?
VJ
or to
\
-jr)
VJ is
according to the axes of reference chosen.
moving with
v
velocity
when
Since the
referred to S'
we
obtain
a>=fl>'(l+47
Hence
V=
V
H-
v -
I/
I
=F
+ __ c
tJ
x
( ( \
1
+
V
v \
'
V'v\
1 ==? ) ( I/ V J/ \\
x^"
6
)
/
approx. -"-
*
/
1
=
V+
(I
v -J
approx
(15).
This gives Fizeau's result to the order of accuracy with which
THE PRINCIPLE OF RELATIVITY
307
Electromagnetic Equations.
The transformation of the electromagnetic equations of Maxwell variables referred to the system 8' has to already been made in The equations assume the form the last chapter.
c\ct'
i(?|f
+w
y)=^-^L '
i
dHx
c
dt'
8/ c
dt'
das'
dy'
where
u c
ui^
v, p'=/3p, etc., but the above Previously we wrote values of the velocities relative to S' are the values required by
the theory and are consistent with the chapter, provided
we put
//
= /3(l V
V -
c /)
equations in the last P-
This value of p
is
thus the value required by the principle of relativity.
Since according to the principle of relativity the physical laws are independent of the motion of the system as a whole it follows
THE PRINCIPLE OF RELATIVITY
308
the system as measured by an observer who with his apparatus If the system is moving relatively to the is moving with it. observer the forces will have a different value from, the above in accordance with equation (o). It follows that the so-called electro-
motive forces acting on a moving charge in a magnetic field are measured by an observer moving nothing but electric forces when with the charge. From this standpoint the distinction between
becomes indefinite. By a suitable may be made to vanish, involving change For instance with a the other. in a corresponding change are there important magnetic uniformly moving point charge forces if the motion is relative to the observer, but if the observer
electric
and magnetic
of
forces
moving axes either
moves with the charge the
forces are all electric.
(/) that if a body is at rest, relative in reference to the system S' is measured to S' in reference to S at e measured its total the same as charge total For the to S. a definite instant t referred charge e' referred It follows from equation /
its total
charge e
to S' is
= jjj
Now
at
any particular instant
t
referred to
S
follows from
it
equations (4) that
ftdxdydz.
dse'dy'dz'
Also since the body M! = v.
is at rest
referred to &', u'
=
0,
and therefore
Hence
So that
6/ /
P dtt'dy'dz'
1
1
III
P
dxdydz =
e.
any material system is set in motion the charges, as determined from axes at rest magnitudes with reference to the initial state of motion of the system, are It follows that
if
of the
unchanged.
The equations
(a)
(/)
suffice for
the solution of
all
problems
in the electromagnetics and optics of moving systems in which accelerations do not play an important part. As an illustration
we
shall use
them
to
determine the amplitude A' relative to the
THE PRINCIPLE OF RELATIVITY system S' of a wave of light which
is
309
specified relatively to
S by
the equations
c
Let
Ex Ey E ,
components of
E
,
r
tj)
= 0)
,
/, (t
V
we
J
E
l3 EO, E$ the components of
see that the values of
-- ZV +
and
for the
,
Referring to the equations on p. 307 the vectors referred to S' are
The values
E
denote the components of with a similar notation
z
,
my~ + riz'\ c
.
J
7
have already been determined in m, n and v, and the consequences which follow from them have been discussed. terms of
o>,
of w^
Z",
m^
T?
I,
Let us determine the amplitude A' referred to S' for the case which the electric vibration in the wave referred to S is parallel
in
Then in free space (or in any isotropic medium) to the z axis. the direction of the ray will be in the plane of soy, and, if 6 is the angle between this direction and that of the x axis,
#i=0, H,
=-A
sin
0,
J02
=0,
jET2
= - J.
I = A, 9
cos
#
3
<9,
= 0.
Thus
H '=-A sin X
Therefore
sin
J.',
^
JT/
= /3 f- cos + - A cy)
\
the amplitude
referred
given by
1--C080
A'-A
to
sin
$,
H
the system
'
z
= 0.
S', is
OLV cos sin
since
As the
2
JL
r
c
4-
direction of the z axis
is
arbitrary this formula must be
true generally.
Mechanics of an Electron or Material Point.
Let an electric charge e, of infinitesimal dimensions, move under the influence of an electromagnetic field. We shall assume, in accordance with the principle of relativity, that its equations of motion, referred to the system of axes S' with reference to which it is
instantaneously at
where p
The
is
rest,
are
a constant which we shall
call
the mass of the electron.
introduced to indicate that the moving point at rest with reference to the S' axes. instantaneously suffix
is
is
Let us now deduce the equations of motion of the point charge when referred to the system S relative to which S' is moving with velocity
v.
We
have
- -C
-v
(df
Hence
v c~
,.
- v)
dt
,
THE PRINCIPLE OF RELATIVITY 1
=
Similarly
311
dy.
-rp-
with a similar equation for
.
"
CLlQ
If
A?' is
=V
point #
)
instantaneously at rest with reference to the moving = 0, ZQ = 0, so that ^/Q i
""0
-t
u
"'(i
2
c (1
Referring to equation
#o
q
(c), p.
-
"
2 2
2 ?;
/c
we
307,
)
therefore get
These are the equations which hold for the instant when = 0, Z Q = 0. We may on the left-hand side replace v by yo
= v>
V&'
2
-f 2/
suffix
equations
2
+ zd
2
and on the right-hand by ^
and adding the other terms
may be
-H c
z
,
.
Leaving out the
-H c
y)
the above
written as a particular case of the symmetrical
equations
d dt
'
J (\/l
JJM
|
__ jf
'IT^sMJ .(16),
where
c
c
L
J (17).
/i is
a constant coefficient in
d{
fJLx
/
ft
}
three equations and since
all
[
_
with similar expressions for the terms in y and z, it will be seen at once that equations (16) reduce to the equations on p. 310 for the
=
z=Q. Since the particular case when y when transformed to (17) retain their form
equations (16) and
any new
set of axes
but differently directed in space, if any one set they will be true for any other.
at rest referred to the first
they are true for But we have seen that they are true
when y = z =
;
they are
therefore true in general.
The
Kx Ky K
vectors
,
z
,
we
shall call the
components of the
force acting in the electron. When q 2 /c* is negligible the equations of motion are identical with those of Newton; otherwise they
are not.
We
shall extend the scope of equations (16) so as to embrace the case in which the forces are of gravitational origin. As their
applicability in this case is a sheer regarded as a definition of force.
assumption they can only be
Energy and Momentum. If we multiply each component of the universal equations (III) and (IV), Chap, xni, p. 286, in turn by X9 y ... z add them together and integrate over a space at whose boundaries the electric and magnetic forces vanish, we obtain
E E
r
IP J
where
L=
\Ex u
Jj [(Ej
-f-
EyVi
+ Ez Wi) dT H
H
,
dL = j
at
,
0,
+ Ef + E *) + * (Hj + Hj + H/)] dr z
the electromagnetic energy of the space considered. If the electric density p is due to electrons of charge e, the integral is
is
equivalent to
2
e
(Ex x +
Ey y + Ez z\
where 2L,
E...
#~
*
t.h*
THE PRINCIPLE OF RELATIVITY
313
from the electron itself) acting on the electron. Thus, if the of energy is to hold, the rate of working of the
conservation
E
E
on the electron is e(Ex x + y y -f z z). It will be xx + from z z. seen yy + equations (17) that this is equal to has the same relation to the defined rate above as force Thus the of working as in the Newtonian Mechanics. electric field
K
--
in the form
The equations (16) may be written jX 4"
-
== -K-xCvt,
^
"O"
K
K
with similar expressions in y and z. Multiplying each of these in turn by x, y and z, and adding, we get
/
The kinetic energy being equal be
\
gVc*
work done by the external
to the
forces will therefore
(xKx + yKy + zKz)dt = The constant
will of course
have
-..
+ const.
.
be determined by the
to
initial
=
If the kinetic energy was zero when q we should 2 The value of the kinetic energy for the have const. = //,c conditions.
.
*
system whose velocity
is
q
is
therefore
^ ,
** '
If p.
we multiply the second and
307, in turn
Chapter
x,
by H
H
z,
y)
E
z
^q
r ;V
""' -
agreement with the i
;
-
add,
dE,
-
.
---
we
.
;::;
(6),
obtain, as in '
,
*
>"
........ . ....... (is).
-
r;
;
te'^W*""!: dEx
in
Ey and
Ly
TTT
4
~ *
third of equations (a) and
and -
omitting the dashes,
.
2
( *.
//,
r;./.
For small values of q this is equal to value of ordinary mechanics.
fi ?
.
'
z
"
-
1
/
*
(
rr
.
*~
JR. a
y
~
^ xz
_
x
,
J ,
Integrating throughout a closed space over the boundaries of = and forces vanish, and remembering that div
H
which the div
E = p,
the whole of the right-hand side vanishes except
and we obtain
hence
- (Hz E -
%- j C Out J
y
HE y
t)
dr
+ ^Kx = 0,
so that
with similar equations in y and Since c
(Hz Ey
HE
z ) is
y
z.
the # component of electromagnetic
momentum
per unit volume of the system, the equations above express the law of conservation of momentum if '
f g
=
-p=^= Vl-f/c
is
the
momentum
........................ (20)
attributed to a point charge whose mass r)
We
2
have as in mechanics
= Kx
is
//-.
.
ot
These values of the kinetic energy and
momentum
enable
the equations- of motion of an electron to be written in the Hamiltonian form. The student who finds any difficulty with this
may be
referred to Einstein (Jahrbuch der Radioaktivitdt
und Elektronik,
vol. iv. p.
435 (1907)).
THE PRINCIPLE OF RELATIVITY
315
Experimental Test
For small values of q the laws of motion deduced from the of relativity are identical with the Newtonian Laws, but principle 2 2 no is this longer the case when # is comparable with c By .
observing the effect of externally applied forces on electrons moving with very high velocities we might expect to make a There are three functions of test of the principle of relativity. the velocity q relative to the observing apparatus which might under favourable conditions be capable of affording observations
purpose. These are (1) the potential difference required produce the velocity q, (2) the deflection of the path of the
for this
to
moving electron by a stationary electric field, and (3) the correIf e is the sponding deflection produced by a magnetic field. charge on an electron, the potential difference V required to increase its velocity relative to the observing system from zero to q is
given by the equation (see
p.
313) v (21). '
For the electric and magnetic deflections, consider the case which the direction of q is instantaneously along the x axis. If a magnetic force along the y axis and an electric force Z in
M
along the z axis act on the electron,
its
equation of motion will be
:M The path
is
therefore curved in the
taneous radius
R
of curvature
is
scz
plane and the instan-
given by
q/R =
drz
-^Cit"
.
The
and magnetic deflections are therefore measured respecby
electric
tively
M
~
_
p.
c-q
These deflections vary with q in exactly the same way as those calculated in Chapter xi, for Lorentz's contractile electron.
THE MANCIPLE OF RELATIVITY experiments of
Kaufmann and Bucherer.
As the
So far it has not been possible to test the re relation , ion K V and q between experimentally With th* v ,r "' mann F is not under he KaufcltrVof /h "* cathode rays it is not
^^^
^0 $7Z ^
a o/ Energy.
Now,
M1
= JL.
i
Wl
Also
=
.
7*
THE PRINCIPLE OF RELATIVITY
So that
GXX + T v, + Z w^ dr t
e
jdt jp
=
Ida;
jdt
3.1V
fdy }dz
= jdt fdx Hdy'dz'
F (x, y, z, t) F' (at, y',
z', tf)
(Z.V + F.V + Zt'vf) dr'
'
+ pv
f
Idif
fp (z.' \
+
N
'
e
-
^M 6
J
J
Since the principle of relativity must also apply to the system may be written
S' this
fdE is
= ft
fdE'
+
ftv
JtStf,/]
M
.............. (22).
Consider the case in which the motion of the system as a whole it is at rest relative to S', and suppose further that the
such that
velocities of its parts relative to S' are so
small that
2
v /c
2
may
be
considered negligible. The centre of mass of the system is thus at rest relative to S' which, under the further condition postulated, for all values of if. can only be the case provided 2j?y = In not will for this of this innecessarily vanish; ^\^K^"\dif spite tegral is not taken between given values of if but of t, so that in
general the limits will involve #' as well as
if.
If, however, the external forces do not act except interval considered, the parts which would otherwise
during the depend on
This statement will be made clearer in the sequel where the general case is considered. If the forces vanish entirely outside the time limits then we have
a! also vanish.
so that
dE
We
therefore conclude that the energy of a uniformly moving system which is not under the influence of external forces is a function of two variables, namely its energy Q relative to a system
E
THE PRINCIPLE OF RELATIVITY
3J8
moving with
of axes
and
it
Thus
dE
1
^
so that
of translation v relative to
its velocity
the standard system of axes.
7f
E
= 0. This Q evidently the energy of the system when of a case for the material determined been has already point (see A/T
(v) is
p.
313),
and was found
to be
The value for the whole uniformly moving system will therefore be obtained by addition of all the masses and is equal' to
Vl where
S
The com-
denotes summation over the whole system.
plete expression for
E
is
therefore
n'
........... (23) -
'
Comparing namely
this
with the formula
which
for the case in
we
see that, so far as the part of the energy which velocity v is concerned, the effect of the energy
E = 0,
depends on the
E
Q
is
to increase
E
the apparent mass of the system from S/*. to S//, 4- Q /c 2 Thus an addition BE to the energy of any system will give rise to an increase SE/tf in its mass. It is only a short step from this result to the hypothesis that
all
mass
is
.
simply a manifestation of
confined energy.
The question whether of confined energy
portance, and it is the test of experiment. is so
or not
mass
is
simply a manifestation
obviously a matter of the very utmost imvery desirable that it should be submitted to is
The energy liberated in chemical "dead" masses involved
small compared with the
actions
that
it
THE PRINCIPLE OF RELATIVITY
319
supposing that weight as well as mass is proportional to /-& + E/C-. In the case of radioactive change the matter is more hopeful. The decrease of mass of a system due to the loss of. 1000 gm. calories Now 1 gm. atom of radium in radioactive 4*6 x 10~ n gm.
is
4 equilibrium evolves about 3*024 x 10 gm. its diminution of mass per hour would be
3-024 x 10 4 x 4-2 x 10 7 9
_
,
n
cals.
per hour.
Thus
A
x 10 20
This would amount in a year to '012 mgm. or in 100 years to It would be worth the while of anyone who could 1*2 mgm.
make
afford the necessary capital to
observations of this character.
Momentum.
We
have seen that
,
the x component of
charge, satisfies the relation dt
-
= Kx
Let us apply this result to
.
the system, previously considered, which
boundary impervious
to
external electric forces
radiation and
X Y Z e,
e
ej
,
momentum of a point
is
is
surrounded by a closed
subjected to the action of
Then the
etc.
total
momentum
gained by the system in the time during which the forces act will
be x dt
Idt fd
J
v ;+ -
IP'
N; (Z.V + F.V + Zlvf) dr'
Let the system move as before, so that its centre of gravity ' remains at rest referred to S', then 'S X = 0, and if in addition the forces are zero outside the time limits considered, the second I
integral vanishes
and
K
and therefore
d^=B-dE' ........................ (24).
JrxviJN
JLJtmj
L/
therefore also a function of the energy referred the moving system and of the relative with moving referred to the standard system. latter the of We velocity q (= v) have in fact
The momentum
is
to axes
sothat
T/T'
(v) is
obviously the
been determined
E = 0.
momentum when
for a point charge,
Q
and
it is
This has already clear that for the
whole uniformly moving system
where
S/u. is
complete
sum
of the masses contained in the system.
expression for
the
momentum
is
The
thus
EQ on the momentum of an increase of the mass by the amount in agreement with the change we have found it to produce
Thus the
effect of
the system EQ/C~,
the
the internal energy
equivalent to
is
in the energy
itself.
Forces Continuously Operative.
Let us now consider the case referred
to
on
forces continue to act outside the limits of the
fining ourselves to the case
p.
317 when the
time
t,
still
con-
where the centre of gravity of the
moving enclosed system remains at rest relative to S', so that for any instant tf ^K^ is still equal to zero. Under these circumstances the integral f&K^dt' would vanish if the limits of but as they are given values integration were given values of of t it does not. Let the time limits with respect to S be ^ and &>; then the limits for tf are determined by the equation (see p. 300) t
',
They
are therefore
| p
- -2 x' C"
and
-|
p
-. x'. C"
THE PRINCIPLE OF RELATIVITY
The
limits for
We may
t'
thus depend on the
of
321
coordinate as well as on
split the integral to be evaluated into three parts thus
f J
ti
__
V_ j
t.
:
+/*V J
ti
J
cz J?j
The middle x.
We
integral vanishes, since its limits do not depend on are not able to evaluate the other two integrals in the
general case in which K^ varies in an arbitrary manner with the time. However, the most important case practically is that in
which the change of K% during a time comparable with vx'[c? is negligible, and in this case the integrals assume a very simple Under these conditions we have form.
f
J
The sum
ifj
'
-V
of the two integrals
may
evidently be written
calculation of the energy and momentum is now easily carried out in the manner previously employed, and we- obtain
The
*
vr^
,
K
In these formulae x is to be interpreted as the of component of the external force and x' the of coordinate of its point of applicaIn the particular case where tion, both referred to the system S'. arises from a uniform external hydrostatic pressure acting on the boundary of the system, equal to p Q when referred to the system of axes S', then if F" is the volume of the enclosed space referred
KX
to the
and
same
axes,
V
V1-03/C
2
Vl-0 / caJ 3
(29).
THE PRINCIPLE OF RELATIVITY
322
Examples. If the
moving system
consists of electromagnetic radiation
enclosed in a massless closed boundary, the energy referred to axes relative to which the system is
and momenta
moving with
velocity v are (30), ' s
tfc
where with
E
Q
is
v
-_
g
__
E =
-.Q
Vl - ^2 / c2
v .
p
jjj
.................. (ol),
c
the energy of the system referred to axes moving
it.
These are the values on the assumption that the boundary walls are rigid. If, however, the boundary is perfectly flexible the radiation pressure will have to be balanced by an external pressure
p
given by the relation
EV =
T
In *uthis case
,-
Vl -
The
principle of relativity leads to the conclusion that almost all physical quantities are functions of the velocity v of the system relative to the axes of reference. It would lead us beyond the
scope of this book to go into the matter in detail, but the following table of corresponding values is instructive :
Value
Physical Quantity ^ J
refe
d to
xes
f moving with system
Pressure of hydrostatic type
pQ
Confined energy
#
Temperature
FQ
V alue
The reader who further
is
451, 1907).
...... (32),
E=frE<> T^ ~ T
...... (33),
Q ......
V^rto
may
(34),
...... (35).
be referred
Planck ("Zur Dynamik bewegter kgl Preuss. Akad. der Wissenschaften,
to
Systeme," Sitzungsber. d. 1907) and Einstein (Jahrbuch der Radioakt. p.
:
>S
p=po
interested in this subject
information
efen ed
system
% for
1
;
f to
u.
Elektronik, vol. iv.
323
THE PRINCIPLE OF RELATIVITY The Principle of Relativity and
the Aether.
Before leaving this part of our subject it is desirable to review the bearing of the principle of relativity on the question of the We have seen that if we existence of the luminiferous aether.
from the hypothesis that electromagnetic actions have their seat in a medium, the aether, which is at absolute rest, the known
start
facts
can be explained on the hypothesis of Fitzgerald that bodies when in motion relative to the aether. This contraction
contract
can be shown to be a plausible consequence of the motion through the aether. If this contraction is the only change due to the motion, the effects in moving systems would not be exactly correlated with those in fixed systems, although the differences, so far as the writer is able to judge, would not have been detected
been carried out up to the present. which were not accounted for by arise which Any might be such a scheme might explained by making the velocity of light a function of the motion of the system through the aether. On the other hand, the principle of relativity, which is in
in any experiments which have effects
accordance with
all
the
known
facts,
describes
them
in a simpler
and more symmetrical manner. It is clear that if the principle of relativity and its consequences are valid, electromagnetic experiments can never yield any information as to the state of rest or uniform motion of an aether. This follows since, in the last analysis, all the effects are then made to depend on the relative motion of matter. It is, in fact, quite unnecessary ever to bring the word aether into the discussion.
From
this
standpoint
it
is
desirable, perhaps, to state the
To specify any physical event explicitly. necessary to locate it in time and space, that is to say, to determine the four coordinates t, #, y and z of the time and place
matter somewhat more it is
which it occurs. The question arises as to whether we can be sure that two events which appear to occur at the same place at Can we successive times t and t' really occur at the same place.
at
be sure that the place which appears to be the same has not
changed in terms
It is clear that such a position in the interval ? until we have fixed on a set of axes Ox, Oy, Qz of which we can specify the position of the points
its
discussion
is futile
THE PRINCIPLE OF RELATIVITY
324
Let us suppose that we have fixed upon such a set of axes and that they are so chosen that the physical system in which the events occur is at rest as a whole when referred to these axes.
Having marked we may suppose
our axes in terms of measuring rods, which to be part of the system, we should naturally turn off
This could be done our attention to the measurement of time. of a series of clocks which we could compare with one
by means
another at some fixed point, let us say at the origin. It would, however, be necessary to have some means of comparing them
when they were moved away
so
as
This
one another.
distance from
be at a considerable be done by sending
to
could
The simplest assumption we could make would be was the that propagated in spherical waves. By considering light the case of a wave propagated from the origin we see that x, y, z light signals.
and
would
t
satisfy the relation
# 2 + y 2 + z2
c
2
2
= 0, where
c is
the
The coordinates as thus determined velocity of light in space. would be consistent with each other and would be the simplest ones in terms described.
If
of
which the events in that system could be
we had no opportunity
to investigate other systems at rest relative
we should probably conclude that our system was to the medium in which the light was propagated.
Now
suppose that we have another system, let us say a distant solar system, which is moving with the velocity v relative to the
An
first.
investigator located in the second system would be able framework of axes and a set of coordinates xl ylt z
to discover a
and
ti
,
in terms of which light
would
system be propagated
in his
in spherical waves. He would find these coordinates the simplest in terms of which he could describe the events occurring in his
own
system, and he would have the same reason for concluding that the second system was at rest relative to the aether that the
first
observer had had for concluding that his was.
that one of the two conclusions
nothing to favour one rather
The
coordinates
x, y,
z, t
must be
fallacious,
if
we
and there
is
than the other.
and
os l}
yu zly ^ which
refer to a given Each set is
event are of course different for the two systems. preferable to the other for describing events in its
but
It is evident
own system
;
are to describe events in a universe in which both
systems exist there
is
nothing to choose between them.
It
is
THE PRINCIPLE OF RELATIVITY reference corresponding to one is as good as another.
325
the possible values of
all
In
fact, if
we wish
v,
and each
to represent the
whole universe in the simplest and most elegant manner, we cannot thus arbitrarily separate .time and space, but we must rather consider the x,
y,
z,
and
whole as a four-dimensional manifold of
t.
Four-dimensional analysis has now been devised with the It is found that the object of effecting such a representation. electromagnetic equations then assume a more symmetrical form than that in which we have considered them. The reader who is interested in these questions authorities
may be
referred to the following
:
Minkowski,
Raum und
Zeit
Leipzig, 1909.
Sommerfeld, Ann. der Physik,
xxxn.
vol.
p.
749,
and
vol.
XXXIIL
1910. p. 649. M. Laue, Das Relativitdts Princip.
To sum up we may say that the Principle of Eelativity furnishes no evidence either for or against the existence of an aether. It denies the possibility of determining the if it
exists.
In so
far as Relativity is
motion of such a
fluid
a Universal Principle
finds the aether a superfluous hypothesis.
it
CHAPTER XV RADIATION AND TEMPERATURE is a very familiar fact that when material bodies are heated of thermal, emit electromagnetic radiations, in the form they luminous and actinic rays, in appreciable quantities. Such an effect is a natural consequence of the electron and kinetic theories
IT
On
the kinetic theory, temperature is a measure of the violence of the motion of the ultimate particles and we have of matter.
;
theory, electromagnetic radiation is The calculation of this emisa consequence of their acceleration. sion from the standpoint of the electron theory alone is a very
seen that, on the
electron
complex problem which takes us deeply into the structure of matter and which has probably not yet been satisfactorily resolved. Fortunately we can find out a great deal about these phenomena by the application of general principles like the conservation of
energy and the second law of thermodynamics without considering special assumptions about the ultimate constitution of matter. It
is
to
be borne in mind that the emission under consideration
occurs at all temperatures although
it is
more marked the higher
the temperature.
The problem which we
set before us is that of finding the
nature of the radiation which
is found in an enclosure containing material bodies and maintained at a constant temperature. Sufficient time is supposed to have elapsed for any secular changes in
the enclosed matter to have come to an end, so that the nature and condition of any element of the matter does not vary. Special radiations of chemical or radioactive origin are therefore eliminated in so far as they involve progressive material changes. Even in
the steady state the interchange of radiation will be accompanied
by an interchange electric emission
of electrons arising from thermionic (see
Chap. xviiiY
The
effp.o.t
nf
and photo-
t,Vn'
nn
+>><*
RADIATION AND TEMPERATURE calculations
327
may be
eliminated by special devices, for instance by the surrounding radiating surfaces by an envelope of ideal matter
which
is
perfectly transparent to radiation but perfectly opaque and as it makes no difference in the final results we
to electrons,
the sake of brevity, leave
shall, for
it
out of account.
In considering a train of plane electromagnetic waves, the intensity is defined as the amount of energy which they transport in unit time across unit area of a surface perpendicular to their direction of propagation. If the normal to a surface is inclined to the direction of at an angle propagation, the amount of energy
would receive per unit area, in unit time, is equal to the In the case of the radiation in an intensity multiplied by cos 0.
which
it
enclosure the specification of the intensity
is
not so simple;
here the wave trains are travelling indiscriminately in
for
all directions.
do) be an element of solid angle and dS an element of area described normally about the axis of dco. If i (co) dSdco is the of the on in incident dS unit time, whose radiation, energy
Let
direction of propagation
is
comprised within the element of solid
then the intensity of this radiation. employed to indicate the possibility that may depend upon the direction of the axis of dco. dco,
angle i (a))
i (co) is
is
The notation this quantity shall see
We
that one of the characteristic properties of the radiation present in an enclosure maintained at a constant temperature is that i (co) is
independent of this direction. It is clear that if the normal to is inclined at an angle 6 to the axis of dco, the energy incident
dS on
dS
in unit time
which
is
propagated in directions lying within
dco is i (co) cos
The
6 dSdco.
radiation under consideration involves other elements in
composition beside the solid angle dco. We know that by of a prism or other analysing device it can be split into elements having different frequencies (v). The result of this
its
means
independent of the instrument used provided the latter The process or transform the energy in any way. absorb does not series. Fourier's is, in fact, a physical resolution into the equivalent analysis
is
It is therefore legitimate to express i as or as an equivalent definite integral,
possible frequencies between
and
co
.
an
infinite
sum
of terms,
extending over all the To complete the speci-
JtWlJJIAllUJA
-tt.lNJLf
iJCJlVJ.jr.lllXVA.JLlJ.rV.nj
This will be determined if the intensities of plane of polarization. the equivalent beams, polarized in any two mutually perpendicular direction of propagation, are given. We planes containing the them of one that which fix these contains planes by making may the normal to some arbitrary fixed plane as well as the axis of is the perpendicular plane which propagation. The second plane shall distinguish the contains the direction of propagation. intensities of the two corresponding plane polarized beams by the suffixes 1 and 2 respectively. In the light of these explana-
We
tions
we may
express the energy, incident on
whose frequency propagation axis
lies
between
lies
v
and
v
-f
dv,
dS per
and whose
unit time,
direction of
in a cone of solid angle dco described about an
making an angle 6 with the normal to dS, in the form {tj
(v&)
+ iz (VG))}
cos 6
dS dvdco
............... (1).
We
shall now consider what happens to the radiation which on any small material object placed inside the enclosure at constant temperature which contains the radiation whose properties
falls
we
are investigating.
the surface and part
and undergoing
Part of this radiation will be reflected from
may
escape after penetrating the interior and so on, but the
refraction, internal reflection
By absorption of radiation we underconversion into some non-radiant form of energy, so that denote the becomes temporarily stored in the matter. Let
remainder will be absorbed. stand it
its
A
proportion of the incident energy which is not absorbed. will depend on the nature of the substance, general
A
In the
frequency of the radiation, the plane of polarization and the azimuth 0. It might also depend on the incident intensity,
although this is usually assumed, on experimental grounds, not to be the case. For the whole surface the net absorption of
energy in unit time f
J
may
evidently be written
r['
SJ
J
The radiant energy emitted from the whole
surface
may be
written in the form
............... (3),
f
J
where
el
and
e2
sJ
o
may depend on
the nature of the substance, the
RADIATION AND TEMPERATURE frequency v, the angle on the temperature.
329
and the plane of polarization as well as and e.2 must be independent of ^ and i2 provided the enclosure contains radiating matter other than the body S and provided the area of S is vanishingly small compared <9,
el
with that of the other matter.
The only way in which the body S can either gain or lose energy is by interchange of radiation, so that the net rate at which its
energy increases f
J
S
TOO
J
/'27T
fa (1
J
is
- AJ -
l
+ i, (1 - A 2) - e2
}
cos d
dSdvda
.
..(4).
The value
of this integral must always be zero, otherwise the This would contravene the temperature of S would alter.
second law of thermodynamics
;
since the difference of tempera-
ture thus established could be used to furnish available work, which would then be obtained from a source at a constant
temperature.
This conclusion must be true whatever the shape,
and
size
nature of the body S, whatever its position in the enclosure and whatever the nature, shape and size of the enclosure and the other matter contained in it may be ; provided only that the enclosure
maintained at a constant temperature.
is
The fact that the integral (4) vanishes under all these conditions enables us to establish many important properties of the functions
i,
A
and
e.
In the
first
place the radiation must be same value at a given
to say, i must have the isotropic It cannot in space. for all directions point ;
that
is
be a function of
.
If
were, the intensity of the radiation in some directions would be By taking S to be a flat object it could stronger than in others. it
be turned so as to receive more or
less of the stronger radiation of Differences in different positions. temperature would thus be This result must law. second the set up which would contravene
planes of polarization. This is be a substance which absorbs some frequencies
frequencies and
be true for
all
clear since
S may
all
and transmits others, or it may be a plate of a material like tourmaline which has much more intense absorption for light polarized in certain planes than in others.
between the direction of the ray and the normal to angle may write (4) in the form
dS we
considering bodies of varying composition to make up S we and A 2 independently of ix or i2 as functions can vary e lt e2r l
By
A
of either S,
v,
x or the plane of polarization.
equation
i(l-A) =
It follows that the
........................ (6)
identically true for every wave-length, plane of polarization and element of angle. Thus for any assigned range of each of these
is
quantities the part of the radiation incident on dS which is absorbed is equal to the similar radiation which is emitted from
dS.
Next consider the case of a body which absorbs all the radiation falls on it. Such a body has been called by Kirchhoff " The term black body is convenient, although black." perfectly
which
may be rather a misnomer, as such a body may be very bright when the illumination is due to its own temperature radiation.
it
= 0, so that i = e. It follows For a black body, then, we have that the intensity i of the radiation in the enclosure is equal to the emissivity e, as defined by the preceding equations, of a per-
A
fectly black body.
It
is
clear that e
must have the same value
the same temperature so that i must be independent of the nature of the materials present in the enclosure. Thus i is a function only of the frequency and plane
for all perfect absorbers at
;
T
of polarization of the radiation and of the temperature of the the enclosure. so that in which the } By symmetry way
iii
The plane of polarization enters into i is a very simple matter. determination of i as a function of v and T will be considered later.
If a denotes the proportion of the incident radiation which is for each and so on. Whence absorbed, then a = 1
A
=i /a
has the same value for
wave-length
substances at the same temThus the emissive power divided by the absorption perature. coefficient for any substance depends only on the frequency and plane of polarization of the radiation and the temperature, and is independent of the nature of the substance. This result, which all
RADIATION AND TEMPERATURE was discovered by
Balfotir Stewart, is usually
331
known
as Kirchhoff 's
Law. results require amplification when the nature of varies from one point of the enclosure to another.
The foregoing
medium
the
The
relation
i
=e
no longer proves that
has the same value
i
everywhere in the enclosure, since the emissivity 6 of a black body may depend, and in fact does depend, on the nature of the surrounding medium.
If
we
call
/
the function which has the
same value at all points in the enclosure it is clear that the relation between i and I must be determined by properties of the
medium which have nothing For
emission.
there
is
if
we
directly to do with absorption or
medium where
consider a portion of the
no emitting or absorbing matter, but which
is
characterized
by a particular -value of the velocity of transmission, the approIn order priate value of i will somehow have to be established. to find
/
it is
not therefore necessary to concern ourselves directly
with the way in which e may be modified according to the nature of the medium in which the emitting system is embedded. All that is necessary is to consider the passage of radiation across the interface
between two portions of the medium characterized by
different velocities of transmission.
Let us imagine that one of the regions in question is separated from the other by a perfectly reflecting interface. The introduction of this cannot
The
radiation.
make any
interface
is
a small opening of area dS. velocity of transmission is F,
difference to the nature of the
punctured at the point A, leaving In the upper medium, where the is
a perfectly reflecting hemisphere
with equal and symmetrical apertures at equal infinitesimal solid angles da at A.
B
and If
G
AD
which subtend is
the direction
of the refracted ray corresponding to the incident ray BA, it is clear that with this arrangement the only radiation which can get
from the lower medium to the part of the upper
medium
outside
the hemisphere is that comprised in a small solid angle dot' about and the equal beam which is symmetrical about the normal
AD
AF.
The
rest is all reflected
by the hemisphere and returned
through dS.
The
ratio of the
two elements of
solid angle is
and the law of refraction gives
If i v dv denotes the intensity of the incident radiation of frequency between v and v -f dv, in the upper medium, then the energy of this range of frequency which is incident on dS in unit time is iv
dvdS cos
where p
6
The energy reflected along pi v dvdS cos 6 dco,
dco.
the coefficient of reflection.
is
reflection for the lower
which
is
refracted
is
medium
is
the upper
where
'
iv
A G is
is
If p is the coefficient of the proportion of incident energy
medium
1
p',
the value of
{
so that the
for the
energy transmitted into
lower medium.
If the distribu-
tion of energy in the spectrum of the radiation in the is
to
remain invariable, we (1
- p) i dvdS v
shall
cos 6 dco
two media
have
= (1 - p') i dvdS cos & do)'. '
v
333
RADIATION AND TEMPERATURE
Thus
iv
COS
dto
i f
cos
& dco'
t
I
p '
1
p
or If the composition of the radiation in either medium were to change, the temperature of a material particle in it which possessed
This would contravene the selective absorption would be altered. second law of thermodynamics, so that equation (7) must be true.
Now the left-hand side of (7) is independent of the direction and plane of polarization of the radiation, since the latter is Thus if we can determine the value isotropic at any given point. of the fraction on the right-hand side for some particular angle of incidence and plane of polarization we shall have determined it for all values. If 6 is the polarizing angle and the light is polarized perpendicular to the plane of incidence no light is reflected at the boundary and p = p = 0. It follows that p must always be 2 equal to p' and V i v must always be equal to F'H'/, so that the quantity which is invariable, which we previously denoted by /, 2 is not the Thus V*i v must be a universal intensity i v but i v x F function F(v, T) of v and T which we shall seek to determine. Also, if L v is the energy per unit volume of the radiation whose frequency lies between v and v + dv, we observe that i v oc VL V so that V*L V is also a universal function of v and T. -
.
,
Stefan's
We
shall first find
how
Law.
the density of the complete radiation Suppose that at some
depends upon the absolute temperature T.
point of the wall of our enclosure at constant temperature T there a cylinder in which a piston may be made to work up and
is fitted
clown. Both the walls of the cylinder and the face of the piston are perfect reflectors of radiation. The effect of an outward motion of the piston is to increase the volume which is filled by
the radiation and, in addition, work is done by the pressure p of the latter on the piston head. If the increment of the entropy of the system which is produced by a small displacement is denoted by dS, we have (8),
& A.LJL
AiX JL>
JLVJIM
J\.Ji.
JL
JiUTAJ. O
dU is
the increment in the internal energy and dv is the If L is the energy of the increment in the volume. complete have seen radiation per unit volume, then dU=Ldv. (Chap. the pressure of isotropic radiation on a surface x, p. 212) that Since the process contemplated is a reit is p ^L.
where
We
bounding versible one dS must, from thermodynamic principles, be a perfect differential
;
so that
dTdv'
dvdT
dU\
and
*
8 (*L\
3i and
377 OJ.
or
log
Thus
=
1
[1 fiH\
= I at
1
.L
^T ,
/fu (
1
L=4
9)
?
T + const.
log
i4P
(10).
It follows that the energy per unit volume, in vacuo, of the radiation in equilibrium in an enclosure at the absolute tempera-
ture T,
is
equal to a universal constant
A
multiplied by the fourth
the intensity of the power of the absolute temperature. Since radiation is equal 'to the energy per unit volume multiplied by the that the former must also be provelocity of light, it follows fourth to the power of the absolute temperature. Moreportional over, if
body,
E is the
we
total emission
see from p.
versal constant.
from unit area of a perfectly black
330 that
This result
E=A'T\
is
usually
where
known
A
is
as
Stefan's
a
new
uni-
Law.
was suggested by Stefan*, in the inaccurate form that the total radiant energy emission from bodies varies as the fourth power of the absolute temperature, as a generalization from the results
It
The credit for showing that it is a consequence of experiments. of the existence of radiation pressure combined with the principles of
thermodynamics
is
due
to JBartoli~f"
and BoltzrnannJ.
* Wiener Ber. vol. LXXIX. p. 391 (1879).
t
Bartoli, Sopra
Ann. der Phys.
i
movimenti prodotti dalla luce xxn. T). 31
vol.
i
etc.,
Firenze, 1876.
RADIATION AND TEMPERATUEE Since L, it
i
and
e
may be
split
up
into their spectral
components
have to satisfy equations of the
follows that each of these will
type ro
(L v ,i v
Jo
,
ev}
dv
= const, x T 4
Reflection of Radiation at a
There
is
another
plated, which
moving
is
effect, in
produced when
................ (11).
Moving Mirror.
addition to those already contemlight or radiation is reflected at a
This results in a change in the wave-length or
surface.
frequency of the light, which is akin to Doppler's effect. By considering the simple case in which the mirror is moving parallel to the direction of propagation of the radiation, which is incident normally, it is clear that the frequency of the light is diminished after reflection
by an amount which
when
of the reflector,
incident radiation.
beam that
the
this
is
is
in the
If the reflector is
of
frequency
proportional to the velocity same sense as that of the
moving towards the incident
beam is greater than The complete resolution of more most easily by means of the principle of the
reflected
of the incident beam.
complex cases
effected
is
relativity.
We shall consider the case of a plane reflector moving with uniform velocity v parallel to the direction of its normal, which we shall take as the axis of x. In the case of a ray incident at an angle 6 the vectors which specify it will space and time coordinates through the factor
---
/
cos
if
MM
n
the plane of incidence
is
/
,
nit
that of the
these expressions
as
7
and z axes and 7
x d -------z sin
cos
c is
h
The corresponding
\
provided the reflector
contain the
x cos 6 + z sin
stant which specifies the phase. reflected ray is cos
only
at rest.
is
a con-
factor for the
\ h
7
1
,
/
If the reflector is in
must be unchanged when
all
motion
the quantities are
mirror
and
is
t is
The
at rest.
between
relation
f
y
#',
f
t
and
x, y, z
given by the equations
moving with the
referred to axes
Thus
which specify an incident ray f (
x cos
4
t
#' 4-
/ sin ff
+7
\ ,
'
c is
mirror, the vectors
be expressed by
will
f
a cos n' where a
and
z'
,
J'
a constant, the corresponding quantity for the reflected
ray being
acosn
x'
u
cos ff
-z
r
sin
&
In terms of the coordinates referred to axes at rest we shall have for
the incident ray
a cos n
= a cos and
=
,
,
( rt
n' 4/8
( -I
n
p
A, f
1
,
- -v cos
)
vt) cos
8(x
2N
,
vx/G
(t
ff.
z sin
4-
-\
nA
,
*
0'J
/3 (cos 4 ^-^
^
&
&
47
We 2 ) a? 4 z sin
6'
,
.)
47
V
,
for the corresponding reflected ray
l/o f-t 1 a cos n {8 /
(
,
\
The frequencies
5' 4- v/c ^ -----
v
(cos c~^-
ziA ^ cos v )t c )
+-
c
2
--
)- A-
-
5 sin ff -------- -
k
,)
4- ry '
}
and t at rest in space are Thus if vl is the frequency of the incident and 2 of the reflected ray as measured by an observer referred to whom the mirror is moving with velocity v, we have referred to axes x, y, z
proportional to the coefficients of
t.
i/
_
- cos 6'
__+ 1
v%
c
m
'''l-^cos/ c
but from equations
(11),
Chap. XIY, cos
cos
=
-
RADIATION AND TEMPERATURE ai
-=
thus
2
neglecting
(y/c)
337
_i + 2
^jr/c
and higher powers.
The corresponding
relation
between the wave-lengths
is
Wieris Law.
The
foregoing considerations, coupled with thermodynamicai principles, enable us to take another step forward towards the dis-
F (v, T). This advance is due to Wien*. The argument resolves itself into two parts. In the first place, if we start with an enclosure containing only thermal radiation, such as we have seen to be characteristic of some temperature T and then alter the nature of this radiation by means of a motion of some part of the perfectly reflecting boundary wall, we shall be able to show that the resulting radiation is invariably such as is characteristic of some other undetermined temperature T'. Having covery of the function
3
established this proposition, the second step consists in making use it so as to find out as much as we can about the nature of the
of
function
F (v,
T).
Consider the cylinder with perfectly reflecting walls shown in The ends are closed by plates of radiating matter mainFig. 41. T.2 is greater tained at the temperatures T^ and T2 respectively. than T!. The transverse partitions are perfectly reflecting and which can be opened and are provided with shutters 2 l
D
closed at will.
or
The
along the cylinder. We occur 2 is shut and (!) :
D
tion characteristic of of
When
T-L.
(2)
The
D open. Then C is filled with radiaT arid A and B with radiation characteristic
the
B
l
D
J equilibrium has become established C is allowed to push the piston
radiation in
compress the radiation in in
D
partitions can also be caused to slide now imagine the following processes to
B
is
closed.
F
and so
until the pressure of the radiation
On account of the Doppler effect equal to that in C. will have become changed nature of the radiation in is
B
RADIATION AND TEMPERATURE
338
Let us examine the consequence of supposing that the radiation now in B is not homogeneous with that in 0. Since the total pressures in B and G are equal, the pressure, or by the motion.
B
and for others some frequencies will be greater in Place some in G. (3) selectively reflecting material over greater so that it transmits more of the it the opening in 2 choosing in than in G. Open the shutter an whose is density greater rays density, for
D
,
B
instant and then
close
it.
The pressure
in
G is now
greater than
Allow G to expand until the total densities again External work can be done by this expansion. and allow the two radiations to mix. 2 (5) Open the door back to its position after .the the door Leaving open push that in
J5.
become
equal.
(4)
D
F
F
D
back until 2 and push displacement in (2). Then (6) close the pressure in In this step the work is equal to that in A. lost in the second step is exactly recovered.
B
Fig. 41.
In the complete
cycle,
we have obtained work equal to
by the expansion of the radiation
that-
done
mixing without The heat which is
after selective
any transference of heat to the cold body Tlt the equivalent of the work done must have come from the body jT2 , As the process might be repeated indefinitely it is clear that the whole of the heat of T^ might be converted into available work in this way. As this is contrary to the second law of
thermodynamics does not
exist.
it
follows that the
inequality supposed in (2) true: If we
Thus the following theorem must be
with a perfectly reflecting enclosure containing nothing but radiation characteristic of a certain temperature, the nature of the
start
radiation will be changed if the walls are allowed to expand or contract, but the resulting radiation will always be identical with
that temperature radiation which exerts the
same
total pressure.
Now let us suppose that we have a cylinder fitted with a movable piston and filled with radiation characteristic of
RADIATION AND TEMPERATURE
The
temperature T.
339
face of the piston is perfectly
smooth and
The walls of the cylinder reflect completely, perfectly reflecting. i.e. without loss of This energy, but irregularly in all directions. device keeps the radiation isotropic and so makes it unnecessary for us to consider certain complications which would otherwise sectional area of the cylinder is A and its height h. with the position of the piston. Let L (X) d\ denote the energy per unit volume of the cylinder which belongs to wavelengths between X and X + c?X, then J=AhL(\)dX, is the total
The
arise.
h
will vary
energy belonging to these wave-lengths which is present in the cylinder. Owing to the motion of the piston the value of J will tend to change.
It will tend to decrease because the radiation
whose wave-length is
is
reflected at the
near
because other radiation
near
X when
it
X
moving
will join
piston,
have
will
its
The
reflected.
is
some other group when
and
it
will
it
tend to increase
radiation changed to values
rate of loss of energy
by the
clearly equal to the total amount which is incident on group the moving piston in unit time. The proportion of which belongs is
J
whose direction of propagation lies within any small solid = 2?r sin 6 d6 is c?o>/47r, since the radiation is isotropic. da angle to rays
Thus the amount
of energy belonging to these wave-lengths
incident on the piston in unit time in directions lying and within the cones whose semi-angles about the normal are
which 6
is
+ d6
is
iL
(X)
d\
.
sin
Odd cA .
cos
8.
The
rate of loss of energy by the group will be obtained by inteand grating this expression over all angles ff which lie between TT/2.
It is thus equal to
AcL(\)d\
........................ (14).
The calculation of the rate of gain of energy by the group is more complicated. If we consider a ray of wave-length X' incident at an angle 6 the wave-length of the reflected ray will be given by
-y cos0by(13). The change v to
of sign
be positive
for
due to the fact that we are now taking an outward motion of the piston, which
is
RADIATION AND TEMPERATURE
340
by X', d\f and 6 are changed into the group comX and X 4- d\, where between prised characterized
d\'
The energy incident by X', d\' 6 and d6 is
=r
d\
-
(1 V
cos 9
c
............... v(15)'
)
J
in unit time in the group of waves defined
}
%AcL(\')d\'$m6Qosed0 Only part of the energy of the
............... (16).
is changed into energy of the work done by the group the X group. The balance is equal to on the moving piston. The pressure exerted by the X' group may
X'
group
be written cos 2
ZAj The work done by
if
j=
L
this pressure per unit 2
2 Aj cos 6
By combining
this result with (16)
the energy of the group
group
X
(X')
d\'
.
2t
.
time
is
thus
v.
we
see that the rate at which
increased by reflection from the
is
X' is
Ajc cos 6
But j = i
sin
de
- ZAj cos
2 .
v .................. (17).
L (V) d\' and
L (V) = L (\ \
X cos c
0} /
,
from (13),
SZ (X) 2v ^ -= LT (X) cos 6 X + higher .
.
.
,
-
.
C
,
terms,
C/A/
by Taylor's theorem. By moving the piston slowly enough v/c can be made as small as we please, so that only the lower powers of this ratio need be considered. By substituting for d\ from (15) and neglecting higher powers of v/c than the first, we find f
After substituting this value of
AdXx = ~^
j(l sin e cos ^
j, (1*7)
becomes
2-COS0J i(X) rf6>
U
(*)
-
2 - cos
cos0.X-^-l(c (9
^2Z
(X)
+X
2ycos<
RADIATION AND TEMPERATURE to the
same order in
The
v/c.
total gain in energy is obtained to 7r/2, giving
by
integrating this over all values of 9 from
...... (19).
By combining
this with (14) we see that the X in unit time due to an
net increase of
outward motion
energy of the group of the piston is
A
dJ
L r/
.
N
-=.--t,|2I(X) But by
direct differentiation
+
,
,
dL(\)}
X-^ft,
......... (20).
we have
Thus, from (20),
1)
where K
= ^y-
is
a function only of the time
t.
The differential equation (21) gives the relation between L (X) and the time t as the piston is moved. As we have seen, the the motion
change the radiation in such a way that it is always identical with the radiation which characterizes some temperature T. Thus for each time t there is a corresponding effect of
is
to
temperature T, and T must, therefore, be a function of t only. Thus it must be possible to replace t in (21) by some function of
T
only.
This can be determined
if
we make use
of Stefan's
together with certain simple properties of the radiation which be regarded as given by experiment.
Law may
,3C
Consider the total energy density L
=
L (X) d\. J o
both sides of (21) with respect to d\ from
Now
to oc
,
Integrating
we have
the form of the experimental curves connecting L(X) and \L (X) vanishes at both limits, so that
shows that the product
dL
X
But by Thus
Stefan's law
Z = 4iT*,
where
A
a
.is
a universal constant.
or so that (21)
may be
written (22).
If
are
variables
the
),
changed (22) becomes
to
T,
= logT, X^logX
and
^ Now change the independent variables from 2\ and Xj to T! and TTJ, where ^ = ^ 4- X Denoting the new partial derivatives by S; we x
.
have
so that
L
and
an arbitrary function of TTJ where ^(TT^ Substituting that logarithms and multiplying both sides by d\ we find is
.
for
the
(23)
(24),
where
and % are undetermined functions of the product
T\
only.
It is known from experiment that L(\) has the value zero when X = or oo and a single maximum between these limits. Differentiating (24) by X we see that the maximum and minimum
values of
L (X)
satisfy the equation
3i(X)
,.
RADIATION AND TEMPERATURE
343
The value zero at X=oo is clearly indicated, but the value at X = cannot be foreseen without further information about the function
%
Presumably the
.
maximum
value
is
given by (25)
If this equation has a single root
which
L (X)
X T = b, then
the value
Xm
-
of
vary with the temperature in such a way that the product \ m T is always constant. This result, which has been well established by experiment, is known as
X, for
is
a
maximum,
will
Wien's displacement law. It is probable 'that the relations (23) and (24) which involve the universal undetermined functions and % of the argument <
we can
\T
from such very general considerations as have been employed above. To determine these functions more
are as far as
get,
it is probably necessary to consider the constitution of the radiating matter in a manner more explicit than we have done
particularly hitherto.
The Formula of Rayleigh and Jeans.
We
have seen that the properties of the radiation in an at a uniform temperature are determined by the
enclosure
temperature alone and are independent of the nature of the material bodies which are present in the enclosure. Thus if the enclosure contains nothing but radiation, the latter will have exactly the same constitution at a given temperature whatever the walls are made of, provided they have even the smallest power of emitting every possible kind of radiation. On the other
hand
it
is
evident that the nature of the contained radiation, established, must be determined as
when equilibrium has been
the result of an equality between the emission from and absorption by the walls. The constitution of the radiation ought therefore to
be determinate
if
we can
calculate the rate of emission
and
absorption of different types of radiation in any particular case. Since the nature of the radiation is independent of that of the
matter
it
makes no
difference
what constitution we assume
for
the matter, which we make use of in carrying out the calculations, Drovided that it is a Dossible tvr>e of matter and also is one which
has some capacity for emission in every part of the spectrum. To facilitate the calculations one naturally assumes the simplest type of hypothetical matter which is compatible with these requirements.
Of the attempts to solve the radiation problem which have been based on the principle thus outlined, the earliest to rest on a substantial foundation, and the most successful, is due to Before considering Planck's theory we shall briefly method of attack which has led to results that
Planck.
indicate another
are quite inconsistent with the experimental data.
known
result in molecular
It
dynamics (see Chap, xvn)
a well-
is
that
if
any
self-contained dynamical system possessing sufficient complexity is provided, with a certain amount of energy and left for a sufficient length finally
terized
which
become
by the is
of time, a state of statistical equilibrium will This state of equilibrium is charac-
established.
fact that each degree of freedom, or
required to specify completely the
each coordinate
whole energy of the
Now system, possesses the same average amount of energy. consider a perfectly reflecting enclosure containing a small amount of matter. The matter contains a finite number N, let us say, of molecules and each of these will have some finite number, p on the average, of degrees of freedom. On the other hand the aether
which the enclosure contains of
modes of
vibration.
will
be capable of an infinite number
These are determined by the geometry of boundary and extend from the gravest
the perfectly -reflecting mode of vibration to vibrations of infinite quickness. There is thus an infinite number of degrees of freedom in the aether,
whereas the number in the matter remains
finite. Consequently, each degree of freedom receives equal energy, all the energy will be found in the aether, in the final equilibrium state, and none in the matter. Moreover, for a given range d\ there are
'since
many more
possible
modes
of vibration the smaller
X
is,
so that
the energy tends to accumulate in the waves of infinitesimal wave-length. By calculating the number of natural wave-lengths between X and X 4- d\ and by supposing that there is an infinite -amount of energy in the whole system, so that each all
wave-length
RT
which is appropriate to two degrees of acquires the amount freedom in the matter at the same temperature, we can find the
amount
of energy -which occurs in the stretch of radiation
between
RADIATION AXD TEMPERATURE
X and \ + d\.
By
345
carrying out a calculation of this kind, Jeans*
has shown that D/77
q
L(\)d\,=
^-d\
(26),
or in terms of frequency instead of wave-length
L(v)dv-fyr*RTdv c
A
(27).
rather similar conclusion had previously been reached
by
Lord Rayleighf.
From
this point of
view the radiation problem reduces to a
determination of the number of modes of vibration in the aether
comprised within given limits of frequency. A simplified form of Jeans's calculation which is due to H. A. LorentzJ may be stated in outline as follows
:
Consider the temperature radiation in equi-
librium with a small amount of matter in a rectangular box, whose sides are parallel to the coordinate axes and of lengths d1 d>, d%. ,
The walls is
be
are smooth and perfectly conducting so that no radiation absorbed or emitted by them. In the steady state the box will
with stationary waves, satisfying the condition that the tangential electric intensity vanishes at eveiy point of the filled
boundary. Any parallel beam of radiation which is travelling in any one of the eight directions given by the combinations of n will always after reflection be the direction cosines 1, in, }
The totality travelling in one or other of this group of directions. of such groups will therefore represent the number of modes of
On account of the vibration of the aetherial part of the system. two planes of polarization which are required fully to specify the beam of radiation, each mode will have associated with it an amount of
of energy %RT equal to four times that of a single degree If X is the wave-length of the radiation under
freedom.
immediate consideration, the condition at the conducting boundary requires that __ ~
*
Phil.
Mag.
vol. x. p.
.
v
91 (1905).
t Phil. Mag. vol. XLIX. p. 539 (1900)
;
Coll.
Papers, vol.
iv. p.
483.
Since
where '.kl9 &2 and ks are integers. ,
If 2
1
.
4_
Z 2
2
2
_1_
I
+m +
2
2
ri*
= 1, we
have
J,
3 m
df*df*df~\* Thus this
and ks can have any integral values which satisfy Since the equation is that of an ellipsoid whose equation. A],
7c 2 ,
coordinates are kl} k2
and ks
,
,
and whose semi-axes
are
2c?1 /\ >
2d3/X, the number of vibrations whose wave-lengths lie 2cZa/X,-and between \ and X -f d\ is equal to the number of points having coordinates which are positive integers which lie in the volume between the
\ + d\
whose parameters are determined by X and
ellipsoids
respectively.
The number
of such points
is
equal to the
volume lying in a single octant between the two ellipsoidal shells. The number is therefore ^ rrdl dAd\l\4 and the energy per unit volume of the box and wave-lengths between the limits V and r
\+d\
'
4
It has been found that every purely of method calculating the radiation formula leads to dynamical is
87rjRjPc?X/X
.
(26) and (27).
Although the value of L(\) given by (26) satisfies Wien's functional relation (24) it is quite inconsistent with the results of experiments on the complete temperature radiation. The experiments show that L (X) has a maximum for a value of X whose position is governed by Wien's displacement law and has the value But (26) would make (X) increase indefinitely zero when X = 0.
L
There are a number of ways in which this Jeans has suggested contradiction might conceivably be avoided. that the final state of equilibrium, if it could ever be attained,
as
\ approached
zero.
would be given by (26); but that the rate of emission of the energy of short wave-lengths by matter is so slow that the would only become established after and in any finite time would not even bei approximately realized. According to this view the observed distribution is one of intermediate equilibrium due to less
distribution given by (26) an indefinite lapse of time,
fundamental causes.
Nevertheless the causes which establish the
intermediate distribution must be sufficiently fundamental to be common to all types of matter; otherwise the degree of consistency which has been observed radiation would not be found.
Another way of escape
is
to
in
experiments on thermal
deny the applicability of the
RADIATION AND TEMPERATURE
347
theorem of equipartition of energy to aetherial, as opposed to There are a number of reasons why this is not material, systems. a very satisfactory alternative. In the first place the same kind of objection may be made against the applicability of equipartition
Thus the specific heats of gases are not what would be given by a naive application of this theorem, par-
to material systems.
ticularly when one considers the large number of electrons present in the atoms and that each electron ought to have its quotum of On the other hand the law of equipartition has been so energy.
successful in other directions that it is difficult to believe that its
deduction from dynamical principles involves a fundamental error. In the second place the dilemma in question does not seem, to be connected with the law of equipartition alone, so that denial of the validity of this would not really remove the difficulty. It appears that a number of calculations which use the first method, that of equilibrating the absorption and emission of energy, lead to Eayleigh's formula (see for example Chap, xvn, p. 433). In fact this
formula appears inevitably to arise whenever the emission of radiant energy by matter is assumed to be a
and absorption
continuous process subject to dynamical and electrodynamic laws.
Although
it
may
appear very revolutionary to some,
it
seems
to the writer that the only logical way out of these difficulties is to deny the adequacy of dynamics and electrodynamics for the
explanation of the emission or absorption of radiation by matter. we leave the familiar guidance of dynamics behind it is necessary
If to
find
some other
set
of fundamental
principles
to
rely on.
Although the matter cannot yet be regarded as fully and satisfactorily worked out, a valuable start in this direction has unquestionably been made by Planck and his followers. Planck's Law.
We have stated that Planck has shown how theoretically to deduce a radiation formula which is in good agreement with the results of experiment. It would take up too much space adequately to consider the various vicissitudes
at the hands of Planck
and
which
this theory has
his critics, so that
confine ourselves to the discussion of the
we
undergone
shall practically
most recent form* of
it.
this
Although
involves special assumptions which
seem rather
is free from self-contradiction and from strange at first sight, it of the discontinuous nature of as that such assumptions, energy, which appear to do violence to the fundamental ideas of
physics.
So much could not be said of the earlier forms.
.
We
shall suppose the radiating properties of the matter to from the presence in it of an indefinite number of minute To fix our ideas we may take the picture electrical oscillators.
arise
by the electron theory and suppose each
afforded
oscillator to
consist of an electron in equilibrium in a. certain position, when free from the action of external forces, and subject, when displaced, to a restoring force
ment from the equilibrium
which
is
position.
proportional to .the displaceIf s denotes the moment of
equivalent to the displaced electron, then the kinetic and energies are equal respectively to \Ms* and potential the doublet which 2
|JVs
;
where
is
M and N are constants.
The
U=$Mf + $N**
total
energy
is
..................... (28).
If the component, along the axis s of the doublet, of the external the differential equation force due to the incident radiation is 8
E
satisfied
by
,
s is
Ms + Ns=E8
........................ (29),
we
neglect the reaction arising from the radiation. frequency i/ of the oscillator is
if
When
The
natural
the small force arising from the reaction of the emitted is included (see Chap, xir, p. 266) the equation for s
radiation
becomes
=E
s
.................. (31).
The
first step in the solution of our problem is to find the rate of emission and absorption of radiation by any particular oscillator.
The method
originally adopted by Planck is an immediate application of the principles of electrodynamics which we have already developed. The radiation is determined by the acceleration
of the electron, the rate of emission at any instant being proto the of in $ accordance with formula (11} of portional square
RADIATION AND TEMPERATURE
349
Chap. xii. The energy absorbed is equal to the work done by the external force s on the oscillator, so that the instantaneous
E
rate of absorption of energy
component which
E
is
Now
s s.
E
the force
8
is
the
along the axis of the doublet of the electric in However complicated the radiation the radiation. intensity it be will be to s as a function of the time t may possible express lies
E
by means
of a Fourier's series in the form
(32),
n=l
-L
/
where the (7's and S's are undetermined constants and T is an arbitrary time which is greater than every t. By substituting this value of Es in (31) and solving for s, we can show* that the
mean value
Ms
2
U
of the energy of the resonator,
since the kinetic
average,
which
is
equal to
and potential energies must be equal on the
is
__
Q~3
(88)
'
where (7no2 denotes the mean value of all the coefficients Cn for which n lies near vQ T. It happens that the distant terms do not r=o
contribute anything to the energy.
per unit volume of the radiation electric
of
and magnetic
when expressed
E,f + Ey + E + H + H y + H. *
mean value
(v)Av A?i/T,
Jo
L(v)dv
mean value
Since the radiation
is iso-
of each of these terms is the same, so that
;
By putting dv =
I
in terms of the
intensities is equal to half the
*
tropic the
But the energy
= ZE?
..................... (34).
a large
number, the integral on
o
where A?i
is
can be expressed as an infinite series, and by comparing with the value of Ef which results from (32), it follows that the
left
whence from
(33),
Z(*8 ) = 8ir^Z7 c
..................... (35).
This result is in formal agreement with equation (27), since the oscillator has two degrees of freedom and the energy per -I
__ *_
l-tf\t\f*\
,
RADIATION AND TEMPERATURE
350
when equipartition holds is %RT. Thus it degree of freedom if equipartition of energy is established that appears among the by actions going on in the matter (these may be be independent of radiation) we are again led to
oscillators
to
supposed
This formula in fact appears to result from Rayleigh's formula. which makes both the absorption and of calculation method every matter take place in a continuous the emission of the energy by manner.
It
is
well
to
point out
that
the "Rayleigh formula
results satisfactorily when v is small expresses the experimental and T is large, that is to say when the energy of the radiators is
There
large.
is
therefore at least
an element of truth in
it.
In order to arrive at a formula which does not make L(v) when v is infinite, it is necessary to introduce discontinuity
infinite
somewhere, and thus bring probability and entropy considerations In his earlier papers to bear on the state of the radiant energy. to were made the assumptions equivalent postulating that the itself
energy
had a discontinuous structure, but Planck has now
shown that equivalent
may be obtained by merely
results
sup-
emitted by jumps, the absorption As the emission of radiant energy
posing that the radiant energy
is
taking place continuously. might be expected to be conditioned by the breaking up of some structure present in the matter, this seems a very natural hypothesis.
investigation which follows involves a number of additional of hypotheses. In the first place we assume that the energy
The
U
the oscillators that
is
determined entirely by interchange of radiation
:
interaction of the
the influence of any direct dynamical if it occurs, has negligible consequences.
oscillators,
We
shall
assume that the absorption of energy by the resonators is continuous and follows the requirements of the classical laws of dynamics and electrodynamics. It is thus determined by the conditions laid down on p. 349. On the other hand we shall assume that emission of energy is not a continuous process but one which never takes place except when the energy U of an osdillator is an integral multiple of a certain element of energy
=
/?i/
and A
In this expression
.
is
z/
a universal constant.
is
the frequency of the oscillator The element of energy is thus
proportional to the frequency of the oscillator. that whp/n p.rmssifm
We
also
assume
.
RADIATION AND TEMPERATURE of
its
It is clear that the oscillator
energy.
emit when
351
must not invariably
an integral multiple of
e, otherwise the energy would never exceed We shall assume integral multiple unity. that whenever one of the critical values is reached, the ratio of the
its
is
probability that no emission takes place to the probability that emission takes place is proportional to the density L (VQ ) of the radiation surrounding the oscillator. If 77 is the probability that
omission takes place, the probability that emission does not take so that 77 place is 1 ;
(36),
where p
is
a constant quantity which we shall determine later. of an oscillator in the steady state may now
The mean energy U
be determined in terms of
77,
as follows
:
N
oscillators selected at random that have completed Out of an emission NTJ will emit when they have accumulated a single element of energy, N(l 77)77 when they have accumulated two n~l elements of energy, y when they have accumulated (I y) n elements of energy, and so on. Thus, in the steady state, out of oscillators selected at random simultaneously,
N
N
Ny = NP
where
Pn = (1
oscillator lies
will possess
n
energy between
and
e,
e
2e,
(w-le
we,
the probability that the energy of an between ne and (w-fl)e. The mean energy of an rj)
77
is
oscillator is therefore ............ (37),
since the average value of the fractions of an element of energy which intervene between any two consecutive integral multiples is Je.
Thus from
(36)
U={pl(v )+%}6 ..................... (38).
We
determine p so that (38) agrees with the corresponding equation (35) of the former theory when U is large. In this case the -| in (38) can be neglected, so that, by comparing with (35), shall
RADIATION AND TEMPERATURE
352
Thus the mean energy
U of the
the division of the energy completely determined.
oscillators,
among them
and
also, since
in the steady state, is
introduce the temperature T it is necessary to This is equal (see Chap. calculate the entropy S of the system.
In order
to
xvn, pp. 400 and 407) to E times the logarithm of the probability of the system, denned in a similar way to that introduced by Boltzmann* into the kinetic theory of gases. The present case is a little different from that contemplated in the kinetic theory, inasmuch as it is only the part of the energy of the resonators which is an integral multiple of e that is a matter of chance and
The probability therefore subject to probability considerations f. in which the resonators can be sought is the number of ways
N
arranged so as to have the given distribution of energy units, subject to the condition that the same distribution arises whenever the resonators which
interchanged
among
have a given number of units are This probability may be
themselves.
calculated as follows:
Along the positive energy axis lay off marks at the points From these marks draw lines perpendicular 0, le, 2e ... me ... oo to the axis and on them lay off JVP NP ly JfP m NP M = dots. there are -AT and their dots SZ\TP W Altogether equidistant .
.
,
distribution over the
diagram
is
.
.
NP
.
.
.
a geometrical representation of
the way in which the resonators are distributed about the energy of the system. Since the dots which have equal numbers of units are considered to give rise to systems which are indistinguishable, dots can be the number of independent ways in which the
N
arranged to form the given distribution
where
II
from
to oo
is
denotes the continued product for .
all
Thus
*
Vorlesungen liber Gastheorie, p. 41, Leipzig (1896). t Planck, Sitzber. der K. Pr. Akad. dor IF-/** i
the values of
m
RADIATION AND TEMPERATURE
and since we may use
when
N
is
Stirling's
approximation
353 for
the factorials
large
8 = R]ogN\-RZ {NPm (log NPm -
1)
+ $ log ^NPm
}
= R {log N - N log N + N -$S log ZirN} - NKZPn log Pn = 1 and remembering that the neglected term using 2-P I
,
N
vanishes compared with the others when is indefinitely large. The first term on the right depends only on and is therefore
N
constant.
It
may
be assimilated with the undetermined constant
always included in the entropy. The value of this constant does not enter into our calculations, so that, leaving it out of account, we have for the entropy SN of the resonators
which
is
N
= N8. And
since,
by the
definition of entropy,
U 1
and
e
U =
= hv,
Finally, from (38)
R,
dS
and
1
+2
^
(39),
zw
^(?-JU STT
and
e
hv*
RADIATION AND TEMPERATURE
354 If
iv
dv
is
the intensity of a plane polarized constituent of the
radiation, travelling in a
direction, given direc
since the radiation is isotropic.
If the
medium
is
from the value
c
we have
iv
o
=*L(v\
Thus
V
of radiation differs one in which the velocity in a vacuum the right-hand which it possesses
have to be multiplied by c*/V2 since, according to the conclusions on p. 333, V*i v is the function which has a universal value. Corresponding changes would have to be introduced side of (46) will
in (44)
and
(45).
,
We
shall,
however, confine our discussion to the
case of the radiation as it is found in a
vacuum.
Formulae (44) and (45) are those which are known as Planck's radiation formulae. They are in agreement with the functional relations (23)
and (24) demanded by Wien's argument.
These
formulae have been derived theoretically in other ways* which differ in important points from that which has just been given. But in order to obtain them it has always been found necessary to introduce discontinuity somewhere, either in the constitution of the radiation itself or in the mode of its absorption or emission.
When \T is
small, or
v/T
large, it is evident that unity
may be
neglected compared with the exponential in the denominator of the fraction, so that for small wave-lengths, or high frequencies,
and low temperatures, 3
~ h
(47),
(48).
When
v is large the exponential factor in (47)
much
diminishes
more rapidly than v 3 increases, so that L (v) when v = oo For = the same reason L(\) Q when X = 0. Thus the conditions referred to on p. 341 are satisfied by Planck's formula. On the .
BADIATION AND TEMPERATURE other hand
when v/T
is
\T
small or
355
large, (44)
and (45) de-
generate into
x
J
They are then identical with Rayleigh's formulae (26) and (27), which, as we have already pointed out, are in agreement with the results of experiments which are made subject to these conditions. In fact throughout the whole range of the variables X (or i/)
and T, which has been tested, the differences between the experimental results and those given by Planck's formulae lie within the limits of experimental error.
Numerical Values of
the Constants.
For the density in space of the black body radiation in a
vacuum we have
=
* dv /0
(50), C*i
where
a
= 1 + ~ + .T + ji + L O4
.
..
= 1-0823.
TB
The absolute constant a which Stefan's law
is
occurs in the expression of thus defined in terms of R, c and h. Accurate
measurements of a have been made by Kurlbaum*, who a
=
7-06 x 1C"
15
3
erg cm."' deg.~
finds f
4 .
Another relation between the constants is given by the value X for which L (X) is a maximum. If we differentiate L (X) by X and equate to zero, the equation which corresponds to (25) is
of
5(e-*-l) + ff= *
Ann. der Phys.
t
Some
vol. LXV. p.
than 20
759 (1898).
recent values of a are
finds a value about 10
%
(51),
much
% higher than Kuiibaum's.
this. Thus Gerlach (1912) and Drecq (1911) find a value more
higher than
higher, and Fery
Most
of the determinations, however,
have
RADIATION AND TEMPERATURE
356
where # = --& jRX./
x=
(51) has two real roots, viz.
Thus the maximum value
is
X
and x = 4'9651.
given by substituting #
= 4*9651
T=-~ = b
in
...(52)
of Lummer and Pringsheim* According to the measurements and h we find b = 0*294 cm. deg. Solving (50) and (52) for
R
R = 1-346 h = 6-548 Since for
R
a single
x 10~
16
1
erg deg.-
x 10- 27 erg
sec.
)
j
=
is the constant in the gas equation pv R^T reckoned enables us to deduce the molecule, this value of
R
value of N, the
number
of molecules in a cubic centimetre of a
and the charge gas under standard conditions, well-known data. We have pv = NRT, where j>
= 76x13-6x981
2 dynes cm.-
N= 2'76 x 10
Whence
Since the charge which
is
v=lcm.
,
19
per cm.
:l
on an
e
and
from
ion,
T= 273 deg.
3 .
required to liberate half a cubic centimetre
of hydrogen, measured under standard conditions, in electrolysis 0'4327 E.M. units (see Chap, i, p. 6) it follows that
Ne = 0-4327 e = 4*69 x 10" = 18*9 x 10~
10
whence
10
is
E.M.XJ.,
electrostatic units in our units.
These values of 'N and e are in excellent agreement with those which have been found by Millikan, Rutherford and others, using This agreement must be regarded as supinore direct methods. porting very strongly those assumptions in Planck's demonstration which are necessary to produce the formulae finally obtained.
We shall see that Planck's radiation theory has recently received unexpected support in two other directions. be considered.
One
of these will
now
Radiation and Specific Heat.
From
the
phenomena exhibited by absorption bands, the and so on, we know that something like Planck's must exist in actual matter and possess natural
residual rays, oscillators
RADIATION AND TEMPERATURE
357
frequencies not far from the frequencies of the radiation forming the visible spectrum. From (42) we can calculate the energy
corresponding to any one of these natural frequencies at a given temperature. Einstein *
suggested that practically
all
the energy stored in
simple bodies might belong to a few frequencies, and, on this hypothesis, was able to calculate the specific heat as a function If this hypothesis were not true it would be of the frequency. to necessary suppose that the number of systems having a given
frequency is much smaller than the number of atoms present in the substance otherwise the specific heats of bodies would be ;
much
larger than those which they actually possess.
N
Let us suppose that there are oscillators per unit mass of any substance, and that they all have the same natural frequency v. The total energy of the oscillators in unit mass of the substance at the temperature
T
is,
by
(42),
hv
)
~
If all the heat
is
+ "2
in this kind of energy, the specific heat
be given by differentiating
this expression with
Cv
will
respect to the
temperature, so that Jtv
^\ =$-&2 J
]lv *.
dT[ ~*L RT \e -l
J
At low temperatures, according
since
we can
denominator.
AT
*!i
T
V
"*
(e* -l)'
to this formula, 7
av ~
RT*f
ll
o *
o
v~
hv
~wf
RT*
neglect unity compared with the exponential in the Thus Cv rapidly becomes extremely small with
At high temperatures Cv approximates
decrease of temperature. Jn^
NRe RT = NR
approx. It thus becomes independent of the if we suppose that there is one natural freand temperature; is the number of atoms so that per unit mass quency per atom, to
N
of the substance, the value
and
Petit's
Law.
NR agrees quantitatively with
Dulong
RADIATION AND TEMPERATURE
358 ;'
An
number
elaborate investigation of the specific heats of a large of substances, particularly at low temperatures, has
Nernst* and his pupils. They find recently been carried out by that the relation between Cv and T is of the same general character as that called for
when
by
(54),
but that a better agreement
a slightly different equation
is
used,
is
obtained
viz.
The second term, which was introduced by Nernst and Lindemann, was obtained empirically.
The agreement between the results of experiments and (55) is shown by the following numbers, which represent typical cases selected at random from a paper by Nernst and Lindemann f.
Cp
the specific heat of the solid substance at constant pressure and is the quantity which is given by the experiments. The calculated values of Cp are obtained from the values of Ov is
given by (55), together with the known dynamic considerations,
<7,-C.(l *
+
relation,
based on thermo-
9|gr)
Zeits.fur EUUtrocliemie, vol. xvn. pp. 265, 817 (1911).
(56),
RADIATION AND TEMPERATURE where a
is
359
V
the coefficient of linear expansion,
the atomic
is
K
the compressibility. The difference between volume, and and Ov is only appreciable at the higher temperatures.
Gp
'It
will be observed that (55) contains only one adjustable constant, the frequency z/, so that the agreement shown by the table is quite convincing. In fact in the case of KC1, KBr and
NaCl the
value of v has been taken from the experiments of
Rubens and Hollnagel on the residual rays from these substances. Thus in these cases the complete thermal behaviour may be predicted from the determination of a single optical constant, and shows an excellent agreement with the observed results. In other
Y
01
0,2
0,3
0,4
0,7
0,6
0,5
0,8
0,9
1,0
Fig. 42.
cases, for instance mercurous chloride, it is necessary to use a formula involving a summation over two different values of v\ but there are good reasons for assigning at least two frequencies for
compound
substances, so that this cannot be considered to be
an argument against the general
The general
position.
character of the relation between
Gv and
-5777,
given by (54) and (55), is shown by the accompanying diagram The ordinates represent (Fig. 42) taken from Einstein's paper. the values of the right-hand side of (54) and the abscissae those hv i s taken = 5'94. The calculated values of Cv are of ot
RT
.
.
shown by the broken
The
curve. -
/-v4-'
circles /.-
vT-v/-vi,-
represent the rt-
o r^v\v/-vv\Ti o
i-
older
a ATOM 10
KAD1ATLOJN AiNU TJliMJflfiKATuRE of v being assumed.
(55) than with (54),
The newer
observations agree better with of the curves is tie
but the general character
same.
* has deduced formulae for the In a recent paper P. Debye heats at various temperatures which agree even "better specific than (55) with the experimental results. Debye's method, Avhich is a very interesting one, brings the specific heats quite directly
into relation with the elastic constants of the substances concerned,
The formulae obtained are also quite simple and general, and only contain the same number of parameters as those of Einstein and This also represents an important advance, known to be inaccurate and (55) has never received a
Nernst-Lindemann. as (54)
is
satisfactory theoretical explanation.
Another direction in which Planck's theories have received be considered under the heading of photointeresting support will electric action in *
Chapter xvni. Ann. der Phys.
vol.
xxxix. p. 789 (1912).
CHAPTER XVI THE THEORY OF MAGNETISM
WHEN we regard magnetic phenomena from the point of view of the disturbance produced by the material^ media in which the effects take place we are struck by the great variety of phenomena manifested as compared with those in the electrostatic case. When a plane slab of dielectric is placed in a uniform electric field in free space, so that the lines of force are perpendicular to the face of the slab, the electric intensity is invariably smaller inside
the dielectric than in the surrounding space. In dealing with magnetic phenomena an effect of this nature is by no means the invariable rule.
In comparing the behaviour of magnetic media they are found to belong to one of three distinct classes.
Diamagnetic media. Substances of this class are characterby a permeability which is less than the value unity attributed to free space. They therefore tend to move into the weakest 1.
ized
parts
of
the
magnetic
field,
as
potential energy of the system a
this
arrangement makes the
minimum.
Paramagnetic media. The permeability is constant and These substances tend to move into the greater than unity. 2.
Their behaviour is thus analogous to strongest parts of the field. that of homogeneous dielectrics in the electrostatic case.
Ferromagnetic media. The permeability is greater than The polarization (magnetization) tends but is not constant. unity to reach a saturation value as the magnetic intensity is increased. The high value which the polarization may reach is a characteristic 3.
feature of this class of bodies.
There is another very important and fundamental difference the magnetic and electric behaviour of matter. It is never possible to separate the positive and negative magnetic charges in
Til Jb
m
IIJ&UJK,J:
different portions of
matter in the way in which the
electric
This would be extremely unlikely to charges can be separated. be the case unless the fundamental magnetic element contained both positive and negative magnetism and were therefore similar to an electric doublet rather than to the positive and negative is constituted. charges of which such a doublet This
difference
basic
receives
a ready explanation on the magnetic forces can
to that theory
electron theory.
According from the motion of electric charges and, in the last from the motion of electrons. Now there is no possible
arise only analysis,
motion of electrons which can give rise to an isolated magnetic of motion which gives pole; but there is a very simple type rise to a system which is the magnetic analogue of the electric doublet, i.e. a system which has positive magnetism on one side
and negative on the other. Many years ago Ampere built a up theory of magnetic media on the assumption that the atoms were the seat of circular electric currents. As is well known, such
of
it
a current behaves like a small magnet, and the hypothesis is therefore all that is required to account for magnetic polarization and hence, from the analogy with dielectrics, for magnetizable media. Now we shall be able to show that an electron revolving in a closed orbit
is
equivalent to a small magnet in the same way that This theory, whose developwere.
Ampere's atomic currents
is due largely to Weber and Langevin, will be shown to give a simple explanation of diamagnetism as well as paramagnetism. With certain further assumptions which do not seem improbable,
ment
it
made
can be
phenomena
good account of the more complex
to give a
of ferromagnetism as well.
The Magnetic Force due
to
a Moving Electron.
We
have seen (Chap. XI, p. 221) that the components of the magnetic intensity due to an electrically charged particle moving with the uniform velocity
w
parallel to the z axis are ft 1 ,.
u
where
/3
=
w/c.
When
the case in the motions
/8 is
we
= e/r,
_ =
small,
&
--_===_
Vl-
9
s
which we
are dealing with, #!
=x
and
yl
rr ,
JJL
>
=
n U,
90i
,8
=
shall suppose to be
we can put y.
THE THEORY OF MAGNETISM Tims
w = --
'.-
e
dr
,
c
We may
~5
T~
-"y
j
~"
c
r*dy
"2 5~~ r^ dec
'
therefore write the resultant magnetic intensity as
The magnetic
force
is
(2).
3
cr
evidently distributed in circles about the
axis of motion.
The value
of the magnetic force above is that for a charge e It will, however, give the instan-
in uniform rectilinear motion.
H in
motion provided the acceleration is not too great (see Chap. xn). We shall suppose that this condition is satisfied by the intra-atomic motions which give rise to the magnetic quality of bodies. Let us seek an expression for the component in any direction of the magnetic taneous value of
intensity at
cases of curvilinear
any point Q due
an electron moving in a closed
to
orbit.
Let
it
intensity
be required to find the component Q (Fig. 43). Resolve
H at
H
z
of the magnetic
the velocity at every point of the orbit
two components, one parallel to Oz and the other perpendicular to Oz. The components parallel to Oz con-
into
tribute nothing to the value of
other components
H
z.
The
whole can, be represented by the projection of the original orbit on a plane perpenfor
the
y
orbit,
PMN
Let be Let the velocity at
dicular to Oz. jection.
this proin the
P
projected orbit be along PR, and take the origin so that it lies in the plane of the original orbit. and then
PMN
Join PQ.
The instantaneous
due Let
it
QR
perpendicular to
SO.
OR
to a
moving charge
be equal to
and
OR
at
PR
Fig. 43.
lie
in the plane xOy.
H
resultant magnetic intensity
P is
perpendicular to the plane
QS and draw SZ
PR
perpendicular to Oz. Then all the lines OZ, ZS, t.n PR M/^.*. fk, or^l,^
and join OR.
are nornp.nrlimita.v
k
and
QOR
are right angles, so that the triangles ZQS, are equal. Thus
SZQ,
SQR
QRO
are similar and the angles ZQS,
H /H = ZQ/QS = Z
where
p
is
QRO
R/RQ = p/r
sin 0,
the length of the perpendicular drawn from on the and r, are the coordinates of Q
line of motion, projection of the
P
with respect to the point
as
origin and the projection of the line of motion as axis. But we
have seen
that
the
^
.
.
same notation
We
shall
resultant
H
magnetic intensity
in
ev sin
is
cr 2
the
, ,
now apply
whence
this re-
sult to find the average value of
the components of the magnetic force at any point arising from
P
the motion
of an
electron in
an approximately circular Let QRS (Fig. 44) be the its
centre and
ON
orbit. orbit,
the polar
Let the angle PON=0. We shall call the average component of magnetic intensity along OP the radial component and the average force at right angles to OP in the plane of the paper Pio
.
axis.
44
the tangential component.
The Radial Component Consider the motion at any point Q of the orbit. It may be resolved into two parts, one parallel to OP and the other in the
QRS
direction of the tangent at Q' to the projection Q'RS' of on a plane perpendicular to OP. The component of velocity parallel to contributes nothing to the radial component of the force
OP
The
direction of the force arising from the component of a in direction perpendicular to will always be pervelocity to the radii SP and so on. will thus lie along It pendicular QP,
at P.
OP
THE
TJi.fcU.KJC
ur
iiiAvji
-,/,*,
*.-
PP PP
and so on. The radial component arising in the same sign as we proceed round the have this way will thus Provided the dimensions of the orbit are small enough it orbit. not matter if we take the moving point 8 to be at will the lines
19
2
evidently
S',
Q
at Q'
and
so on.
To the same order
PQ =PS = r
treat the lines
as constant.
of accuracy
Hence from
we can (3)
the
Now Hr^^pv. c
the
>
average value of the force along
OP
is
area of the curve average value pv is clearly equal to twice the Q'ES' divided by the periodic time T of the orbit, p being the on the tangent to this curve. Hence if 8 perpendicular from is
the area of the original orbit
QRS, pv
= 2S
cos
ff/r
and (4).
The Tangential Component. This is not so readily found, but a similar method of treatment may be applied with success. Take OP as the axis of s. The tangential component of force sought is the force in a direction
PT
(Fig. 45) perpendicular to in the plane PON. Call this the axis of y. The axis of x is thus per-
OP
Resolve pendicular to the plane PON. the velocity at any point S of the orbit into
its
components
parallel to
PT
#, y,
z.
will contribute
y being nothing
to the tangential
component at P. Let us consider the effects of the z and x Let the (k tted curve represent the projection of the; original orbit on a plane containing OP and perpendicular to the plane NOP. The dotted curve will thus be a representation of the x and z velocities and its area = ft sin (9. Consider the 3 com-
components separately.
ponent of velocity arising from this
The y component of magnetic force at P always in the same direction whether thr
first.
is
particle is above or below the plane of the paper, on aeeount of the change of sign of the velocity.
The magnetic
Q
motion at
is
P arising
force at
equal to
,
ez
~ -n T>
.
is
PT
.
which
: ;
QR '
cQP*
to the plane
perpendicular
the component along POQ'. This is equal to
ez
QPR
sin
This force
from this component of the
POQ.
What we want
is
perpendicular to the plane
is
ezQE is the line of intersection of the planes QOP and Q'OP. of the effect being in the same sense all the way account On we may to the first order in OQ/OP put orbit the round
since
PR
PQ = OP = r = const. 'The component arising in this
way
e
[
thus the average value of
^,^dz
e
1
is
.-0. - sm
dz
r cr**\-ji at J
or
Now
r
consider the y component of the force at
P
which
arises
from the x component of velocity in Let VQR (Fig. 46) be the
the orbit. orbit,
V'Q'R'
its
Let
plane of xz.
projection in the be the dia-
SOU
meter perpendicular to OP and ON] this diameter will be perpendicular both to the element of the actual orbit at S and to its projection. Contwo points Q\ R' symmetrically U. situated with respect to the axis The corresponding points in the orbit
sider
are Q, R.
Fig. 46.
d)
motion at
Q and
the coordinates of
It will be seen that the
P
arising from the tangential force at If is oppositely directed at the two points.
R R are x,
y, z,
those of
Q
are
x,
y,z.
More-
over
If
OP
is
sufficiently great
compared with the dimensions of the
orbit the average force in the
motions at
R
Q and
ex
{
2C
[(OP "z?
P
arising from the
z
^
is
1
(OP
IV +
ex
2
z)
(A
2cOP* JV
j
= The average value
y direction at
c70P
x
as
'
OP,
OP"
of this taken all round the orbit
is
The sign of this force is determined by the direction of the motion at the side of the orbit nearest to P and is evidently The balance of opposite to that arising from the component z. tangential force
is
thus
*";'" The average value
< 5 >'
component of the magnetic force and PT vanishes. Because if AB is
of the
perpendicular to both OP with the plane of the the line of intersection of the plane orbit the latter can be divided into pairs of elements dS dS.2 which are symmetrical about and are equidistant from P.
NOP
} ,
AB
These elements produce equal and opposite effects at P so far as the component under consideration is concerned. The orbit can thus be divided into mutually interfering pairs of points yo that the average value of this component It
is
is zero.
evident from formulae (4) and (5) that the average value due to the revolving electron is
of the magnetic field of force
exactly equivalent to that which would be given
whose moment
by a small magnet
is
^
cr
and whose axis coincides with the axis of revolution of the electron.
An atom may in
general contain a
in closed orbits as well as others
about a position
of
static
numerous and distributed so
number
of electrons rotating
which execute small oscillations
equilibrium.
The
orbits
may
be
azimuths inside the atom, as to furnish no resultant magnetic moment: or they mav in various
THE THEORY OF MAGNETISM
368
an axis of symmetry with a resultant magnetic moment. In every case we shall see that a phenomenon analogous to diamagnetism occurs but in the case of the atoms for which the
possess
;
revolving electrons have a .resultant magnetic moment there are reasons for believing that this is marked by the paramagnetic or
ferromagnetic effects which supervene.
In developing a theory of the action of an external magnetic on the revolving electrons it is necessary to make some hypothesis about the nature of the forces which hold them in
field
their orbits
and which determine the orientation of the
orbits
with respect to the atoms. We shall suppose that the forces are determined by the structure of the atom and that the planes of the orbits are determined by the symmetry of the atom. When an external field
is
applied, forces are
brought to bear on the
re-
volving electrons which derange the previous state of motion. This displacement will give rise to a force acting on the neighbouring
atom which will, in general, cause the axis of the and so change the plane of the orbit. The state of are we things imagining is in fact much the same as if the electron parts of the to turn
atom
were revolving in a channel cut in a rigid non-conductor (the atom). If in the absence of an external magnetic field the orbits are arranged so that the atom has no magnetic axis, these forces will not give rise to any tendency to change the orientation of the atom as a whole. In such cases we shall see that the effects
phenomena like those exhibited by diainagThe same results would follow if the atoms were by interatomic constraints but as a number of liquids
produced give
rise to
netic substances.
held rigid
;
are diamagnetic such a supposition would not help in explaining diamagnetism. The hypothesis that the revolving electrons can
be treated like currents flowing round their orbits, which are more or less rigidly attached to the atom, simplifies the mathematical calculations very considerably.
Diamagnetism.
Let us suppose that, for reasons of symmetry or otherwise, the external field has no tendency to alter the orientation of the orbits in space. can calculate the magnetic permeability of a substance
We
if Wft
Pfl.n
f*A.l
mi
1
J^.t.A
t.llP
r-.Vianava
in
t.V*i
ormiTral^-nf.
w-k
rmn on f.
A-f f.ViA
THE THEORY OF MAGNETISM
369
the application of a revolving electrons produced by given external of the orbit projected on a area the Let S' be field H. plane Then the establishment of an external field perpendicular to H. will cause a flux HS' of magnetic force through the orbit. If
H
the orbit
is
circular this will give rise to
an
electric
intensity
E
where
tangential to the orbit,
G dt
CT
2<7TC
H
and the normal to /S, r is the the angle between the to angular velocity and 8 denotes the change periodic time,
where
is
per revolution.
But the moment
moment
of
of the force
momentum
Ee
will cause
an increase
in the
Thus
of the electron.
m dM -c
e
eS =e M = CT C ,
since
dM =
So that
SM = -^
and
If
we
~Erdt=
2mc
co
8M
r
e
M.
4>7r
mc
to correspond
magnetic
C)
Eds,
6),
Hence
8(HScos0) S
S compared with
correspond to
.(8).
S,
and
if
we take
lines, since
x 10 7 xcV47r,
of the order 10~ 9 x SH. fields
.
zmcco
with spectral
m
is
^--
2
during one revolution.
-=1-77
SM/M
cor-
-
neglect the change in
= 10~15 sec,
,
Eds = - -r-^^ 8 (HS cos
I
neglecting the change in
T
r
dt
Since the greatest attainable 5 it follows that the effect we
H < 10
are considering will not change the magnetic moment of the 4 by more than about 10~ of its value.
orbits
THE THEORY OF MAGNETISM
370
We
consider the conditions under which the
now
shall
field
H cos 9 will
change the area S of the orbit, limiting ourselves to If this is the case of a circular orbit under a central force. f(r) motion in the absence at distance r, then the condition for steady of the
magnetic
field is
After the magnetic field
is
applied this becomes TT
mo)V
-h
meo 2 Sr + 2ma>rSco
H
-- cos 6 ear =f(r) + f (r) Sr, c
of Sr and neglecting squares and higher powers - o>r
Thus
So>.
H cos
6.
c
--^-
But '
since
we
as small.
4mc
4-Trmc
e
are neglecting terms involving the product of
H and or
Thus 4mo> 2 Sr
p
=
+
2mcorSco
- cor
H cos
0.
G
{f (r) + 3mw
Whence
2 }
= 0.
Sr
It follows that either
or
/i\ (1) v J
Sr
(2)
f
*
=A (r)
j and
sj
Sco
= - 3me
2
=
He --cos
-----
s
2mc or
/ (r)//(r) = -
/( r) = constant x
and
,
r~"
3/r,
3 .
Thus, except in the special case in which the force varies inversely as the cube of the distance, Sr = 0, and there will be no change in the area of the orbit. to
change the angular It is clear that the
The only
effect of
the magnetic
field will
be
velocity.
H
sin 0, component of the magnetic intensity change the area of the latter, as
in the plane of the orbit, will not it only gives rise to displacements
perpendicular to this plane.
THE THEORY OF MAGNETISM
371
Calculation of the Magnetic Permeability.
The
effect that
we have been
considering involves a diminution
component of the magnetic moment of the elementary orbits resolved along the direction of the external magnetic field. The in the
is, in fact, precisely analogous to that of the induction of currents in linear conductors occupying the same positions as the orbits. The net result is a creation of polarization of the elements
phenomenon
medium in a direction opposite to that of H. Since the creation of positive polarization in the electrostatic case leads to a dielectric constant greater than unity it is clear that the present effect will lead to a magnetic permeability less than unity. In other of the
words the
effect
we
are
not to paramagnetism.
now considering leads to diamagnetisin and The value of the diamagnetic constant can
readily be calculated.
We
have seen that the increment in
establishment of
where H'
H
M
per orbit due to the
is
= H cos 6
H
is the component of along the normal to the Suppose that the atoms of the body considered possess no resultant magnetic moment, then the only effect of the field will be to produce this change BM. Suppose that there are v atoms per unit volume, each of which contains n electrons executing
orbit.
H
ON
Let the normal whose areas are Sl} 8, ... Sn P to the an make d with the of orbit direction of H. On angle p any plane the average all directions are equally probable for the line p orbits
.
ON
so that, out of any
which
lie
number
between 6 and
-
,
of orbits considered, the proportion 4-
d6
will
be -
-
.
For each
4-7T
orbit of type
p
H
we have to to get the resolved part of this parallel to if there are vp orbits of this cos Thus P type multiply again by
and
.
in unit volume their contribution to the magnetic
medium
will
be
-^- vp Sp r sin 6 cos Mp = - 8-7TMC ^Jo 2
1
&H
2
6de
moment
of the
THE THEOBY OF MAGNETISM
372
To all
of the polarization I we have to get the intensity Thus the different orbits in the atom.
The
sum
this for
force in a cavity perpendicular to the lines of force is /
'>
n*
\
B=>JLH=H+I=:H(Ip= 1
so that
127rmc2
-
& 2 VpSp
^ 1-^=-^
and
2
........................ (9).
do not It is necessary to show that the known values of 1 JJL Of the known lead to absurd values of Sp the areas of the orbits. substances bismuth has the largest value of 1 ^, namely 3*1 x 10~ s Let us suppose each atom to contain one orbit of each kind, then .
vp
becomes
estimate
As a
number of atoms in unit volume. sum of the areas of all the orbits approximation we shall take
We
v the
2$p
sufficient
the
efm
= 1'8
and
x 10 7 x v
=
e
c V47r,
= lO" x 20
c \/47r
can now
in the atom.
5
Q.ITO -
x 10 24
=
5 x 10 22
5
9'78 being the density of bismuth, 207 its atomic weight, and
10-- 4 the
mass of the hydrogen atom.
Thus
2^ = 10~
14
for
i
bismuth.
If the area of the orbits
were comparable with the
atom we should have S = 7rxlO~~ 16 so that, treating all the orbits as of equal area, ?i = 30 approximately. In the case of other substances, whose diamagnetism is less pronounced, the values of n would be smaller than this. The atomic weight of bismuth is 207, and there are good reasons for believing that the number of electrons in each atom of the different
cross section of the
elements
,
is comparable with the atomic weight. It is thus clear that the diamagnetic coefficients are of the order of magnitude
THE THEORY OF MAGNETISM
We
373
have seen that the
effect of applying a magnetic field is to the of the orbits projected into a plane perpenchange periods dicular to the magnetic field, the component of the motion
the
to
parallel
latter
Since
being unaltered.
the revolving
electrons are accelerated they will in general be radiating, and the frequencies of the radiation will be determined by the periods of revolution. There will thus be a change in the frequenej^ of the
radiation produced by the magnetic field. The displacement of the spectral lines due to this cause was discovered by Zeenian in 1896. The foregoing method is not a satisfactory one for determining quantitatively the changes in the frequency and character of the radiation which arise in this way, as it does not sufficiently
consider the motions perpendicular to the orbit. This deficiency will be remedied when the theory of the Zeeman effect is con-
Without going more deeply into the matter at this stage, it is evident from what has been said, that the magnetic displacement of the spectral lines and the phenomenon of diamagnetism are very intimately related, on the theory sidered
we
is
Chapter XX.
in
are discussing.
It appears that the occurrence of electrons revolving in orbits quite unnecessary to account for diamagnetism. The same kind
of effects occur even
if
field is applied.
This
vestigation which
is
We
the electrons are at rest before the magnetic is shown very clearly by the following in-
due to Lorentz*.
shall confine ourselves to
arrangements of electrons which
are isotropic with respect to three mutually perpendicular directions. Let the coordinates of any particular electron with respect to any is the centre of mass of the set of rectangular axes, whose origin
system, be body.
Let
I=2mk.
y, 2.
its
Then 2#
moment
= 2^/ = 2^ = 0,
of inertia about
taken over the whole
any axis through
be
Then k
Let
oc,
= Sar =
5t/
E be
2
= 2^ and 2#y = Hxz = ^yz = 0. 2
the resultant electromotive intensity of external origin any point, then the force acting on an electron has the comhas the components ponents eEx eEy, eEz and the couple about at
,
,
THE THEORY OF MAGNETISM
374
If the whole system is very small to point of it, so that we can
E
:
+ .W* + yJJjL -t-s & B-f+JIz x -x x + x
where ,
much from
vary
put as a sufficient approxi-
point
mation
will not
F
etc.
is
the value of
E
,
Hence since 2# and the couple become
at the origin.
vanish, the components of
f-^ dzj'
dy or
c
kH~,
kH
kH,,,
z.
c
c
There is thus an angular acceleration about the axis of
and the creation of a
field
H therefore
H= - ~
,
H,
results in the creation of
H.
angular velocity
-^
If we take the system considered by Lorentz to be one of the atoms of the substance we see that the effect of placing it in a
H
will be to set all the electrons in the strength atom in rotation about the axis with the uniform angular velocity
magnetic
field of
P
^
H.
This rotation, whose axis
is
parallel to
H,
will give rise
an intensity of magnetization in the same sense as that given by Langevin's theory. Moreover the magnitudes are the same in both cases provided we replace Scos<9 by Trr 2 where r is the to
,
distance of an electron from an axis, parallel to H, which passes through the centre of figure of the atom. The resulting value of
the permeability
where is
v is the
easily
number
extended over
magnitude of
is
1
all
p
is
shown
to
be
of atoms in unit volume, and the summation the electrons in the atom. The order of
evidently the same as before, and
is
equally
THE THEORY OF MAGNETISM agreement with estimates of the number of atom which are derived from other sources*.
in
375 electrons in the
The theory outlined brings diamagnetism. into agreement with the fact that the Zeeman separation results in sharp lines. The alteration in the frequency of the emitted radiation depends
H
and is independent of its direction. only on the magnitude of In general the Zeeman effect is much more complicated than that outlined by this theory which, nevertheless, probably contains the gist of the matter.
Two if it is
important deductions about diamagnetism may be drawn admitted to be identical in nature with the phenomenon
recognized optically as the Zeeman effect. In the first place the Zeernaii effect is exhibited by practically all the spectral lines of every substance, so that we should expect every substance to have
This does not really involve any contradiamagnetic quality. diction with experience, as the diamagnetic property is necessarily is therefore easily masked by small paramagnetic There seems to be no valid reason for supposing that the same kind of actions which produce the diamagnetism of bismuth do not occur and produce similar effects even in substances like
very feeble, and effects.
iron.
The other point
is
that
//,
1 is proportional
to a universal
2
constant
-
e' -
multiplied
2 by z>2r
.
Now, provided the unit
of
symmetry we have considered is the atom, we should expect v and 2r2 both to be independent of temperature within the order 1 can be measured. Of the substances of accuracy to which 1 with the temperature was measured for which the variation of by Curie f, water, quartz, KN0 3 and molten bismuth showed no /JL
/z,
In the case of solid bismuth on the other p, which was large at ordinary temperatures, fell off in a linear manner as the temperature increased to the melting point, when there was a sudden drop to the small value detectable variation.
hand the value
* It
is
of 1
not necessary that
communicated
to all
all
the electrons in the unit considered should be able
symmetry. The momentum calculated above will be of them, but some may be prevented from rotating by
to rotate about the axis of
THJKi
TJti.BiU.KX
UJt
characteristic of the fused metal.
changes
are
connected with
the
seems probable that these
It
crystalline
structure of the
substance.
Paramagnetism.
To explain the magnetic
qualities of bodies, other than dia-
shall make the hypothesis that, in some cases at any inagnetic, we resultant magnetic moment of the atom is not zero, the when rate,
the applied magnetic field is able to turn the planes of the orbits of the constituent electrons of the atoms towards the plane perpen-
We
do not know the precise nature of the mechanism by which the turning is brought about but in defence of the hypothesis we are able to urge that such a rotation tends to dicular to its direction.
;
make the
potential energy of the system a therefore tend to occur if there is any means
accomplished.
Superposed on
minimum, and will by which it can be
this there will in
every case be the
already discussed; in many cases, however, diamagnetic these are insignificant compared with the effects which arise from effects
the turning of the orbits.
The couple which tends to turn the axis of an orbit depends on the mutual energy of the external magnetic field e and that Let the magnetic force Hi, due to the of the revolving electron.
H
revolving electron at any point, consist of two parts, an average value and an oscillating part H>2 Then Hi Hi and Z = Q,
^
H
.
where the bars denote mean values of the vectors taken over a complete revolution. point
The energy per unit volume
+ where
at
any
is
(H + H^ + H*) e
The mean value
2
(HtHJ +
(H,Ht)
sum
of these vectors.
of this taken over a revolution is
since the average of the other terms value of the mutual unit
energy, per (HeHj), and does not depend on which on the average produce a orbits will
2
denotes the vector
H
2
.
is
zero.
Thus the mean
volume of the medium,
is
It follows that the forces
given rotation of the planes of the be the same as those which would produce the same
THE THEORY OF MAGNETISM
377
on a similarly situated equivalent current. The revolving electrons can therefore be replaced by the equivalent currents effect
an external magnetic
so far as the rotational effect of
field is
concerned. arising from the
If there were no other actions than those
the orbits would
all
set
themselves with their
magnetic fields, normals parallel to the external magnetic arrangement in which the potential energy
force, as is
a
this is
minimum.
the
The
tendency to complete alignment will however be resisted by the thermal motions of the atoms and also by the interatomic forces of other than magnetic type. The latter embrace the forces which give rise to cohesion and to chemical effects and which in all
One effect of the application probability are mainly electrostatic. of a magnetic field will be to convert the mutual potential energy of the orbit and the field into kinetic energy of the matter which moves with the orbit. In the simplest case, as perhaps in a gas where the elastic and chemical forces may be neglected, the mutual energy will be entirely transformed into the kinetic form. This, however, will change the distribution of kinetic energy among the different atoms or sub-atoms so that it is no longer that which was characteristic of the substance at the original temperature.
One of the effects of magnetization then the temperature of the substance.
may be
that of changing
H
is in a field of and strength is / / from to of + dl, the changed intensity magnetization work done by the magnetic force in increasing the magnetization is Hdl. If heat dQ is communicated to the substance it will
If the magnetizable substance
its
U
partly be used in changing that part of the internal energy of the substance which does not depend directly on the intensity
and partly
of magnetization, shall
In general absolute that
in
altering the latter.
Thus we
have
to
is
H
U
as well as of the and I may be functions of For processes which are reversible, T. say, where there is no hysteresis, the increment of
temperature
entropy ,~
dQ
1
fdU
rrSI\
,
n
1
fdU
dl\
THE THEORY OF MAGNETISM
378
must be a perfect
dH
Thus
differential.
\T (dT
or
+H
~ dT)\
dT (T '
dT+^dR^TdH^ V
H
on account of the strains probably involve field. With gases produced in the material by the magnetic
In general -.
^\
1
will
yy
w ill
oH
be zero and in most cases
we obtain
so that
as
M+ ~
n/YT
is satisfied
probably be very small;
an approximation covering a majority of
cases, the equation
This
it will
^
r rn o rr
=0w
by /=/(J?/T), where /is any
function.
In the case of paramagnetic substances / is directly proabove are satisfied, portional to H, so that, if the conditions
I=H where
A
number
........................... (12),
Curie found that for a a constant independent of T. the susceptibility I/H substances of'typical paramagnetic is
was inversely proportional to the absolute temperature. For such substances it would seem to follow that U does not depend appreciably on H, that the only important change produced by is in the potential energy of the elementary magnets and that the
H
energy of the accompanying
strains, if any, is negligible.
The
result contained in equation (12) was first discovered by experiment, and is often referred to as Curie's Law. The quantity
A
is
also
sometimes called Curie's constant.
A
considerable
number
of exceptions to the law have been found *, particularly at low temperatures. Some of these have been attributed to changes
in the crystalline or other configuration of the material. * Of.
Kammerlingh Onnes and
Proc. vol. xiv. p. 115 (1911).
Perrier, Koniitk.
Akad. Wetensch. Amsterdam
THE THEORY OF MAGNETISM
379
Paramagnetic Gas.
An
instructive case, which has been considered
by Langevin,
that of a paramagnetic gas such as oxygen. In this case the kinetic theory of gases enables us to calculate the form of the is
/=/(-_ which
function/ in the relation of thermodynamics.
is
j
We
required by the laws
know from Boltzmann's Theorem
(see
Chap, xvn, p. 403) that if the molecules of a gas in a closed vessel occupy positions in which they have varying amounts of potential energy, then there will be a greater number per unit volume in is less. In fact the two points where the w RT where R isjbhe gas cone~
the positions in which the potential energy ratio of the concentrations of the gas at
w
potential energy differs by stant for a single molecule.
In the present instance
w
is
'
,
be the potential energy of the i.e. MHcos 0, where d is the
will
in the field
H,
equivalent magnet angle between the magnetic axis of the molecule whose moment is and the field H. The number of molecules dn whose magnetic
M
axes
lie
within the two cones whose semi-angles are 6 and 6
respectively
4-
dO
number
N
is
MH COS0 where
A
is
The
a constant as yet undetermined.
of molecules considered
is
total
evidently
MH
n
r Jo
MH
n
Thus
N MH
A A
I
=~f7rRf/
.
,
smh
MH RT'
The resultant intensity of magnetization / H, by symmetry, and is given by r
/=
I
M cos Q dn = ^irAM (cosh a
is
r+i I
sinh a)
xe
in the direction of
M& x d&
THE THEORY OF MAGNETISM
380
where a
= MH/RT.
But
A = Najkir sinh a, ,
/cosh a
-
_ 1\
\sinh a
Since
M
so that
a/
determined by the structure of the molecules, we
is
see that for a given density of gas / depends only on a, i.e. / is a function only of HIT in accordance with the conclusion already also note that / is. reached by thermodynamic reasoning.
We
proportional to JV, It has not
i.e.
to the pressure of the gas.
been possible as yet to test this formula by experi-
ments on gases on account of the smallness of the intensities of magnetization which they acquire. It has, however, been extensively used by P. Weiss in building up a theory of the behaviour of ferromagnetic substances, as
we
shall see.
Ferromagnetic Substances.
The main
difference between the ferromagnetic and the the very high intensities of substances lies in paramagnetic attainable the former, in the fact that they by magnetization are capable of permanent magnetization, and that the magnetiza-
not in general a definite function of the external magnetic The magnetization does not change reversibly with force H. tion
is
H
and substances of hysteresis.
when
this class therefore exhibit the
It results that the heat
H
phenomenon
Q =fHdI, which
is
of
developed
is made to pass through a cycle of changes, no Ion gelvanishes but has a finite value.
Weiss* has attempted to explain the facts of ferromagnetism on the hypothesis that the only forces which it is necessary to consider as acting on the elementary magnets or revolving electrons are (1) the impressed magnetic field
H
l
of external origin, and
(2) the molecular field H% arising from the elementary magnets of the neighbouring atoms. It is also necessary to take into account the kinetic reactions arising from the thermal agitation of the molecules, just as in Langevin's theory of a paramagnetic gas.
The assumption of a molecular
field
uniform throughout the
substance will naturally be no more accurate than the similar assumptions as to uniformity which are almost invariably made in *
Journal de Phusiaue. vol.
vi.
r>.
fifil
THE THEORY OF MAGNETISM vary greatly from point to point but we may reasonably expect to get valuable indications of the way in which a real body would behave by dealing with molecular physics.
This
field will
H
a uniform value equal to its average value 2 assuming for The theory entirely omits to consider the substance. throughout interatomic forces of non-magnetic type. This is clearly legitimate until it is shown that such forces do play an important part in the phenomena under consideration.
Permanent Magnetization. Weiss's theory accounts for the existence of permanent magnetiBearing in mind that the molecular
zation in the following fashion. magnets are in equilibrium
under the influence of thermal and and the intensities l 2) the value of / for a given agitation will be determined by the equations value of
H
H
H
-
-
o
smha
it
;
_
TT __ TT JJL
1
RT
y-j .7,
a
and
JLJ.
TT H-
i
JLL -' s) .
MN
is the maximum possible intensity of magnetization where / = which is attained when the axes of the elementary magnets all Moreover H.2 is proportional to the point in the same direction. and of may be written 2 = \I, where magnetization intensity X is some constant. Permanent magnetization corresponds to the
H
absence of external
field,
so that
H= l
0,
and we have the two
independent equations
/ r
7
1 a -= cosh T
sum a
a
1
ana
/ T
7
=
RT
r-rrr a ............ (14). ' \i}lJ Q
The values of / which satisfy these equations may be found most readily by a graphical method (see Fig. 47). Let ORP be 1 coshet / RT +u of = =-r-=----- and OOP that of T = ~ vrf-.-a. the graph Ac& r J sinha a / \MIQ
i*^
.
-
-
M
cording to the simplest hypothesis which can be made R, X, and / are constants independent of T and H, so that OQP is a straight line which
a=
Tt
makes an angle a with the
axis of x,
where
r
r proportional to the absolute temperature T.
The
possible values of /// are given by the intersections of ORP. There are thus always two possible values of /, one of
and
tan
T^~T
is
OQP
which
THE THEORY OF MAGNETISM the point this, let
to
an unstable condition of the substance. To show means of an external field the in-
us suppose that by
tensity of magnetization is so that it is now determined a. is
made
to
undergo a slight decrease,
the line
by
ST
parallel to the axis of
This state of magnetization gives rise to an internal field which U, whereas to overcome the thermal tendency proportional to
to disorganization it is only necessary to have a field proportional V. The magnetization of the substance will therefore increase to
P
is automatically "until the state corresponding to the point if the is The reverse happens reached. magnetization given a virtual increase beyond that which corresponds to the point P.
Thus P represents a stable configuration of the material. In the same way it can be shown that represents an unstable condition, The permanent magnetization exhibited by the paramagnetic metals is not as definite as this theory would lead one to expect.
The
indefiniteness
may, however, be due to the fact that these
materials are not microscopically homogeneous, as well as to the occurrence of microscopic local reversals of magnetization. There are also, in
all
probability, complications arising
from the crystal-
line character of these materials.
As the temperature until at a certain
rises
the slope of the line
temperature
the curve at the origin.
T
it
OP
increases
coincides with the tangent to
At higher temperatures than
this the
/=
0, so that the substance would only possible solution would be be incapable of permanent magnetization in the absence of an
external magnetic
field.
The temperature
T may
therefore be
THE THEORY OF MAGNETISM To determine the origin we have
to the curve at the slope of the tangent
(I\__d / cosna _l^ da\Soha a) da\JlJ d
i
.
a'V/S+2-r+l 6/
2
ct
= 4 when ^aV/o^o TO
this theory if
where / is the same
a ,o
for g
<
is
/^ a Since
for all substances.
///
XMI
'
;
we write /(a)
so that
where the function
=
;
3
and
On
= 0.
,-iflWJ^ Uoa
!
Thus
a
~ ~ we
nave
H = Xl we have 2
= 0(r/T ) ........................ (16), ()
the same for
all
Thus
substances.
if
we
express I, the intensity of permanent magnetization, in terms of the maximum possible intensity of magnetization I and T the ,
absolute temperature in terms of the absolute critical temperature T we obtain a characteristic equation for the intensity of ,
permanent magnetization which
is
identical for all ferromagnetic
substances. It is probable, as we shall see later, that the magnetic. properties of the ferromagnetic metals are too much complicated by various secondary causes to afford a satisfactory t<\st of the above theoretical conclusions. The properties of various crystalline
ferromagnetic minerals, though complex enough of themselves, some respects simpler than those of the ferromagnetic metals and are better suited for carrying out a test of this character. The mineral magnetite has been Found to be are in
especially
suitable for this
nnmose
hut,
Vw>f'm*ii
Jiufiiouirwr *K,
,^^,...,".^^,^..1
THE THEORY OF MAGNETISM
384
data which have been obtained
with this substance we
shall
consider briefly the remarkable magnetic properties of another mineral, pyrrhotite, which seem likely to shed much light on the nature of ferromagnetic substances in general.
The Properties of Pyrrhotite.
The magnetic properties of this ferromagnetic mineral, which a sulphide of iron having a composition very near to FeS, but with a slight excess of sulphur, have been investigated in detail
is
by Weiss.
The
crystal has three mutually perpendicular planes
may be indicated by the axes Ox, Oy, Oz. more easily magnetized parallel to one are much crystals in than of these axes Ox any other direction, and furthermore the of
symmetry which
The
susceptibility parallel to The plane xOy is called
Oy is much greater than that
parallel to Oz.
by Weiss the magnetic plane.
The magnetic phenomena exhibited by a uniform
crystal when to are in a field Ox characterized placed magnetic parallel by remarkable simplicity. If the crystal shows no magnetic polarity to start with, the intensity of magnetization remains zero until
H
suddenly assumes
+H
when the intensity of magnetization e saturation value +/, which remains constant
reaches a critical value its
for all positive values of
than
H
As soon
G.
tion suddenly or
becomes
=
phenomenon
as
becomes
>
-f
H
c
.
,
H H c
and is
for all negative values greater reached the intensity of magnetiza-
I8t and retains that value until the field The /, curve is thus a rectangle and the
H
is irreversible.
If the magnetizing field is inclined to Ox the phenomena are the curves for / obtained very different*. For different values of
H
H
in the magnetic plane are shown in by rotating the direction of the accompanying figure. If // exceeds about 12,000 gauss the
value of /
is
constant for
all
values of the
field,
but unless
H
is
very great I is not in the same direction as H. This effect is well = 4000. The short lines represent exhibited by the curve for
H
the direction of the resultant magnetization for a field of intensity if =4000 gauss inclined to the axis Ox at an angle given in degrees by the numbers alongside. *
P. Weiss. Jnnrnn.l HP
P basin 11.0
For a variation vnl
TV
r>
ziftfi
/
1
in
QOM
H of
5
'
THE THEOEY OF MAGNETISM
385
from the axis Oy the direction of the intensity of magnetization varies more than 45. The component of the intensity of magnetization parallel to the field is a minimum when the angle between
H and
Ox
minimum
is
a
maximum,
For larger
for small fields.
fields
the
occurs at intermediate values*.
The curves
48 can be represented quite closely by formula. If 6 is the angle between
in Fig.
a simple trigonometrical
Fig. 48.
the axis
Ox and
that
where n
H
}
and
<
H sin (6 is
that between
<)
a constant quantity.
nl sin
<
Ox and J then ;
cos
<
=
The phenomena
it
appears (17),
in the plane
xOz
may be represented by a similar formula but with a different value of n.
A
simple physical interpretation can be given to the foregoing the elementary magnets will In addition to the field equation.
H
be acted on by forces which depend upon the magnetization of the *
P. Weiss, loc. cit. p. 487.
These forces
medium.
will certainly
from the intermolecular magnetic necessarily, be entirely magnetic,elastic or other type, to
be partly magnetic; arising They will not, at any rate
field.
They may
arise as reactions, of
the displacement of the elements of the.
normal equilibrium position, which is produced by the magnetic field. Since I measures a displacement of the elements from either a more normal or a less regular arrangement, the potential energy of the system which is due to these forces material, from the
be proportional to 7 2 even though the forces are not actually On account of the aeolotropy of the medium the magnetic.
will
7 a will be different along the different axes. Thus and / make angles 6 and in general when respectively with the axis of #, the x component of force per unit magnetic moment cos of an elementary magnet maybe represented by + \ I cos and the y component by Tsin0-fX 2 7sin<, where Xj and X2 are
coefficient of
H
H
constants depending
on the structure of the material.
moments about the centre
taking
of
Thus mass of the elementary
magnet
(H cos 6 4-
H sin(0
or
This
is
= (H sin 9 + X / sin cos <)- (\ X ) rsin
Xx/ cos <) sin
the same as
<
2
)
2
.
whereas the
For the
2
X2 by X x X3 plane ocOz we have only to replace \ l from the experiments that (\ 1 \ 2 )I= 7300 gauss and (A*
<j>,
i
- A )7=
maximum
8
1 50,000 gauss,
(saturation) value of
7
lib
appears
;
is
about 47 gauss.
The great difference between these numbers is somewhat surprising. If they were really magnetic forces one might expect them all to be of the same order of magnitude.
One
of the most striking features of these
phenomena
is
the
very small field which can reverse the magnetization along the axis Ox compared with the fields which are required to produce any appreciable magnetization along the perpendicular axes. This the more striking as the reversal of the magnetization would appear to involve the intermediate passage of the elementary magnets through the perpendicular orientation which is so difficult
is
throughout the mass of the material, at any rate, by field. There is not, however, any essential difficulty here. It seems clear that there are in general
to produce,
the application of an external
THE THEORY OF MAGNETISM
387
two stable positions of the elementary magnets, namely the positive and negative directions along the axis of x. In the presence of an external field which exceeds one of these e in
H
magnitude,
becomes unstable. As the equilibrium is a kinetic and statistical one, as is shown (see p. 388) by the variation of the intensity of permanent magnetization with the temperature, there will be a continuous passage between the two states, so that in a very short time the molecules will have arranged themselves with the axes of the magnetic atoms all oscillating about the position of stable This condition will ultimately be reached, no matter equilibrium. the forces which oppose the intervening motion. great
how
Permanent Magnetization and Temperature. Broadly speaking the properties of other ferromagnetic crystalline minerals, such as hematite
general features as pyrrhotite.
and magnetite, exhibit the same The other minerals have not been
so thoroughly as pyrrhotite, and there are important in detail but they all possess different magneticaxes of symmetry, and the three the different properties along minerals referred to all possess one axis of conspicuously easy
examined
differences
;
The phenomena in the direction of this axis magnetization. enable some of the most important consequences of Weiss's hypothesis of molecular
fields to
magnetic
be tested
We
have seen
that the hysteresis curves for magnetization in this direction are very simple compared with those of the ferromagnetic metals. In
the case of the minerals there
is
one stable value of the intensity
of magnetization Ic which is practically independent of the external The direction of Ic may be positive or negative, depending field.
on the previous treatment of the specimen, but otherwise it is = 0, which quite definite. The evidence for regarding the value Ix is also permanent within a more limited range of treatment, as Ic seems quite satisa mixture of equal amounts of 4- Ic and most the important argument being that there is no factory :
Ic continuous change but a sudden jump from /=0 to 7= The minerals thus possess a definite value Ic of stable intrinsic .
magnetization which is different from with the requirements of the theory.
zero.
This
is
in accordance
The reason why the
ferro-
expected to be less definite in this respect
magnetic metals maybe be briefly referred to
will
later.
We
382 that this theory led to a simple relation between the intensity of permanent magnetization and the tempeThis relation is the same for all rature (equation (16), p. 383). also
saw on
p.
substances, provided the intensity Ic is expressed in terms of the value of Ic (the" value at 0) and the temperagreatest possible at which the ferromagnetism the of ture T in terms temperature Q
T=
T
is to say, the maximum value of Ic and the disappears: that are to be taken as the units of intensity absolute temperature
T
This theorem has of magnetization and temperature respectively. been tested experimentally by Weiss* in the case of magnetite.
He
finds that the theoretical curve is followed
The
Even
in these regions the deviations are not 79 C. to the critical experiments extend from
low temperatures. very large.
with great accuracy Q and at very
T=T
immediate neighbourhood of
except in the
r = + 587
When
one considers the wide range of temperature covered, and the fact that there are no disposable constants in the formula, this agreement must be regarded as a
temperature
C.
o
remarkable confirmation of the theory.
There
one point which seems to call for further discussion at Equation (16) is derived from the equations
is
this stage.
/
__
cosh a
1
MH
a
RT
-
.
o
smha
-.-
.
and.
JLZ
_
H
H
= X 7, the 1 equal to zero and used to eliminate the constants. equation (15) being = it should also Since 7 is a continuous function of T when l be a continuous function of is allowed to vary. 1 and T when
by putting the external
field
derived
H
H
H
At
1
sight this appears to be contradicted by the experimental results. For, so far as the experiments have shown, there is no first
H
appreciable change in the value of 7 = 7C as the external field is increased from zero to the highest values available in the This would seem to be a fatal objection to the theory laboratory. l
unless the values of the internal field
XZ were so great that the values of In attainable were H-^ largest negligible in comparison. that case the behaviour would be = l sensibly the same as for even in the highest fields which can be obtained. shall see in the next paragraph that it is possible to deduce the values of the coefficients X by an method, from
H
We
independent
*
Journal de Physique, vol.
vi. p.
experimental
665 (1907).
THE THEORY OF MAGNETISM
The
data.
values of
X
389
show that XI
so obtained
is
extremely
large compared with the magnetic forces at our disposal this objection falls to the
:
so that
ground.
Properties 'near the Critical Temperature.
In this neighbourhood we have seen that
a
H-.
/
Thus
The
critical
temperature T
trr \
f
It follows that at
M
1
is
T= G
p
Thus
,
OJLl/
TT
TT ni
/77
temperatures sufficiently near to
T
the product
TT
of the intensity of magnetization I, and the difference G between the actual and the critical temperature, will remain constant when the magnetizing force l is maintained constant.
H
The
/,
T
curves will
730
be portions of rectangular hyperbolas whose
740
750
760
770
to the field strength H. That this is parameter is proportional at least approximately true is proved by the accompanying diagram, which represents the results of measurements by P. Curie on a of the critical temperature. specimen of iron in the neighbourhood
The values
of
H are written alongside the corresponding curves.
It is evident that equation (18) may be used to determine the values of the coefficient X, since T, T I and HI are given by From measurements of this kind the following the experiments. values of X and JT2 have been deduced by Weiss ,
:
X=
Iron
3,850
X = 12,700 Magnetite X 33,200 Nickel
H = Q'5Q x 10
6
2
E.M.u.
= 6'35xl0 #3 = 14-3 x 10' 6
J?2
The values of J5T2 are enormous compared with the magnetic fields which can be obtained in the laboratory, so that the peculiar result that Ic does not appear to vary with the external field is accounted for satisfactorily.
Abrupt Changes
At high temperatures
in
Magnetic Properties.
iron exhibits a
number
of more or less
Below 756 C. it abrupt changes in its magnetic properties. exhibits the characteristic ferromagnetic properties usually assoBetween 820 C. and 920 C. it appears to be incapable of permanent magnetization, but it exhibits the rapid diminution of susceptibility with rising temperature which, as ciated with the metal.
we have
should characterize ferromagnetic substances in the neighbourhood of the critical temperature. Between 920 C. seen,
and 1280 C.
it
behaves like a typical paramagnetic substance,
the susceptibility varying as the inverse absolute temperature. Above 1280 C. it has similar properties, except that there is a sudden increase in the susceptibility at this temperature. This
varying behaviour has been attributed to the existence of different modifications of iron within each of the limits of temperature specified.
and Fe
These modifications are denoted by Fe
a,
Fe
/3,
Fe 7
8.
been pointed out by Weiss* that the constants which determine the magnetic behaviour of these different forms of iron It has
*
Journal de Physique, vol.
vi. p.
C85 (1907).
THE THEORY OF MAGNETISM exhibit simple numerical relations. follows
These
391
may be determined
as
:
Fe
.
In the neighbourhood of
when / and a tend towards
jT
zero
7where
N
is
the
of the material.
r
a_ ~
SET
_ ~
number of magnetic molecules per unit volume But NM = IQ and NRT is the pressure p which
would be exerted by the substance if it were gaseous and occupied the same volume at the same temperature. Thus if 8 is the density of the substance the constant
This formula
derived on the
is
supposition
that each kinetic
molecule forms one magnetic molecule. If however n of the latter go to make up one of the former the right-hand side will have to
Putting in known values of / and value of p derived from obvious data, one finds that
be multiplied by
n.
<7
Fe/3. of this
We may
body to the
= % r=
attribute the quasi-ferromagnetic behaviour
and CS
the true value of Curie's constant
r = i _ g ffi + xi^jL I ~X G~X" is
H
l is helped by the Thus if %S is the true susceptibility (S = density)
fact that the external field
field \I.
where %'S
and the
0-00165*.
molecular is
S,
xg
the measured value of the susceptibility.
Comparing
this with the equation
X we
see that, since
H
l
is
arbitrary,
Substituting the experimental value of one finds
% = x
4 5*13 x 10~ at 820 C.
(7=0-00164x2. Fe7.
The experimental values
of the susceptibility of this At 940 C.
values for C. paramagnetic body give the following 0'00182 x 2. 0-00172 x 2. and at 1280 C.,
0=
C=
5
Fe
S.
is
body 1336 C.
In a similar way the .value of C for this paramagnetic found to be 0*00198 x 3 at 1280 C. and 0'00173 x 3 at
all the values of C above are interesting point is that of limits experimental error, either to twice or equal, within the to three times the common factor O'OOlT.
The
The occurrence
of
sudden changes in the magnetic
qualities
is
Similar features are presented by the other ferromagnetic substances which have been examined. In fact abrupt changes also occur even with diamagnetic substances.
by no means confined to
iron.
Thus Curie found that there was a large drop in the value of the diamagnetic constant of bismuth when, fusion occurred. Moreover the diamagnetic constant of the molten bismuth was independent of the temperature, whereas this was not the case with solid bismuth. In the case of tin, which is sometimes diamagnetic and at others paramagnetic, according to the temperature, a number These facts support the of transition points have been observed*. view that a considerable part at least of the magnetic properties of bodies are determined by the occurrence of systems of considerable size rather than the atoms or sub-atomic structures^. It is
of interest to form
molecular of iron
is
an estimate of the
local strength
of the
on the hypothesis that the apparent saturation due to the equilibrium between the internal fields and fields
the kinetic energy of thermal agitation. Weiss (loc. cit. p. 688) estimates that for iron at ordinary temperatures 7/J = 0*91 about, whence, making use of Langevin's formula, a = 11*3. Thus
.-
RT~ NRT where p
_ ~
_ "-
"
p
the pressure exerted by the magnetic molecules, supposed gaseous, and filling the same space as the metal. Putting jjfi/=2000 gauss and p = 2 x 10 dynes per sq. cm. one finds is
H= about 11 x 10
6
lines per sq. cm. This rough calculation agrees magnitude with the more accurate estimate which was obtained on p. 390, and supports the conclusion to which we have already been driven that the behaviour of simple
as to order of
*
Du
Bois and Honda, Versl. Kon. Ak. van Wetenscli. Amsterdam, vol.
p. 596 (1910),
t Of. Oxley, Camb. Phil Proc. vol. xvi. p. 486 (1912).
xn.
THE THEORY OF MAGNETISM
393
ferromagnetic substances, in the fields available in the laboratory, = 0. be practically the same as when l
E
will
Other Properties of the Ferromagnetic Metals.
Whatever the ultimate explanation of the very interesting properties of ferromagnetic crystals like pyrrhotite and magnetite may be, it is probable that they furnish an indication of the direction in which we should look for an explanation of the behaviour of ferromagnetic metals. It is well known now that all metals are complex aggregates of minute crystals. When the
metals are impure, as is the case with most specimens of iron, for example, the crystals may vary considerably in composition as well as size. It is therefore reasonable to expect the behaviour of iron to resemble that of
an irregular matrix of small
crystals
us say, pyrrhotite. The behaviour of such a matrix can If, for the moment, we neglect the small readily be calculated. susceptibility, parallel to the y and z axes, of any crystal selected at of,
let
random, the latter will not develop any magnetization until the # component of Thus all the reaches the critical value c
H
H
crystals, supposed initially in
random azimuths,
.
will not
become
magnetized simultaneously, and the intensity of magnetization will not approach saturation suddenly, as with a single crystal, but This agrees, of course, with the behaviour of iron. gradually. On the other hand there would on this view be no magnetization
H= H
whereas iron has a definite susceptibility for H=Q. This can be accounted for when the small magnetizations parallel to Oy and Qz are considered, and by considering such additional until
e,
factors as the lack of
homogeneity of the material and the possi-
that bility of local inequalities in the magnetization, it is probable the behaviour of any particular specimen of iron could be imitated exactly
by a model
For these reasons the study of probably of fundamental importance
of this kind. is
ferromagnetic crystals towards the understanding of ferromagnetic materials in general. It is worthy of remark that in the case of iron deposited electro-
magnetic field, hysteresis curves have been obtained which are almost rectangular like those given by pyrrhotite in the lytically in a
magnetic plane*. *
Manrain. Journal de PJmsiaue.
Ser. 3. vol. s. n.
123 (190U.
The Ultimate Magnetic Elements. In the case of magnetite above the
temperature, Weiss* not proportional to T~ l as in the case of the typical paramagnetic substances, but that the graph of x"" lines at different inclinaagainst T consists of a series of straight
finds that the value of
%
critical
is
1
G = mI^j3R where C is Oiirie's constant per unit molecular weight, and I I the saturation intensity of the mass, magnetization per unit mass, this result might be interpreted as Weiss rejects or l lt or both. arising from a variation of either Since
tions,
}
m is
m
m on
the ground that it does not lead to simple is assumed to be constant, the results. On the other hand, if the experimental values of C from I calculated values of 1 resulting are to each other, within the limits of experimental error, in the
the variation of
m
8 and 10. To explain this Weiss is led that the magnetic properties of substances " arise from the presence of an ultimate unit, the magneton," in
ratio of the
to
numbers
4, 5, 6,
make the hypothesis
the atoms of the substance.
apparently necessary that these elements should be capable in some way of annihilating each other temporarily, as the same substance may contain different numbers of
magnetons at
It
is
different temperatures.
In the case of other substances the number of these elements per gramme a bom or molecule may be determined in different In the case of ferromagnetic substances it may be deduced from the saturation intensity of magnetization at the absolute zero. In the case of solutions of paramagnetic substances all that is ways.
In these ml-f/SR. required is the value of the constant G ways values of this number have been deduced for nickel, cobalt,
and a large number of salts of iron, cobalt, manganese, chromium, copper, uranium, vanadium and the rare earths. The In the integral numbers vary from 4 to 56 in different cases. case of two salts of vanadium there is no indication of an approach
iron
to simple integral multiples but the agreement in a great many cases in which the integral number is small is so exact that there ;
seems
little
room
for
a substantial basis of
doubt that the idea of the magneton rests on fact.
It is worth while to add that a theory of the behaviour of the atom which has had very considerable success in some other *
Le Radium,
vol.
vm.
p.
301 (1911).
THE THEORY OF MAGNETISM
395
branches of physics leads to the existence of a magneton. This theory supposes the atoms to be made up of rings of electrons, revolving round a positive kernel of small dimensions, having a charge equal and opposite to the sum of the .charges on the The steady motions are subject to dynamical laws and electrons. do not give rise to radiation in appreciable quantities. Radiation occurs
when the
another, and
its
electrons
frequency
move from one v is
stable configuration to
determined in accordance with the
quantum hypothesis by the equation hv = W, where h constant, and
change
W
is
is
Planck's
the change in energy which accompanies the
Nicholson* and Bohrf have shown T is the
in the configuration.
that under these circumstances the value of T/v, where kinetic energy of the electron and v its orbital frequency,
an integral multiple of
is always have seen that the average
Now we
A/2.
magnetic moment of the magnet, which
is
equivalent to n electrons
M=
neS'r, where revolving in a circular or elliptic orbit, is the area of the orbit and r the time of description. Thus if
S
is
m is
the mass of an electron
jf-i. 2-Trm v
from this type of theory that in integral multiples of Q> where e h &* Jfo = -r-
Thus
it
follows
Jtf will
M
m 4?r
Putting e/m
= T77
x
1
7
E.M.U. Jl/o
=
always occur
.
and h = 6*55 erg 9-23 x 10-*
sec.,
this gives
This is nearly sixt times as large as the value of the magneton found by Weiss from experimental considerations. The experi-
mental value of the magneton
is
T64 x 10~ 21
in the
same
units.
Mechanical Reaction caused by Magnetization. the theory of magnetism which we have been discussing that we might expect to observe a it appears to the writer rotational mechanical reaction when a bar of iron is magnetized.
On
*
Dr
Monthly Not. Roy.
Astr. Soc. vol. LXXII. p. 679 (1912).
t Phil. Mag. vol. xxvi. p. 1 (1913). $ Since this was written I have learnt from a conversation (July 191B) -with Bohr, who had made similar calculations, that a more exact experimental
value of the magneton makes this ratio exactly
five.
For simplicity suppose that the magnetization arises entirely from the orbital motion of negative electrons whose charge is e. Let
them per unit volume, and let A z denote the average value of the projections of the areas of the orbits perpendicular to the axis of magnetization, divided by the times in which they are
there be JV'of
Then /z the magnetic moment per unit volume, is The resultant magnetization is taken to be
described.
given
,
by Tz = NeA z
.
parallel to the axis of
We
shall
now
z.
calculate
moment
the
of
momentum
of the
Consider any approximately revolving electrons about the z axis. circular orbit, the coordinates of whose centre are given by # yQt z. ,
Let the coordinates of the revolving electron referred to this centre at any instant be The moment of momentum of ??, ,
the electron about the z axis /
is
then
-(yo^) ,
Averaging this over a complete revolution the mean values are
9
i-cfy
i
area of projected orbit
Hence the average moment
of
__,
momentum about any
axis
independent of the position of that axis so long as its direction The moment of momentum the same. It is equal to z
%mA
per unit volume
.
is
U
z
is
=
It is
is
2~I
Z
(19).
thus equal to the intensity of magnetization multiplied
by 2m/e. of
By the principle of the conservation of momentum the moment momentum thus created must be balanced by an. equal moment
about the same
This reaction might conceivably occur either (1) on the electromagnetic system producing the exciting field, or (2) on the atoms of the magnetic material. In the former case In the effect should depend simply on Iz and not involve e/m. the latter ease the effects observed depend on the looseness of the atoms.
axis.
If they were free to rotate without affecting the neighbourmoment of momentum of the orbits might be
ing atoms, the
THE THEORY OF MAGNETISM compensated by
local rotations of
397
the non-magnetic matter.
In
that case the magnetic material would not have any tendency to turn as a whole. On the other hand the fact that the magnetic properties
of the
temperature makes
material are very susceptible to changes of it very unlikely that the connection between
the magnetic atoms and the neighbouring matter is a loose one. It seems therefore most probable that the moment of momentum created in this
way will be compensated by a motion of the magnetic material as a whole. Experiments which have been made to detect this effect have not led to a decision as to whether
it
exists or not.
Specific
If a substance
strength H, and
/
Heats of Ferromagnetic Substances. is
is
magnetized to an intensity I in a proportional to
field of
H, the energy of the system
changed by an amount ^HI. On account of the very large magnitude of the internal fields this energy is comparable with the whole thermal energy in the case of the ferromagnetic substances. is
This
true even when the average intensity of magnetization is As the energy of the molecular fields is a function of the
is
zero.
temperature, a very considerable part of the specific heat may The molecular magnetization diminishes with
arise in this way.
rising temperature so that additional energy must be supplied in order to overcome the attraction of the elementary magnets. The effect
thus involves an increase in the specific heat of the sub-
stance.
Since H.2 =.\I this part of the specific heat
is
-y
^,
where J is the mechanical equivalent of heat. Since / disappears suddenly at the critical temperature so will this additional specific Weiss and Beck* have shown that the part contributed by the internal magnetic energy in this way accounts quantitatively for the anomalous specific heats of ferromagnetic substances.
heat.
* Journal de Physique, Ser. 4, vol. vn. p. 249 (1908).
CHAPTER XVII THE KINETIC THEORY OF ELECTRONIC CONDUCTION Thermodynamics and (i)
WHEN total
the Kinetic 'Theory
Entropy and Probability.
a material system
energy
U
and
total
is
isolated in such a
volume v remain constant,
state will nevertheless, in general, is
of Matter.
way
that
its
its physical
change with lapse of time. This U and v do not completely
evident since the two variables
the condition of the system if condition of the system the way in which
specify
we understand by it
the
reacts to instruments
which measure such quantities as pressure, volume and temperature, which characterize matter in bulk rather than the individual molecules of which we believe it to be composed. For example,
we might have two systems having the same material composition and the same values of
U
and
v,
but the temperatures of
corre-
sponding points of the two systems might be different. The two .systems if left alone would then change in different ways as time
The changes which ensue are not capricious but definite. elapsed. So far as physical measuring instruments are sensible of them, they tend to the establishment of a definite end condition. The state is characterized by the fact that a certain function
final
called the entropy (S) of the system has attained the
maximum
value consistent with the imposed conditions.
The entropy may be considered
be defined by the
to
dif-
ferential equation
where
T is
temperature and p pressure.
the second law of thermodynamics,
it
From
may
this definition
and
be shown that any
THE KINETIC THEORY OF ELECTRONIC CONDUCTION reversible change occurring in of S unaltered, whereas
any
.
399
an isolated system leaves the value irreversible
change increases
S.
Thus the final state is that for which S is a maximum. It may also be shown that the change in S produced by any reversible action depends only on the initial and final states of the system and not at all on- the way in which the change has been effected, provided it is reversible throughout. Thus S is a perfect differential with respect to the independent variables in terms of state of the system is described.
The
which the
final state is
only steady as regards quantities like pressure, temperature and so on, which are usually taken to be sufficient to describe the behaviour of matter in bulk. If we made use of
instruments fine enough to determine the motions of the individual molecules there is every reason to believe that the system would be found to be the seat of very lively, never ceasing changes. The final
steady state
is
therefore one of statistical equilibrium merely.
The
actual spatial distribution of the individual molecules and the distribution of the momentum and velocity which they possess are
both constantly changing. On the average, however, and actually if the system contains an indefinitely large number of molecules, the distribution of the molecules in space and of momentum and velocity among the molecules is definite. to discover what this law of distribution is.
We
shall
now attempt
The number of ways in which a given amount of energy may be distributed among an indefinitely large number of molecules is clearly infinite to a very high order.
Some
more probable than others and there
be one distribution which
will
of these are
much
the most probable. Jeans* has shown that in the statistically steady state which is independent of the time the most probable distribution is infinitely probable compared with the others; so is
if we can find the most probable distribution we shall have obtained the actual distribution for all practical purposes.
that
Boltzmann pointed out the intimate connection between the probability of a given state of a system and the entropy of the In the final state of an isolated system we have seen system. that both the entropy and the probability of the system have attained a
maximum *
value.
In the intermediate stages both
Dynamical Theory of Gases, Chap.
in.
quantities are
moving towards the maximum.
Let us now
intro-
duce the hypothesis that the entropy of a system is a function This course is evidently permissible, since of its probability only.
we have not
yet defined the probability of a system precisely. care, when we come to do this, that our Let us see what definition does not conflict with the hypothesis.
We
shall
have to take
conclusions
we may
arrive at from
the general conceptions of
entropy and probability with the aid of our hypothesis. Consider two entirely separate material systems,
let us
stars so far apart that the interaction of their radiations
say two may be
disregarded. Then we have, by hypothesis, Sl =f(wi) and $2 =/(w 2 ), where S and $2 are the entropies and w l and iv2 the probabilities of the systems separately. If S and w are respectively the entropy L
and probability of the two systems considered together, we have
w=w w
But
l
2)
/ (w^w. ) =/ (w^ +f (w.>),
hence
2
S = klogw
so that
........................ (2),
where k is a universal constant. Thus the difference between entropy and probability is only that one combines by addition and the other by multiplication.
Of
the total
number
of molecules under consideration let us
suppose that in the steady state the fraction
/
(it*,
y, z, p, q, r)
dxdydzdp dqdr
............ (3)
+ dx, y and y + dy, and z and and their momenta between the components p and p + dp, We shall suppose that the six q and q + dq, and r and r + dr. variables x, y, z, p, q and r completely describe the state of the. have their centres between x and x
z
+ dz,
equivalent to treating them as massive points subject to the action of forces, and although not general enough for many problems in dealing with gases, is sufficiently so for the
particles.
electrical
electrons.
that
it is
have
its
This
is
problems in which the particles under consideration are Another interpretation which may be given to /( is ) the probability that a particle selected at random should within the assigned limits. We shall
six coordinates
suppose that /is a continuous function which can be differentiated,
THE KINETIC THEORY OF ELECTRONIC CONDUCTION and proceed
to
4OJL
determine the probability of the state of the gas to any assigned form of the function/.
which corresponds
We may
consider the gas to be represented by a series of points in a six-dimensional space, lengths measured along the axes of which give the values of x, y, z and p, q, r, the components of the distance of each particle from a fixed centre and of its Each particle is represented resultant momentum respectively. point, and if we know the density of such points at every of the six-dimensional diagram we shall have a complete part of Now divide up the whole of the the state of the gas. picture
by one
space so that the six-dimensional elements of volume
dxdydzdpdqdr = da are everywhere equal. as the distribution
/
We
shall define the probability of a
number
of
ways in which the given
given distri-
bution may be constructed by distributing the total number n of particles among the different elements of volume. shall consider the elements da to be so small that the state of the
We
which are defined by the limits of da is to be considered This would introduce a precisely the same for all of them. we if considered the to difficulty diagram represent an actual instantaneous state of the system. The difficulty can be overcome particles
by considering
to represent a large
it
number
of successive states.
In the former case the number of particles in each element of volume is necessarily limited in the latter case it may be made ;
as large as we please by contemplating a sufficiently large number of successive states. Since / is given, the number of particles in
the element
da
Since the particles in any element are any rearrangement of them within the element will not give rise to a fresh distribution. The problem therefore is to find the number of ways in which n like objects to
is
be treated as
nfda.
alike,
may be distributed among the totality of the compartments da which make up the whole of the space, when the same distribution is considered to arise wherever the same particles occur in the same elements of volume, no matter how they are arranged within the element of volume.
The number
of
ways
nl~U(nfda)l where
is
clearly (4),
denotes the continued product taken over all the elements. As an example we may consider the number of ways in which all II
the particles may be given the same position and momentum. This is evidently equal to one, which is also the value given by (4), the one in which the particles since, for all the elements do- except lie,
we have
(nfd
= 0! = 1,
and
for the
(nfdcr)\
remaining element
= nl.
Since the size of the elements dcr
is
arbitrary the expression
for the probability above will only be the value in terms of some not necessary, however, for us to determine arbitrary standard. It is
the value of the standard, since
it is
possible to arrive at results
which are independent of da without doing By combining (4) with (2) we have
so.
S = klogn\-k^]og(nfda)\
Now we
make nf do-
can always
N
............... (5).
we please by taking any very large number we have as large as
When n big enough. Stirling's well-known approximation is
or
=
N (log N -1)
with sufficient approximation, since we may neglect logN comis Thus (5) may be written pared with ^V" when large.
N
S=k log n\
&
= k log n
k
\
= k [log nl since all the
das
2 nfda (log nfda 1) 2 nfda (log nf+ log da
n (log da
are equal and
1)]
k
1)
2 nfda
5 nfda = n.
.
log nf,
Since n
is
constant
we obtain
S=
const.
k
I
nflog nfda
If the particles under consideration system then (7) will be the complete
............... (7).
make up the whole
of the
expression for the entropy.
Without any more elaborate analysis we may add to S a part which is independent of the particles under consideration and therefore independent of
/
With
this
understanding we
may
put
= S -k nflognfda Q
(8)
THE KINETIC THEOEY OF ELECTRONIC CONDUCTION where S
403
made up
of the entropy of the foreign parts of the with an system together arbitrary constant. is
The
(ii)
Law
of Distribution.
Now if the total energy and volume of the system are constant, the final steady state is characterized by a maximum value of S. This fact is sufficient to determine the function nf. We shall suppose the energy (1) a part UQ which
L
energy
W
energy
potential ;
U is
of the system to consist of three parts, independent of the n molecules, (2) the
of the n
molecules, and (3) their kinetic
i.e.
..................... (9),
W= [<(#,
where
L=
and
Thus a
W
T
and
maximum
L
ff
y,
(p*
z)fd
W fd
............... (10),
+ f + i*)fd
Since the entropy
are independent of/.
for the actual function
number n
is
to
be
/we
have by varying nf
=Q also since the total
......... (11).
of particles
......... (12);
constant
is
(13); finally since the total
energy of the system
is
constant
............... (14).
These equations are
satisfied for all possible variations
=log nf
fc
(
F
T
+L
ff
)
+ const.,
where kQ
is
constant throughout the system.
A
is
constant throughout the system.
where
We may enere-v
Lin
ofnfif
Thus
determine kQ from a knowledge of the mean kinetic We have evidently
of the particles.
/////
= 31
3
.
^ = 21 .......................................... (16).
r
2T
Moreover since
IIIHS* ~ JJ T -AJfj
* X dpdqdr
,
J
From equation
(15)
we
see that
with no restriction as to
dr
vr
if
momentum
is
the
number
of particles,
or velocity, in the element of
volume dr=dxdydz, then
(e'^ Wr dr ne~~
k{}Wr
if
}
...(18).
dr
fjf Thus
dr
we compare any two
different elements of
volume dr
and dr log
~~ \
=k
Q (
W
'
T
W
T)
(19).
If we denote the probability that a particle situated in the element of volume dr has components of momentum which lie
between p and p
+
dp,
q^dq+ dq, and r and r + dr, ),
then
nfdxdydzdvdQdr =
v,
q,
by
r)dpdqdr,
dr FdnJnJ*.
^
A ^
~
A'o
(^V-M^)
,
,
,^
,
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
405
and from (18)
F(p, q
}
r)
dpdqdr =
dpdqdr
_ ,
m:
e
'e
k* L "
dpdqdr
(20)
F
from equations (20) and (21) that is entirely of the of the r independent potential energy particles in the element dr. The distribution of velocity among the particles, It
follows
W
and
also of kinetic energy, is entirely
independent of their potential see from that the variation of the potential (19) energy. causes the numbers of energy particles in equal volumes to vary
We
from point to point, although their mean kinetic energy same everywhere.
An
apt illustration
is
is
the
furnished by the equilibrium of a column
of gas, in an enclosure at constant temperature, subject to the action of gravitation. The mean kinetic energy of the particles is the same at every point, as also is the way in which the kinetic
energy is distributed among the particles. On the other hand, the density of the gas is greatest at the bottom of the vessel where the gravitational potential energy is least. The law of variation of density with height
is
readily deduced from equation
(19).
The number
pressure p exerted by the particles at a point where the in unit volume is v is easily calculated. Consider a smooth
impenetrable surface, whose plane is perpendicular to the axis of On impact with the surface each x, to be placed at the point. will have its x component of momentum reversed. colliding particle
The momentum communicated time
is
to unit area of the surface in unit
evidently /co
Too
roc
vp (
]%
\-
_fc
^HJ-J-.mteK "
2
V
"i"
ff" "T"
^
?'
Dpdqdr
L (22).
Since we are neglecting effects which depend upon the size and mutual potential energy of the particles they will behave like a perfect gas and satisfy the equation pv = R l T. If R is the value of Rl reckoned for an amount of gas equal to one molecule, this
may be
written
p^vRT ........................... (23). By comparison with
we
(22)
see that (24),
L and the average kinetic energy make use
=-3 RT. We
shall frequently
of (19) in the form
The Constant
(iii)
Having determined the nature
k.
of the function
nf we can now
write the expression (8) for the entropy of the system in the form
S= S + Q
T + L9 kJA (k,{W ]
log
A) e~
k (Wr
*
L
dxdydzdpdqdr ...... (25).
To determine the
universal constant k
we need only
the simplest case, that of a monatornic gas for which
where
where
S
kV
A(k L -logA) e
/So
+
is
a new constant, which
'
t
consider
T is
every-
Equation (25) then becomes
zero.
8=
W
In the present case potential energy
is
I
L
" * (Wr +
is
L
dp dq dr
independent of
L
and
V.
the total energy of the system, since the zero so that is
;
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
On
the other hand
we
obtain from (26)
_3foi ' whence, by comparison with
and by comparing have
407
fdS\
,
ancl
kn "^
(27),
this with the equation of a perfect gas
we
The numerical value of R is known to the same degree of prewhich we know the value of the mass of an atom. With the accuracy which has recently been reached in this branch
cision as that to
of physical measurement the different are consistent to within about 1 /
methods of determining
.
deduced by Planck (see Chap, xv, the complete radiation formula,
p.
One
R
of the best values,
356) from the constants in
is
R = k = 1-346
.
sec.
2
deg.
The Theory of Metallic Conduction,
The
view, that the transportation of electricity in metallic like the corresponding phenomenon exhibited by
conductors,
electrolytes, is
due
to the
motion of minute charged
particles,
Weberian formulation of electroThe modern development is due to the labours of dynamics. and others. Thomson J. J. Riecke, Drucle, According to the and which we shall the one form of the most prevalent theory, was a
definite feature of the old
adopt, the atoms of the metal are regarded as continually dissociating into a negative electron and a positively charged It follows from the kinetic theory considerations which residue.
we have
all just discussed that the electrons will be moving in directions in the interior of the metal with very high speeds.
They
will
in fact possess the
same distribution
of velocity as
would an uncharged molecule of equal mass, and their mean kinetic energy will be proportional to the absolute temperature. We should rather expect that the atoms and positively charged residues would have an equal distribution of kinetic energy, but it is
coming about.
move
able to
Whether the atoms and positive residues are or not, they are to be regarded as only oscillating
about fixed positions and not travelling from point to point of the material.
We now intensity
which
suppose that the effect of applying an electromotive on the haphazard motion of the electrons,
to superpose
is
from thermal agitation, a velocity of
arises
drift
which,
since they are negatively charged, is in the opposite direction to This motion constitutes the electric current, the applied force.
A priori, is thus carried entirely by the negative electrons. one might be tempted to suppose that the positive residues would also drift along under the influence of the electric field. If any which
such effect exists
it
must be extremely
we should expect an
small.
For
if it
were
current to transport appreciable other across the the into atoms of one metal junction between electric
experiments have been made to detect such effects, but they have always led to negative results. This objection would be removed if we supposed the positive particles
two metals.
Careful
to be of the
same nature
in different materials.
There
are,
how-
In the first place ever, two serious objections to such a view. there is no other evidence of the liberation of such particles from atoms under conditions which can be considered at all analogous to In the second place the hypothesis appears to be incapable of removing such difficulties as are presented by the simpler theory.
those
which hold inside conductors.
The
strongest arguments in favour of the view which asserts that the currents in metals are carried by negative electrons are as follows
:
Conductors when heated or illuminated are found to emit
(1)
electrons into the surrounding space. The mass of the electrons thus emitted from a hot wire may, in favourable cases, be large com-
pared with the number which, on any reasonable hypothesis, may be expected to be present in the wire at any instant, showing that electrons are continuously flowing into the wire from other parts of
the system*.
The
(2)
metals
typical good conductors at low temperatures are all
electropositive elements which are known from other phenomena to liberate electrons from their atoms readily. * ^ IMK /IMO\ Cf. 0. W. Eichardson, PhiL Man. :
i.e.
l
VT
THE KINETIC THEORY OF ELECTRONIC CONDUCTION If the particles which carry the current did not possess(3) an extremely small mass in proportion to their charge we should not expect to get the well-known effects produced by a magnetic
the change of resistance in a magand the Hall effect. These effects are unquestionably very complicated, and so far the electron theory has not been able On the to furnish an adequate quantitative explanation of them. other hand it is the only theory which has been able to account
field in electric currents, e.g.
netic field
for
them
qualitatively.
The Simple Theory of Conduction. In order to illustrate the problem presented by electric and thermal conduction in metallic conductors we shall consider the behaviour of the electrons to be
much
simpler than
it is
likely
to-
be in any real case. A more exact treatment will be given later. For the present we shall make the following assumptions (1) that :
move
same distance X between two that all the have same velocity of thermal collisions, (2) they v and that the an motion of electron subsequent to a (3) agitation
the electrons
collision
is
all
freely for the
entirely independent of its history previous to the shall also assume that the only force acting on the
We
collision.
electrons throughout the free path X is the applied electric intensity X. It follows from assumptions (1) and (2) that the free time t \fv
=
is
the same for
all
To be
the electrons.
consistent with the
requirements of the kinetic theory of matter we take the kinetic 2 energy, mv of the electrons to be equal to the mean value f of the same quantity for the atoms of a monatomic gas at the
RT
,
same temperature.
Subject to these simplified assumptions the metal may be calculated as follows
electrical conductivity of a
:
the electric charge of an electron, the force acting on it its free path is Xe and its acceleration is Xe/m. If the during of the of the to the electric electron, parallel velocity component If e
is
intensity, at the
beginning of
its free
component at the end of the path free time.
The average 1
field
is
therefore
-~
X
path
will
is u,
be u
H
the value of this 1,
where
t
is
the
velocity in the direction of the electric
e t,
since, all directions of v
probable, the average value of u over a large
being equally
number
of electrons
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
410
the direction of the electric field
number
the
may be
--
written
If n is ~^X in unit volume, the number of them across a unit area drawn perpendicular
of electrons
which, in unit time, drift
to the direction of the electric force
X will
be ^
nX -mv
.
Each
.
of
2 these carries a charge e, so that the quantity of electricity transin unit time or, in other words, the electric ported across unit area current density i, will be .
Thus the
It is a
%
e
2
X
,
specific electrical conductivity
a
is
well-known result of experiment that for the pure metals, a- is almost exactly inversely proportional
which are good conductors,
T except at very low Now e and R are universal constants and do not vary
to the absolute temperature
ture. it is
Hence
if
temperatures.
with temperaour assumptions are to be compatible with the facts
necessary that n\v should be independent of T.
Since
v cc
T^
requires that n\ should be inversely proportional to *J'T. present we are not in a position to say whether this variation with T is to be attributed to the variation of n or X or of both
this
At
of them.
Comparison with Thermal Conductivity.
The best conductors of electricity are also the best conductors Under the circumstances it is natural to attribute the two effects to the same cause, viz. the motion of the electrons. From this point of view the problem of the conduction of heat of heat.
in a metal
is
the same as that in a gas having the same number same free path and the same
of molecules in unit volume, the
molecular weight and temperature, as the electrons in the metal In our calculations we shall suppose that the amount of heat which is distributed by radiation and by the dynamical action of the molecules and residues on each other is positive
negligible
compared with that distributed by the rapidly moving The results can therefore only be expected to be true
electrons. for
good
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
411
Under these restrictions the appreciable thermal conductivity. problem of determining the thermal conductivity is mathematically identical with that of finding the thermal conductivity of the
corresponding gas. In the notation we have previously employed the thermal conductivity k of the corresponding gas, which is given in any standard text-book on the kinetic theory of gases*, is
k = $n\vR
Comparing
this with equation (28)
(29).
we have
k/
(30).
Equation (30) is the expression of a simple and very remarkable conclusion. The ratio of the thermal to the electrical conductivity has the
same value
conductors
different temperatures the value
:
for
at the
same temperature for
all
good
of the ratio
is
The first part of this proportional to the absolute temperature. generalization is known as the law of Wiedemann and Franz, by
whom
it was announced as an experimental discovery about 1850 the law of temperature variation was discovered experimentally
by
;
L. Lorenz
somewhat
later.
The more
recent experiments of Jaeger and Diesselhirst enable an accurate comparison with equation (30) to be carried out. They determined the values both of the ratio of the two conductivities at 18 C. and of its temperature coefficient, for a The values they found are large series of metals and alloys.
exhibited in the table on p. 412.
Assuming that the charge on the electron is equal to that by a hydrogen atom in electrolysis the evaluation of R/e does not require any unfamiliar data. If n is the number of
carried
molecules in one cubic centimetre of a gas at pressure, then nR is the value of the constant
and 760 mrns.
C.
R
l
in the equation
= RiT, where v = 1 cm. p = 76 x 13'6 x 981 dynes/cm. and T = 273 abs. On the other hand, since the molecule of hydrogen 2
3
pv
,
is equal to the quantity of electricity which passes C. when -| c.c. of H.2 is liberated at a water voltameter through and 760 mms. This is equal to '4327 KM. unit, so that the value of
is
diatomic ne
7? 2
3 =Y e
T at
18 C.
= 6*5
x 10 10
.
It will
be observed that
for the
pure
metals which are good conductors the experimental value of the
Material
pure pure Silver, pure Gold(l) Gold (2) pure
Copper Copper
(1)
(2)
Eatio, per cent.
7O9xlO 6-99xlO 7-05xl0 10 l
Nickel
Zino(l) Zinc (2) pure Cadmium, pure Lead, pure Tin, pure
6-72 xl() 10
7'06xl0 10 7'15xl0 10 7-35 x 10
10
6'36xlO 7'76xl0 10
Aluminium
l
Platinum (1.) Platinum (2) pure Palladium Iron Iron
Coefficient of this
6*76 x 10 10 6-65 x 10" 6-71 x 10 10 6\S6 x 10 10 7-27 x 10 l
Copper, commercial
7'53 x 10 l
7'54xl0 10 8-02xl0 10 8-38xl0 10
(1) (2)
xlO 10
Steel
9-03
Bismuth
9-64X10
11-06 x 10 10 Constantan (60 Cu, 40 Ni) l Manganin (84 Cu, 4 Ni, 12 Mn)... 9-14x 10
ratio is very close in
all
Temperature
Thermal Conductivity to Electrical Conductivity
at at at at at at at at at ab at at at at at at at at at at at at
18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
0. C.
-39
C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C.
-39 -37
-36 -37
-39 -38 '38 '37
-40 "34 '43
'46
'46 '43
'44 '35
-15 '23 -27
cases to the theoretical value.
The
deviations are greater for the poorer conductors and in almost all cases are in the direction of values greater than the theoretical.
what would happen if the part of the thermal conductivity which does not depend on the motion of the electrons were to become appreciable. Thus the deviations lie in the direction in
This
is
which they would be expected to occur. The behaviour of alloys is In exceptional, but so are most of their electrical properties. * fact Lord Rayleigh has pointed out that the electrical resistance of alloys may be expected to be " " existence of a false resistance
unduly high on account of the arising from the Peltier heating effect at the junction of of the material of varying comparts It is also desirable to mention that Leesf has found position. that the divergence of the values of k/a-T for different substances is greater at the temperature of liquid air than at ordinary
On the whole, however, the concordance of the temperatures. values of this quantity for so many metals over so wide a range of temperatures is more than the differences, when one considers the
number
striking of factors which
The percentage temperature *
might
Nature, vol. LIV. p. 154; Scientific Papers, vol. ccvm. p. 381 (1908V
t Phil. Trans. A.
enter.
coefficient required
by the theory
vol. iv. p. 232.
,
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
413
%
is O366 The experimental values for the good conductors are practically equal to this within the limits of experimental
error.
More Complete Theory of Conduction.
The extent to which the preceding calculations account for the observed relations between the electric and thermal conductivities is the more surprising when one considers the nature of the .approximate assumptions on which the calculations For example, we assumed that all the electrons moved are based.
of good conductors
through equal
free paths with equal velocities.
Now
the results
of the kinetic theory considerations at the beginning of this chapter lead us to expect that instead of possessing equal velocities the electrons would have a velocity distribution in accordance
with Maxwell's law. This requirement is not obviated by the possible fact that the electrons may be subjected to very intense forces; the distribution of kinetic energy amongst the particles is, Moreover seen, independent of the potential energy.
for
as
we have
the fact that the atoms, with which the electrons
collide, are so
massive that the velocity of an electron does not alter in magnitude during a collision with one of them makes no difference, one another will ensure
for the encounters of the electrons with
that Maxwell's law of distribution of velocity
is
established.
The following calculation* is much more general than the one we have considered. It assumes that when no external force acts
on the conductor and
When .
it is all
at a uniform temperature the
distributed according to Maxwell's law. an electric force acts, or when the temperature varies from
velocity of the electrons
is
point to point of the material, bution to be slightly modified.
we
shall
We
suppose the law of
shall neglect the
distri-
immediate
between electrons compared with that of those between electrons and atomic or sub-atomic structures, and shall treat the latter as though they were immovable centres of force. The theory is therefore incomplete, since we assume that the effect of collisions
normal distribution of velocity is that of Maxwell and at the same time that the collisions take place with particles which are immovable and are therefore unable to change the resultant velocity * lernes
Of.
H. A. Lorentz, Theory of Electrons, Copenhagen (1911)
Elektrontheori,
;
p.
266
0.
;
N. Bohr, Studler over metal-
W.
Pdcliardscn,
Phil.
Mag.
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
414
of the colliding particles. Thus, so far as the set of assumptions we are dealing with is concerned, a complete theory would have to take into account the occurrence of collisions
between
electrons,
or something which produces the same effect, in order to account for the existence of Maxwell's distribution under normal condifficulties of a more complete theory, Our method of attack will be insuperable. unfortunately, appear to try to find a law of distribution of velocity, slightly different from the normal one, which will make the distribution at every
The mathematical
ditions.
point a steady one
when
or
electric
other forces act on the
electrons in the metal.
to change the number of have electrons, at any point, which assigned components u, v, w These are two in number, viz. (1) the free motion of velocity. of the electrons from one part of the conductor to another, and
Consider the causes which tend
Let us imagine the distribution of velocity among (2) collisions. the electrons at any point to be represented by a three-dimenThe diagram is drawn so that the sional velocity diagram. resultant velocity of each electron at the point is represented by a radius from the origin. The density of the points which are
the ends of such lines and which dcr
dudvdw
of this
diagram
lie
within any element of volume the number of elec-
will represent
trons at the given point which have velocity
components between
u and u + du, v and v + dv, and w and w + dw. In this discussion " " we mean, of course, by the expression the number at any point z the number in an infinitesimal element of volume dr=dxdydz a),y, which contains the point x, y, z. Let us denote the number of electrons which have velocity at the instant t by a;, y, z
components
f
u,
v,
w
(u, v, w, x, y, z, t)
at the point If the
dcrdr.
X
it will give rise to an intensity in the metal is is acceleration of each electron equal to Xe/m. supposed to be parallel to the x axis. If there were no collisions these
electric
X
would be found in a different element da dr of the Owing to velocity and space diagram at a later instant t + dt. the motion the new velocity coordinates would evidently be electrons
u z
-f
X
+ wdt
points
/j
dt, v,
w,
instead
and the space coordinates of u,
v,
x+ udt, y + vdt
and
w, % y, z respectively. Corresponding in each equally diagram so that
would be displaced
}
THE KINETIC THEORY OF ELECTRONIC CONDUCTION have as the condition
the
for
/
distribution
of
:
\
Q
-f X f(u Til \
of a
existence
velocity invariable with the time, that
415
x
w,
dt, v,
+ udt, y 4-
=/ (u, The occuiTence
v,
w,
vdt, z 4-
x, y, z,
wdt,
t+dt)J
t).
makes
necessary to of electrons in the elements
of collisions, however,
it
The number number in the element dcrdr at time but to the latter minus those which have disappeared from the original group plus those which have come into the group this equation.
modify
dad-r' at
t
+ dt
is
equal, not to the
t,
from other groups, owing to the occurrence of collisions. Since these quantities are each clearly proportional to the range of velocity and space covered by the group and to the interval of time dt
}
we may
respectively.
write them The condition
bution of velocity
u
H-
may
:
w,
dt, v,
=/O,
and bdcrdrdt
the existence of a steady distritherefore be written
6
X
in the form adordrdt for
v,
x
udt,
-}-
w, x,
y
y, z, t)
4-
vdt, z 4- wdt, t
+ (b-
+
dt
a) dt,
and on expanding
/ (u+X m dt, \ J
by
Taylor's theorem,
If
df ox
df ou
we can evaluate
b
J
we have
_ Xm + uf-+v- + e
etc.)
df
oy
df df = + wfdt oz
and a in terms
an equation from which, may be deduced.
if
we
of
/ we
b-a ......... /m (31). 7
shall evidently obtain
are able to solve
it,
the function
/
a particular case of a more general In the present instance we are calculation given by Maxwell*. the electrons collide as treating the centres of force with which
The
evaluation of a and b
is
This will correspond very closely with the facts on account of the small mass of the electrons. The relative velocity of the colliding particles is thus equal to the actual velocity
immovable.
V = Vw + v + w* 2
moving
2
of the
moving
electron.
electron will be influenced
every instant; but
we
shall
by
all
Strictly speaking, any the centres of force at
suppose that
when
the important
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
416
deflections take place the influence of one of the centres is very with that of all the others. As the term collision
great compared often understood to imply the occurrence of geometrical contact, like that between hard elastic spheres, we shall replace it is
by
the more general term encounter, throughout the rest of this The sequence of changes which characterize an endiscussion. is then to be represented mathematically as follows. As the previous history of the electron cannot affect the results of the
counter
calculation
we can suppose
it
to
have been moving
for
an indefinite
V
in a straight line. As the entime with the uniform velocity linear counter begins to occur the path becomes curved, but the As we are neglecting all encounters is unchanged. of
V magnitude more than one centre of force plays an important part, the orbit of the electron will lie entirely in one plane, that which
in which
contains the direction of its original motion and the perpendicular from the centre of force upon it. If the orbit is an open one it
end by becoming asymptotic to a straight line inclined at an to the direction of the angle, which we shall call 20, original will
.straight path.
is
which the orbit
is
evidently the deflection up to the apse, about symmetrical. If the perpendicular distance
from the "deflecting centre to the original straight path is b, the of collisions made in unit time by a single particle whose is V, such that the distance b lies between b and b -f db and .speed
number
lies in an azimuth between ty and -^ -f city, from a fixed plane passing through the measured ^ taken as the axis about which ty is of is direction V, which measured, is nrbdbd^, where n is the number of the deflecting
the plane of the orbit
where
is
centres in unit
volume of the substance.
This follows since the
expression above is equal to the number of centres in the region between two circular cylinders of radii b and b + db, whose height
equal to the velocity V of the moving particle, which is cut off and by the two planes ^r and -^ -f dty which pass through determine the plane of the orbit. Since there are f(u, v, w) dois
V
particles with velocity it
u, v
t
w,
which
^
and
OG,
lie -v/r
components about
number
u,
v,
w
in unit volume,
of particles which leave the group owing to deflections through angles between 20 and 2 (6 -f dO) in an azimuth between
follows that the y,
z in unit time
-f cfyr is
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
The
between b and 9 may be found from the theory of The law of force being Kjd*, where d is distance
relation
central orbits*.
and
K
is
417
positive
when
the forces are repulsive, let 1
where
M
l
is
the mass of an electron and
centres of force.
M
that of one of the
Then
I-'-!. where
a/ is
the least positive root of
Evidently a Since
where
??i
is
a function of 6 and 5 only.
M^M* is
is
very small
we can put (M + l
M M lf = m" z )l
1
2
1 ,
Hence
the mass of an electron. 2
\s-i 7-r
\mK-y
Thus the expression
(32)
may
be written
J_ i__i_ ~ 1 f(u w(\mjV" F s
Now
l
}
v,w)adad^do-dTdt
consider the reverse collisions which bring
..... (36).
new
electrons
into the group u, v, w. Let u', v, w' denote the velocity components, before the encounter, of an electron which, as the result of the
encounter, acquires the velocity components u, v, w, i.e. joins the Consider first the reverse encounters for which specified group. Since 6 and ty lie within the assigned limits considered above. a is collision, the value of the resultant velocity unchanged by
we have
Vw + V + 2
2
u;
2
=
V.
individual components uf, v', -w will be different from -M-, v, w on be account of the rotation of 7 through the angle 20. They may of the that fact the of use component down written
The
by making
* p.
Maxwell, Scientific Papers, vol. u.
198 (Cambridge, 1898).
p.
36
;
Routh's Particle Dynamics, chap.
vi.
THE KINETIC THEUKY
4lO
U.LUUTKUJN1U
U*'
distance has velocity along the apsidal is unaltered. component
perpendicular containing
CONDUCTION
been reversed whilst the
Thus
^=
if
is
the plane
V and u, u'
=u
%u sin2 6
4-
Vv
2
+ w* sin 29 cos
-v/r,
with similar expressions for v' and w'. The volume du'dv'dw' of the three-dimensional velocity diagram which is occupied by the deflected points will be equal to dudvdw, since the new points obtained by reflecting the undeflected points in a plane the orbit and tangential to it at the apse. Thus, perpendicular to in considering the reverse collisions which bring extraneous elec-
may be
change we require w) by f (u' D' w').
trons into the group f(u, v, w) dudvdw, the only to make in (36) is the replacement of f(u, v,
,
,
Hence _2_
F
.,___ " s
I
\{f(u',
v',
w')
JJ
We
shall confine
-f(u,
v,
w)} adad^.
our discussion to the one-dimensional case in
which the state of the material depends only on the x coordinate, i.e. in the electrical problem we shall suppose the electric intensity to be parallel to the axis of x and in the thermal problem we shall suppose the temperature gradient to
lie
Under
in this direction.
these restrictions equation (31), when applied to the steady state which does not change with the time, becomes
w ...
m on
ex
The left-hand
side is the rate of change of / owing to collisions, and the right-hand side that which arises from the displacement of the
group of electrons as a whole.
Now when there
is
/ (u,
v,
ditions,
is
at a
no applied electromotive
the electrons
among if
the material
is
uniform temperature
w) denotes the value of the function
we may
9
A
and
/
under these con-
write
f*(u,v w)**Aer
where
T
the distribution of velocity in accordance with Maxwell's law. Thus, force,
h r'
..................... (38),
and A are constant throughout the material.
It
is
likely
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
419
that this distribution will be only slightly changed by the changed conditions we are contemplating, since the new forces which come into
play are always small compared with the intermolecular Let us see if we can satisfy (37) by a solution of the
forces.
form (38
where the term u%(V)
a),
small compared with Ae~ h ^' and x(F) is a function of V, the resultant There are two velocity, only. conditions which any solution will have to These are is
satisfy.
number of electrons in unit volume of the material, mean kinetic energy at any part which is at an assigned
that the total
and their
temperature, should both have the same values as in the uniform condition to which (38) applies. If we carry out the
integrations
over the whole of the velocity diagram,
we
find that the former is
2 N=A(ir/h)^ and the latter \m V = 3m/4A, whether we use (38) Hence (38 a) satisfies the required conditions so far. or (38 a).
In order to determine %(V) we make use of equation (37). f do not occur on the right-hand side of this equation, we shall neglect the effect of the small term u^(V) Since the differences of
Substituting the assumed solution,
on the right-hand side of (37). the left-hand side becomes
Wlf
Jo Jo s
=n -whilst
the right-hand side
~
V
s
-
- 47TM sin
x ( V}
2
dada,
is
m Thus
.(39) *
\s-l I
mjj
V
. s-l [" sw-vada .
,
Jo
e-*r> f] ox)
..... (40), ^
THE
420
KlJN.nJJL.LU
1 JUJliUJt& X
ur
JL.UEA./.I.JCX/VJLMA-/
I^VJIN
u u u JLJ.UJN
where
(41).
The
definite integral in (41) is a function of 5 only,
and may be
Thus %(F) does not evaluated graphically when s is known. involve a nor -vjr, and only involves u, v, w in the combination F. to satisfy (37) by a solution of the form It is therefore possible of (38 a), where %(F) is
by
a function of
F
only and
is
given
(39).
The electric current density, /, will be obtained if we multiply the number of electrons in unit volume within a specified infinitesimal range of velocities, by their component of velocity to the electric intensity and by the charge each carries, parallel and then integrate the product over volume. Thus
27T^ _ 6 fa
all
the electrons in unit
T+2
A" 1
(42),
remembering that
r Jo
;
= r(p + i)=jpr(p).
The specific electrical conductivity,
x (43). v
W
The thermal current density is obtained by integrating the number of electrons in unit volume, which have
product of the
*
components within a given range, by their thermal 2 and by their velocity component u parallel to the |mF energy temperature gradient, over all the values of u, v, w which occur*
velocity
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
421
Thus
9_
TOO /oo
J o
Q
_L V *""
-1.&
1
^
3
^S " X
When
roo fao
9^,.
V
itx(V)V'dV=*^ml 3 Jo \ ]
'[vfiAY +
g
|_
-m
2
( \s
VU- 9^1 1
/ vanishes, we
9
-
-
ax
9
-
^L
I
\$
n YH
J
h*
9J
\ I
A xl
I
*
a).
have from (42) WJ "
r y rr li ox
.
tTL
h=-~- =
r-^= by equation
^^
hdx~ The thermal
A
3a?
(43
the electric current density
Since, from p. 405,
^
6
m
current, TF
,
when
(24),
we
see that
I'd' there
is
no
electric current, is
therefore
2 |
A^ + thermal conductivity
and the r r
coefficient, &, of
of
in this expression or
is
the coefficient
?)
^
(44)
The
ratio of the thermal to the electrical conductivity is
It shows is very striking. simplicity of equation (45) that the ratio of the thermal to the electrical conductivity is inof the number and mass of the free electrons and of
The
dependent the J?, e
number and strength and
T
it
of the centres of force.
depends only on
~.i.:~~
~.r
4.u^
rtw
.f
^
s,
In addition to
the index which determines the Tvrr f.l-na ppnf.rAs with the
r^r^i-nA
JLJti.fcj
J11JNJUT1U
JLJd.HiU.K.1
We
(p. 411) that the results of experithe value of k/
mutual distance.
have seen
for the
ments show that
occur with immovable particles which exert central forces varying as the inverse third power of the mutual distance.
We
can estimate the order of magnitude of the strength, K, of the centres from that of the electrical conductivity of the metals
by substitution in equation A and <, we have
Substituting the values of
(43).
of molecules in 1 c.c. of any gas at 0C. and do not know either the number, N, of free electrons or the number, n, of centres of force in unit volume but we shall assume that both these quantities are of the same order of mag-
where v
is
760 mms.
the
number
We
;
nitude as the number, p, of atoms of the metal in unit volume. = Bp] then 7 and B are numbers comLet us put
N
parable with unity. The other data required are, in general vR = 3*72 x 10 3 ergs/ C., vni = 5 x 10~8 gin. per c.c., e = 1'6 x lO" 20 :
/ao
E.M.
unit,
T=273C.
and
I
sin 2
which,
=
when
5
x 10~4
E.M. unit
3,
is
equal to
Taking the case of and This result
is
silver as
an
illustration,
cr
=6
of additional interest because
several
other
considerations point to the occurrence of forces varying inversely as the cube of the distance as an important feature of atomic structure.
some would
Thus
Thomson* and Jeans f have shown that which govern the emission of thermal radiation the collisions of the electrons were with centres J. J.
of the laws follow if
of force obeying the inverse third power law. The strength, K, of these centres can be estimated from the constants in the radiation *
Phil. Mag. vol. xiv. p. 217 (1907) t Ibid. vol. xx. p. 642 (1910).
;
vol. xx. p.
238 (1910).
THE KINETIC THEORY OF ELECTRONIC CONDUCTION formulae.
Jeans
The mean
of the estimates given
K = 2'5 x lO"
27
is
423
by Thomson and
This agrees satisfactorily with that
.
given by the electrical conductivity. If the centres of force are
due to the occurrence of electrical possible to have a state of steady motion in which an electron revolves about the axis of the doublets inside the atom,
The energy
doublet.
it is
of the steady motion, as well as that of it, is proportional to the frequency of
the small oscillations about
the rotation.
Thomson* has suggested
this result as an explanathe action of light (cf. p. 469); by shall see in the next chapter, the kinetic
tion of the emission of electrons in
which
case, as
we
energy of the emitted electrons is a linear function of the frequency of the exciting light. Another interesting property of these orbits is that the moment of the magnet to which they are equivalent depends only on the moment of the doublets about which they are rotating. Since the universality of the law con-
necting radiation and temperature suggests that the strength of these doublets is independent of the matter in which they occur, they would furnish an explanation of the atomic magnets, or
magnetons, whose existence has been inferred by Weiss
(cf.
also
p. 395).
Electrical Conductivity as
a Function of Temperature.
Although the electron theory has been very successful in which explaining the laws of Wiedemann and Franz and of Lorenz, are
obeyed by the
ratio of the
thermal to the electrical con-
has not been so successful in accounting for the ductivity, these two properties individually. In point of fact, of behaviour it
indepure metals the thermal conductivity is practically whereas the electrical conductivity is pendent of the temperature, for the
approximately inversely proportional to the absolute temperature, except at low temperatures. Since
K
n \m
even
if
we assume
the index
s
which determines the law of force
to be practically independent of T, the observed law
cr
varies as
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
424
T~ might l
from an appropriate variation of one or more of with T. At present there is no means N, n and
arise
K
the quantities of finding the connection of K,
N and n separately with
T.
Eecent experiments by Kammerlingh Onnes and his collaborahave shown that the resistance of the pure metals becomes low temperatures. This result cannot be, exceedingly small at very
tors
said to be definitely indicated
by the formulae which we have just
in conflict with them. is, however, not necessarily developed. the we centres, with which the elecFor example might identify which vibrators take part in thermal the trons collide, with It
phenomena. vibrators
is to
The most obvious physical interpretation of these The doublets regard them as electrical doublets.
example, by the vibration or steady motion of an equilibrium position. They would exert forces on the moving electrons which would vary inversely as the cube of the mutual distance, a result which, as we have seen, is required by several considerations. It follows from Einstein's theory of
might
arise, for
electrons about
specific heats
that the energy of the vibrators approaches zero
exponentially as the absolute zero of temperature is approached. Now the average moment of a doublet constituted in this way is so that the moment propo'rtional to the square root of its energy or strength of the centres will also approach zero exponentially as :
the temperature posing that the
is
reduced.
Since there
is
no reason
for
sup-
number
of free electrons approaches zero at so fast a rate as this, the conductivity would become infinite at very
low temperatures.
Such a view
is
in satisfactory
mental measurements.
agreement with the experi-
By assuming
that
K
times the product
on the right-hand side of (4*7) varies as T~^, which gives the right variation of o- with T at high temperatures, and that the with T depends upon the contained heat energy, variation of as deduced from the specific heat measurements, in the manner
K
just indicated, I find that the calculated temperature at which the electrical resistance disappears agrees quite accurately with that
given by Kammerlingh Onnes in the case of mercury and is not far from the values given for lead and The way in which gold. the calculated electrical resistance varies with the is
temperature
also
very similar to that found experimentally at these low tern-
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
425
theory of specific heats mentioned in Chapter xv. According to that view the energy of the vibrators does not approach the value zero at low temperatures but the finite value hi>/2. This would -
make
K approach a
finite value, so that
the resistance would not
vanish.
Thermoelectric Phenomena.
As
is
different
well known, if a circuit which consists of wires of
materials
is
two
constructed, a current will flow round it
without the assistance of a battery, if the two junctions are different temperatures. There is no current when the two
at
junctions are at the same temperature. As there is no observable change in the nature or composition of the materials constituting
the
circuit,
the electromotive force which drives these currents
must be derived from the
available thermal energy. By causing the electric currents to perform mechanical work we could evidently construct a heat engine out of a circuit of this kind. As
long ago as 1854 Lord Kelvin showed that valuable information about thermoelectric phenomena might be obtained by the appliof the
cation
principles
of thermodynamics
thermoelectric
to
circuits.
The thermoelectromotive two conductors
A
and
force of
B may be
a circuit consisting of the It has been
denoted by JEAS
.
E
found that for a very large number of pairs of substances AB can be represented within the limits of accuracy of experimental observation by the comparatively simple empirical formula
where 1\ and T^ are the temperatures of the two junctions and The differential coefficient pf A % with
E
a and b are constants.
called the thermoelectric
respect to the upper temperature T% power of the circuit. It is clearly a linear function of the temis
perature within the limits of accuracy of the formula above. The thermoelectromotive forces between circuits of different metals
terminating at the same pair of temperatures are connected by the relation
EA B + E This
is
C
= BAG ..................... (48).
evident since the circuit composed by the addition of
AB
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
426
the temperature single wire of the material of which varies from one extreme to the other and back
inclusion of a
,
again.
in a closed circuit of a inequality of temperature single uniform material does not give rise to any thermoelectromotive force, this
As
cannot cause any difference from the effect given by
AC simply.
In addition to the thermoelectromotive force caused by a difference of temperature there are the converse reversible
heating
which are produced by the flow of an electric We have thus to consider the Peltier effect, which is the current. heat liberated when an electric current flows across the junction between two different metals, and the Thomson effect, which is the heat developed reversibly when an electric current flows
and cooling
effects
along
an unequally heated bar. coefficients
P and
Both
cr.
These are measured by the
respective of these refer to the amount of heat
by the passage of unit quantity of electricity. In has to flow against one degree differspecifying a the electricity ence of temperature, the directions of the electric and thermal
liberated
absorption of heat.
The
application of the conservation of energy to a thermo-
electric circuit gives
R
When -i is made very small is the resistance of the circuit. the Joulian development of heat Hi- vanishes in comparison with the other terms, so that the reversible quantities satisfy the
if
equation (49).
Similarly by applying the second law of thermodynamics in
the form
1-^=0
to the reversible heat,
production, we have
(50).
It follows from these equations thai,
dE
P
,
THE KINETIC THEOEY OF ELECTRONIC CONDUCTION
427
These conclusions have been confirmed in many particulars, and there are no experimental results which can be said with
The values of the different certainty to conflict with them. thermoelectric quantities may be derived from the expressions given on pp. 420
and
421, for the rate of transportation of electricity These expressions were derived on the hypothesis that
heat.
the law of force between the electrons and the centres with which
they collide varied as a power of the mutual distance. will
hold strictly only
if
They
the centres are at distances
apart which are large compared with those within which the forces are The two principal reasons for this are (1) if this appreciable. :
the particles are under collision conditions at every instant and the physical foundation of the calculation disappears, and (2) as the potential energy varies condition
is
not satisfied
all
very rapidly from point to point in the neighbourhood of any electron the quantity A will be subject to corresponding sharp variations (cf. equation (15), p. 403), and the actual A can therefore scarcely be regarded as differentiable.
In order
to
embrace these conditions we
former calculations a
We
rigour.
little at
shall generalize the
the sacrifice of a certain amount of
assume that the average value of
shall
A
is differ-
entiable however complicated the internal fields of force may be is always and that, for the steady flow parallel to the axis of u,
f
Ae~ hv + u^(V\ where ^ involves u, v and w through By working through a calculation similar to that already out, it becomes clear that, provided the medium is
of the form
V only. carried
isotropic,
%(F)
*
will
be of the form
^
involves only the mass, m, of the electrons, of the centres in unit volume, the law of force
where the function
the number, ?i, and V. In this way the equations and heat may be written
for the currents of electricity
:
dh
(53),
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
428
__RT ""
J
e
where
= specific
electrical conductivity
= - 12 m JT l
.
/c
&/
-^ .4 J~
4J
Jt2>
= thermal ~
and
t
-^-
m
" > ~~
T
i
/*
6
*7T
=
2
1
~z
7ft
1
TT = 77
conductivity
f
O
J. -5 Jil
m
2
f
_
Ws
7 JL"
[
JV f" t/4
7" ~
7 \~f
enough apart and act on one another which vary as the inverse 5th power of the distance,
If the centres are far
with forces
we
have, as before,
In the equations above, If
X
o
is
X
is
The thermoelectromotive value of
the mechanical force on an electron.
eX = mX.
the electric intensity, then
I
X d% which
when there
is
n
force
E
round any
required to reduce the current
Thus from
are no batteries in the circuit.
E= X I
()
I
when
dx
_R(< "\
j.
circuit
dlogA i
dx
i
i
is
the
to zero,
(53)
= dT)
LL '--."-(
ax
ax) (57),
after inteeratinp hvnarhs. 1
Tf thp.
int.AOTM.l
i
tn.kp.n
vmind
a,
closed
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
429
part has the same value at the identical limits. The unintegrated part, since the integrand is not necessarily a perfect differential, will not necessarily
are
T and
jf
,
it
be
may
If the temperatures of the junctions
zero.
be written (58),
where the
suffixes 1
thermoelectric power
and
2 denote the respective materials.
The
is
R(
dEl2
A, (o9) "
The Thomson
coefficient,
cr,
may be
obtained as follows
:
The
heat developed per unit area and thickness d-x in the direction of the current is equal to the work done by the electric force inside
the volume
the stream of
the stream of energy flowing in energy flowing out. This is -
-h
* = - -^~ }dx
>
-
+%\
e
d n ~- (log &
dx
a
d x| fj A + ^' P)Y 4- *- (k r- }\dx dw .
,
,
-
^'J
............ (60),
from (53) and (56). The first term on the right in (60) represents the Joule heating effect and the third term is independent of the electric current. The middle term alone reverses with the electric current and effect.
is
therefore the heat production owing to the
The Thomson
coefficient
cr,
which
is
Thomson
the amount of heat
unit rise of temabsorption per unit current per unit time per perature,
is
therefore (61).
The value of the Peltier effect may be obtained by applying the same equation to the passage of a current across the junction between two metals at the same temperature instead of consame metal with varying
a bar of the sidering the flow along The magnitude of the
Peltier
temperature.
coefficient
P
12
is
given by
eP u = -KT (log \
This
is
-0-2
+*-/*.)/
............... ( 62 )-
not quite equal to the work done by the electric force For we see from nf thft sftnaratinp surface.
r.<arv. vV."hrmvihnnrl
1
THE KINETIC THEOKY OF ELECTRONIC CONDUCTION
430
.
/-^/7T
that (53) ' x i
when
there
approaches zero
done by the difference
RTdlogA
,
X
approaches
electric force at the
between
law of force
no difference of temperature
is
is
and eP12
this
is
;
(
\0C
= 0),
so that the
work
A. 1
boundary
due
is
ETlog
to the fact that,
different in different metals, the
amount
as
J
-.
The
--2
when
the
of kinetic
the electrons which carry a given current energy transported by As the ratio of the thermal and electric conductivities
is different.
is very nearly the same for the pure is not metals which are good conductors, it is probable that /^ //, 2 It may, however, be of the same order of magnitude as the large.
indicates that the law of force
Thomson
observed Peltier and
The values given by .
equations
A
mulae
(51)
(49)
different
will
effects.
(61) and (62) satisfy the Kelvin's thermodynamic theory.
(58), (59),
of Lord
method of deducing some of the thermoelectric be considered in the next chapter.
for-
Conductivity for Periodic Forces.
The behaviour
of the electrons in metals under alternating
The following has been considered by various writers. H. A. is an elaboration of a method to due Wilson*, investigation, forces
Jeans f. The electrons instantaneously present originally given by in any given volume may be divided into groups characterized by
The number of collisions of a particular velocity of agitation V. one another with is the electrons regarded as small compared with the
number
of collisions
between electrons and atoms.
Owing
to
relatively small mass of the electrons the latter class of collisions will have very little effect in changing the magnitude
the
Thus the electrons in any given group will of the velocity V. remain in that group throughout a large number of collisions. Let dN denote the number of electrons in the group characterized by the value V and let u denote the average component of velocity of this group in the direction of the electric intensity X. Consider momentum of the group is changing. The
the rate at which the
group
is
gaining *
momentum
from the applied
field at a rate
Phil. Mat). VI. vol. xx. p. 835 (1910).
which
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
431
equal to the force which acts on the electrons in the group. The magnitude of this is XedN. At the same time the momen-
is
tum
thus acquired is being dissipated by collisions at a rate mu/3dN, where /3 depends on F, on the strength of the centres and on the law of force which governs the collisions. The value of ft
can be found by considering the deviation of the electrons produced by collisions*. Thus in general the change of momentum of the group with time will be in accordance with the differential equation
:
(63).
When we are dealing with direct currents a steady motion is soon established which is independent of the time. The value of u
for this case is
thus obtained by putting the left-hand side of The current density is e^udNorX,
equation (63) equal to zero.
the specific electrical conductivity. When the same assumptions are made about the nature of the collisions and about
where a
is
the law of distribution of velocity among the electrons (dN as a function of V), this leads to values of a- in agreement with those
given by other methods
for direct currents.
To find the conductivity for periodic forces let After dividing by dN, (63) becomes du
n ,
As
= Av
X ~X
cospt.
,
cospt.
in the theory of dispersion, the solution of this equation which to stationary conditions is the particular integral
corresponds
e cos
where tan
The mean
8 =_p//3.
(t
"~
ty
value of u for
all
the electrons
is
u^^rludN. the work done by the electromotive force is converted into heat the rate of heat production is the mean value of
Since
all
NeuX = NeuX It is also equal to \crp X<;\
whose frequency
is
p. *
where
p
eospt. is
the conductivity for forces may be taken as the
This statement Cf.
H. A. Wilson,
loc. cit.
THE KINETIC THEOKY OF ELECTRONIC CONDUCTION
432
definition of
which
mean
p
.
The mean value
is large compared with value of
2-Tr/p.
is
be taken over a time
to
Thus
\
cr^Xf
is
equal to the
V
N
and the integral is and /3 are functions of In this expression limits and the V between +00. As an to one with respect the illustration we may consider simplest possible supposition that that the electrons all have equal velocities. electrons for all of The integral then reduces to a sum over the which 13 has the same value. For zero frequency (p = 0) the value
we can make, namely
of
N
then becomes
and
N
on the right-hand side of this All the quantities except equation are known with considerable accuracy, and crp can be deduced from experiments on the optical properties of metals. Schuster* has applied equation (65) to the optical data accumuIn this way he finds that for all the commoner lated by Crude f. metals the
number
of free electrons in a given volume is from one number of atoms present. It may
to three times as great as the
be that these estimates are subject to errors arising from the occurrence of selective optical absorption, a phenomenon which is but it does not seem likely disregarded in the theory above For the experithat the number can be much smaller than this. ;
ments of Rubens and Hagen with infra-red radiation show that
approached rather
much
values
closely.
This would not be true
if
N
had
smaller than those calculated by Schuster.
Th erm al Radiatio n
.
The theory of the motion of electrons in metals is of importance from another point of view because it helps us to form a *
Phil.
Maq. VI.
vol.
vn. p. 151 (1904).
THE KINETIC THEORY OF ELECTRONIC CONDUCTION mechanism
as to the
judgment
433
of the emission of electromagnetic saw in Chap, xv that
thermal radiation from hot bodies.
We
the spectral distribution of energy in the radiation inside an enclosure in equilibrium with matter was independent of the nature of the matter and determined solely by the temperature of the enclosure. It follows that if we can a imagine type of matter which is sufficiently real and at the same time sufficiently simple in its behaviour to admit of our calculating the properties of the radiation which is in equilibrium with it at a given temperature,
we
shall have solved the radiation problem. Such a possibility would seem to be offered by the theory of the motion of electrons in metals.
We
know
accelerated and that
that during the collisions the electrons are
when
electrically
charged particles are ac-
celerated they emit radiation. The accelerations of the other constituents of the system are probably negligible in comparison with those of the electrons; so that it would seem that we are
not likely to be led into serious error
if
we
A
radiation to the motion of the electrons.
attribute
all
the
very general know-
ledge of the nature of the motions enables us to analyse the radiation thus emitted into its constituent frequencies by means
In this way we can arrive at a knowledge of the way in which energy of assigned frequency is being emitted by the moving electrons. In the steady state an equal amount of of Fourier's series.
The absorption radiant energy will be absorbed by the system. of energy occurs through the Joule heating effect of the electric currents established by the electric intensity in the electromagnetic waves.
If the intensity in the
waves of frequency
p
is
Xp the absorption of mean
energy per unit volume per unit time is the value of <rp Xp\ where crp is the conductivity for currents of
frequency p. As we have seen <rp is a function of p. By equating the amount of emitted energy of a given frequency to that ab-
sorbed
we
arrive at
an expression
for
the steady energy density of
given frequency.
This method of calculating the distribution of energy in the black body spectrum was first used by Lorentz*. His calculations were confined to waves of low frequencies. The method has since
been extended by various *
Cf.
_-
Theory of Electrons, .,
,.
.,
TTT
T
writers, including p.
J.
J.
Thomson f,
80; Amsterdam Proc. 1902-3, p. 666. nt
i-r
ftr\r\n\
.
1
rtOO /1A1A\
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
434
It appears that if the
Jeans*, H. A. Wilsonf and BohrJ.
same
carried through the calculations the assumptions are consistently distribution of energy found is that given by the law of Jeans and
As we have seen, this law is not in accordance so that the method is not adequate the with experimental facts the to the solution of problem. In fact, Jeans and McLaren have shown pretty conclusively that any dynamical method leads that a dynamical foundation of the inevitably to Jeans's law, so Lord Kayleigh.
;
||
seem to be possible. theory of radiation does not 'Galvanomagnetic and Thermomagnetic Phenomena.
A
number
of interesting
phenomena
are
observed when a
conductor carrying an electric or thermal current is placed in The effects are conveniently classified field. a to
magnetic according whether they are exerted across or along the primary current.
The transverse
effects are as follows
When
Hall Effect.
(1)
an
:
electric current flows across the
a magnetic field an electromotive force is observed at right angles to both the primary current and the
lines of force of
which
is
field.
magnetic (2)
von Ettingshausen's
Under the
Effect.
like
circum-
temperature gradient is observed which has the to the Hall electromotive force. direction opposite Nernst and von Ettingshausen's Effect. When heat (3) stances
a
an electromotive
flows across the lines of magnetic force there is force in the mutually perpendicular direction.
Leduc and Righi's
(4)
there
is
The
Under the
Effect.
like conditions
a transverse temperature gradient. transverse
sufficiently small
are
effects
product of the intensity fields)
H
all
to
proportional
of the magnetic field (at
and the primary current
i
the
any
vector rate for
of heat or
Thus any of the effects of amount E may be considered to be measured by the coefficients TT, 7ra TT,, 7r4 where,for each suffix, electricity.
,
,
,
E^-jrHi *
Phil.
t Ibid.
Mag.
vol.
vol. xx. p.
xvn.
p.
774 (1909)
(66). ;
vol.
xvm.
p.
209 (1909).
835 (1910).
J Studier over metall ernes Elcktrontheori, Copenhagen (1911).
THE KINETIC THEORY OF ELECTRONIC CONDUCTION The
435
correspond to the numbers which precede the
suffixes
enumerated above.
TT is taken to be positive if, when the primary current flows in the direction positive along the x axis and His in the direction the the positive y axis, along resulting
effects as
electric or
temperature gradient is in the positive direction along the z axis, the arrangement of the axes being such that a righthanded screw travelling along Ox would rotate from Oy to Oz. This choice of signs is not, however, the one which occurs in the
A
literature of the subject. series of measurements by Zahn of the various coefficients for different conductors led to the values
in the following table: Conductor
ir n
TT-,
It is clear
TT,
TT.,
from these numbers that the phenomena are very sign and magnitude of the observed effects
The
complicated. exhibit no obvious relation to any known property of the materials In the case of bismuth, a conducting material which concerned.
can be obtained in the form of large crystals, it has been found that the Hall coefficient changes sign as the orientations of the with reference primary current and the magnetic field are changed to the crystal axes.
The
as the magnetic longitudinal effects are different according of the direction the primary current. With along
field is across or
a transverse magnetic field the following longitudinal been observed. Each corresponds to the transverse the same number prefixed. (1) /
i
i
When
a conductor *
_
i
'4.^
^^^^-C-tirt
is
effects
have
effect
with
placed in a transverse magnetic
-rt/^i^f .-1T1/1/-V
ic?
-P/innr!
f.n
Ar-r-nr
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
436 (2)
There
a Peltier effect at the junction between trans-
is
and unmagnetized material.
versely magnetized
thermoelectromotive force between transand unmagnetized material. versely magnetized There is an alteration in the thermal conductivity in a (4) (3)
There
a
is
transverse magnetic field. All these effects also occur in a longitudinal magnetic field but as a rule they are then smaller. In both cases (2), (3) and
(4)
bismuth and the ferromagnetic The change of resistance in a magnetic field, also, is metals. much larger with these metals than with any of the others. have been observed
only with
Since the change of resistance cannot depend on the sign of H, material is isotropie, it follows that for small fields provided the the change BK in the resistance R must satisfy the equation
A is a constant. This equation is satisfied considerable a over range of values of // by the metals which Tin* following values of A have been exhibit only small changes. 2 SJt/Jt=zAH' where t
determined by Patterson: Cadmium ......... :i'S:ix Hr
1
...............
O-sTxlo
(JoM
...............
r-
0-iitJx 10
l
Silver
The
............
effects are
much
Copper ......... O-2(>xlO~ 12 Tin ............... O-23xlO- 12
'-'
Zinc
*
-
l()
l
-x
4. is
(
li'fJO
\\"ith c-rystals tin- i-ffert (<>
tip*
......
OTixlO" 12
Platinum
......
O'OfJx 10~
larger at low temperatures.
found the following values of
with reference
Palladium
.1
'.
12
Thus Laws
:
+ ;
'To
+100
C.
0'98
depends on the direction of the current and with the ferromagnetic sub-
Tv.stal axes,
intensity of inagndixat i
stances
tin-
electron theory eanitot M said to have been very successvet.., in unravelling these complicated magnetic phenomena. doubt one of the fact>r-s having an influence is the deflexion
The
,
|
ful, as
No
of the paths of the
moving electrons which
is
produced by
THE KINETIC THEORY OF ELECTRONIC CONDUCTION would be expected to follows
way may be
arise in this
437
calculated as
:
We
suppose the Hall
shall
be measured by the
to
effect
external electromotive force which
necessary to introduce
it is
in order to reduce the deflexion
The galvanometer
of. a galvanometer again to zero. connected across two points at unit distance
is
apart which were at the same potential before the magnetic field was applied. Let the primary current i be along the axis of X,
the Hall current j along the axis of y and the magnetic intensity of z. The components of the electric intensity
H along the axis at
any instant are X,
If #,
Y, 0.
y,
mx = eX
TT
my = eY +
'
ey,
c
eXt
(67).
(68),
my = eYt+-ex+mv C
(69),
be the instant at which the and put # = y=0, x = u and y = v when
origin of time to
collision occurred,
From
m =
ex,
c
ey+mu
c
we take the
TT
TT
mx
Thus
if
an
z are the coordinates of
equations of motion are
electron, its
last
= 0.
(68) TT
P
1
cc
-X? m
=^
2
me
eyt y
+ ut,
and from (69)
m If r
is
to be the -r
y y
H
e
(I
e
c
m
(2
m
the time between two
same in
all cases,
v
.
a
H-
...(70).
cm*
collisions,
J
assumed
the average value of y
for simplicity
is
1 / He e TTT H (1 e Tr --e -U--= -1 r ydt= ~Y^+ X^ + ^lu c in m 2 m m c 2\ rJo (6 .
Now
u
7
=v=
0,
since
these
quantities
are
as
likely to
be
TT
ative as jpositive, and negative r,
c
y may be neglected compared with
so that
, y
1 e V = --Ti 7+ (
IH jr
e
_
1
XT\
.
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
438
The Hall current density j
is
= ney,
so that, for this to vanish,
1*1 XT. r- 3cm T
we
if
I 6
y
*=2m Xr
AT
Also
>
neglect y as before.
Therefore
= nex = I~
i
e*
n
m XT,
2
=-Q3 c
------
or
The Hall
coefficient
r
rr l
ne
is
.......
3 if all
1022
Tie
.
............... (71), v '
the quantities are in electromagnetic units.
Several methods of estimating n lead to values of the order and putting e = 10~ 20 E.M.u. approx., we see that the
,
10~2 To the values which are order of magnitude of rr-^ is the on 435 which have been found by different p. following, given .
observers,
Metal Tn
...
may be added
Bi Sn -9*0 -0-00004
for
comparison
:
Pb
Sb
+0-00009
+0-192
Na Te Cu +530 -0-0025 -0-0005
According to the theory above, the Hall coefficient should be negative for all substances. The positive coefficients exhibited by Lead, Antimony, Tellurium, Iridium, Zinc and Iron might be taken to indicate the presence of free positive electrons there were any independent evidence of their occurrence. theory along these lines has been worked out by Crude*, but
if
A it
cannot be considered to have been very successful. It seems more likely that the effects are all due to negative electrons, but that the theory only takes account of part of the phenomena. It seems probable that the magnetic field affects the conditions of equilibrium of the electrons and the way in which they are emitted
by the parent atoms. Suggestions towards a theory of this kind have been made by Sir J. J. Thomsonf It is also probable that .
the current-carrying electrons are deflected *
Ann. der Physik,
vol. in. p.
369 (1900).
by the magnetic
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
439
the
fields of
electrons revolving within the atoms and that the orientation of the orbits of revolution is affected by the external
magnetic
field.
In the case of cuprous
iodide,
a compound substance which
exhibits fairly typical electronic conductivity, it is possible, amounts of in excess of that which iodine adding varying
by is
normally present, to cause the conductivity to vary over a wide range. Presumably this alters the concentration of the free electrons without changing other important factors appreciably ; for Steinberg* has shown that the Hall coefficient for this sub-
stance is directly proportional to the specific resistance. It is therefore inversely as n in accordance with equation (71).
The change of resistance in a transverse magnetic field may be calculated on the electron theory in the following manner :
Using the same notation as in the theory of the Hall effect and neglecting the term Hey/c, because it is small when averaged, we have x
=
e
v
mX
H
e
c
m
(H \
c
I
e /I e
12c 2 If there is no
magnetic
field
H=
n
v
U>~^ m m\2
m
2
and
e?
It is important to notice that in the presence of a magnetic field the free time r will not be the same as T O because the ,
and therefore the average magnetic field affects the curvature affect the orientation of also it may length of the free paths, and collide with which the electrons f. Calling Sr the the systems
* Cf. 123. Baedeker, Elektrischen Erscheinungen, etc. pp. 104,
THE KINETIC THEORY OF ELECTRONIC CONDUCTION
440
the increment change in r thus produced, effected by the magnetic field H, is
_ = n-
gt
2 If p
=
l/o- is
e
2
m
L - 1 H ST r^ -r-
z
e
2
12
(
c
2
)
-
J
?72
2
3 rH
X^ =
,
Scr
in the current,
Xv
.
the specific resistance
__ 1U _ P__8f " "" .Since the difference either of them, this
between r and TO
may
7 we may
_?_
T3
12
T
a-
/>
If
Si
is
small compared with
be written
(T
""T
3 C2
neglect ST/TO this equation
may be used
to deter-
mine n, since all the quantities are known. Using this method, which is due to Sir J. J. Thomson, Patterson* found that his measurements of the change of resistance led to the following values of n: Metal
Metal
It is difficult to judge how much reliance ought to be placed on these magnetic methods of evaluating n until a more satisfactory explanation of the anomalous values of the Hall coefficient is
forthcoming.
A
detailed account of the
modern experimental data relating
to galvanomagnetic and thermomagnetic phenomena will be found in Baedeker's Die Elektrischen Erscheinungen in Metallischen
Leitern, p. 94. *
Phil.
Mag.
vol.
m.
p.
648 (1902).
CHAPTER
XVIII
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS The Emission of Electrons from Hot Bodies and Thermoelectric Phenomena.
WHEN is
bodies are heated an emission of negative electrons
found to occur, which increases very rapidly with increasing
temperature. have revealed
The
salient features of this
themselves
capitulated as follows (1)
The number
peratures
T
experimentally
phenomenon
may
as they
be briefly
re-
:
of electrons*
N
emitted at different tem-
governed by the formula
is
N = AT^e-u T A, \ and b are constants.
A
varies very
(I).
much from one
substance
of the order unity, and its precise value makes difference to the formula. 6 in equivalent volts is always little very
to another.
X
is
comparable with
five.
The emission (evaporation) of electrons is accompanied an absorption of heatf. In the case of the only metals by (osmium, tungsten and platinum) which have been examined the magnitude of this effect is about what would be expected from (2)
the values of
b.
The absorption^ of electrons by a cold metal panied by a liberation of heat. For a large number (3)
is
accom-
of metals
has been found to be approximately what the value of b would lead us to expect.
this
*
0. W. Richardson, Phil Trans. (A), vol. CGI. p. 543 (1903). Wehnelt and Jentzsch, Ann. der Physik, vol. xxvm. p. 537 (1909) Cooke aud We hne Richardson, Phil. May. vol. xxv. p. 624 (1913), vol. xxvi. p. 472 (1913); and Liebreich, Verh. der deutsch. physik. Ges. 15. Jahrgang, p. 1057 (1913). vol. xxi. p. 40 J Richardson and Cooke, Phil. Mag. vol. xx. p. 173 (1910) f
;
;
THE EQUILIBEIUM THEORY OF ELECTEONIC CONDUCTORS
442
and the distribution of velocity among, the emitted electrons is identical with that given by Maxwell's law for molecules having the same mass as an electron at the (4)
The energy*
same temperature
of,
as the emitting metal.
In addition to the points outlined above a large amount of information bearing on this subject in other ways has been accumulated. For an account of this the reader may refer to
The Electrical Properties of Flames and of Incandescent Solids, by H. A. Wilson (University of London Press (1912)), or to an article
by the writer in the Handbuch der Radiophysik (Univer-
sity of Leipzig Press (1914)).
In some
notably those furnished by the alkali metals f this emission may be greatly augmented by the occurrence of cases,
chemical action.
,
In
fact,
some recent writers have taken the
standpoint that chemical action is an essential condition for the This is very unlikely on general grounds. emission to occur. For it amounts to denying that the simplest type of chemical
namely, the decomposition of an elementary atom into a negative electron and a positive ion, can ever occur under the influence of heat alone. In the opinion of the writer, the
action,
facts also are decisively against
which
is
it.
The behaviour
of platinum,
the substance which has been most thoroughly studied,
very difficult to reconcile with any chemical theory, and the currents which can be obtained from incandescent tungsten are
is
much
too large to be attributed to chemical action J.
We shall therefore
assume that there
is
an emission of electrons
from elementary and compound substances which is a purely thermal effect and try to find out what laws we should expect such a phenomenon to obey. It is convenient to have a separate name for an emission of this kind, and we shall describe the emission of electrons which occurs under the influence of heat
by the term thermionic. It is also worth while to consider for a
moment what
should expect an emission to observe which *
Mag.
is
Richardson and Brown, Phil. Mag. vol. xvi. p. 353 (1908) vol. xvi. p. 890 (1908) vol. xvm. p. 681 (1909).
laws
we
conditioned by ;
Richardson, Phil.
;
f Of. Fredenhagen, Verh. der deutsch. physik. Ges. 14. Jahrgang, p. 384 (1912). t fif ft P.i/VhoWla, PJ,,'? MV,/-, xr/V1 WTTT v% Q/fe: /inii>\
W
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
443
chemical action.
The electrons which are liberated are then to be regarded as one of the products, either intermediate or final, of the chemical action, and the extent to which they are formed will be determined the laws which the formation of by govern other chemical Now, the law which governs the rate products. of chemical actions in general is not essentially different from (I), and something very like the other results enumerated on p. 441
would
also follow in this case, so that such considerations will not enable us to distinguish between the two phenomena. The only satisfactory criterion is whether the emission of electrons is
accompanied, pari passu, by chemical combination or decomposimore generally recognized type. There is no evidence that this is universally the case.
tion of the
We
shall now return to the consideration of the theory of the thermal emission. This purely phenomenon may be attributed to the increased kinetic energy of the electrons at high temperatures enabling them to overcome the forces which tend to retain
them
The
in the conductor.
rate of variation of the emission with
the temperature may be calculated in a variety of ways. It will be conducive to clearness if we sacrifice generality to a slight extent
and make a definite hypothesis about the structure of the interior although many of our results will be more than the We have seen that general hypothesis we are making. the electrons in a good conductor behave as though they were of a metallic conductor
:
acted upon by fixed centres of force varying as the inverse third power of the mutual distance. The potential energy of an elec-
must
tron of
its
therefore be continually varying from point to point shall suppose that the potential energy of an path.
We
In other is a function of its position only. are supposing that the interior of a metal may be sufficiently described by mapping it out by a series of fixed surfaces, the level surfaces of W, the potential energy of an electron inside a metal
words, we
electron.
These surfaces are supposed each conductor.
teristic for
motion prevents the
state
of
satisfactory
phenomena.
this
affairs,
account
The
to
be definite and charac-
fact that
the electrons are in
from being a complete representation of but it will be fairly certain to give a of
the
more important features
of
the
444
THE EQUILIBRIUM THEOEY OF ELECTRONIC CONDUCTORS
are considering is defined by the average number v of them per unit of volume at that point at any instant, by the potential energy the electrons at that point, and by the temperature T, which is
W
the average kinetic energy of the electrons. proportional to takes the constant value outside any conductor points just of values are different the In the state of equilibrium
W
W
This definition of
characteristic for different conductors. sufficiently exact. is
At
attracted towards
W
W
is
At
W
Q
.
and not
points very close to the conductor an electron it by its mirror image in the conductor.
increases for some little distance away from the conductor Thus and the points at which it approaches a sensibly constant value
On the other are not immediately outside the bounding surface. surface from the will be affected distances considerable at hand,
W
by the potential of other bodies in the neighbourhood. The true at points at a considerable distance is the value of value of
W
W
from the surface, either in a cavity inside the conductor or outside if there are no other bodies in the neighbourhood. It follows, as
a
result
of our
theory of matter (p. 404), that equilibrium at temperature T,
investigation of the kinetic at any point in a system in
dn~vd>r = Ke- wlRT dT where dn
is
the
number
of electrons, which participate in thermal
volume
in the element of
phenomena,
dr,
throughout the system, being a function of
We
shall
now
and
T
K
is
constant
only.
which determine the electrons and the internal
consider the formulae
equilibrium between the external free electrons
(1),
which can become
Confining our attention to a
free.
single conductor, consider first the special case in which there are a finite number p of finite internal regions each characterized by
constant values v 1}
W
ly
r l3
...
vp
,
W
pt
rp
.
Then
if VQ)
W
,
TO are
the values of the corresponding variables just outside the conductor,
we have
where n
is
the total
number
of electrons which can
that are present in the system. in/
In general
it
will
become
free
be possible
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
445
infinitesimal volume, so that
dn where the integral regard
W
is
n
taken throughout the entire system.
If
we
as a fixed constant,
(4).
The next step is to calculate the absorption of heat when electrons are allowed to evaporate from a conductor. Consider the conductor to be surrounded by an insulating boundary such that the volume of the space between the conductor and the If the boundary
is displaced so that the volume be done by the equilibrium pressure, p, of the atmosphere of electrons. If we admit that such a change of the external volume is not accompanied by any material
boundary
increases
is v.
by
dv,
work p dv
will
mutual potential energy of the positively charged of the atoms which make up the conductor, the increment parts dS in the entropy of the system will be alteration of the
....................... (5),
where
Z7= w(f JZT + J)
and
p=
vQ
RT
......
.
.............................. (6),
................................ (8).
number of electrons in the system which can the integrals in J are extended throughout the and free, n is constant, it follows that when the volume is Since system. heat abstracted is the increased by dv
n
is
the total
become
'
"dv
( 9 >-
"
;
Since r
{
Jr+0
n We~ w/RT dr + SvWe~ WlRT
= Lt dv
-
mr[e/ T
=<)
J r +v\
.+ 0+fi0
- "" '...(10), e -wiRT dr
f J
where
T+V
r+V
\
denotes that the integral
/
taken over the volume r
is
/r+'y
volume v of the external
of the conductor plus the
space.
Let nv be the number of electrons in the external volume v and nr the number, of the kind contemplated, in the metal. Then in these problems nr And since
nv
is
always extremely small compared with
.
_ I e-
rr
_
w RT dr '
(3),
we may
^v
\e-
JV
Jr
by
_
replace
f T+V
e-
w RT dr
in (10)
i
J
by f e" T
w RT dr. l
/
Also since
JT n r
JT
nr r
Jr
v r T'
T
and the following equation
is
true as regards order of magni-
tude,
the order of magnitude of f v
'
f J T
n is
that of -H. ^7
T
Thus
We~ w RT dr may l
J
T+V
also
be replaced by
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS f
in
Whence
( 10)
J r
447
r - J)
............... (11),
e
and from
(3) (12).
By comparison with (9) the loss of heat which accompanies the escape of one electron is (13).
RT
the external work, w = TF latent heat of evaporation of one electron. is a perfect differential, Since
From
(5)
is
and
is
the internal
Moreover, since
/S
(6)
_^/3
fdS\
-/
dJ\
...,
If dJ
fdS\
Thus
and
"
r I
so that
v
'
1
civ
d'JL'
w
w
= AeJ
= Ae
rial
[ 1
9z0
f
(14) v a TS /j -
' i
where
a quantity, characteristic of the material, which independent of T. J.
is
is
Equations (14) exhibit the relation between the equilibrium concentration
z/
T
of the electrons in the external atmosphere, the their internal latent heat of evaporation w.
and
temperature If w were a known function of T, (14) would enable us to comSome information about pletely specify VQ as a function of T. the
way
in which
w
depends on
following slightly different
T may
be obtained by the
thermodynamic argument.
The Relation with
the Specific
Heat of
Electricity.
There is an intimate connection between vQ ,T,w and the specific heat of electricity o- within the conductor under consideration. Suppose- that
A
we have two conductors
A made f
and
of the same
but maintained at the different absolute temperatures T and T'. They are to be of sufficiently great size and are The atmoconnected by a thin conductor of the same material.
material,
A
A
1
are enclosed and separated and spheres of electrons about a suitable insulating boundary (see Fig. 50). from each other by
V
v
r
T
T Fig. 50.
If contact difference of potential depends upon temperature, the potential "V at the surface of A' will not be the same as that at the surface. of
A.
If the value at the latter surface
electrons in the enclosure surrounding
A
is F",
the
be at a different
will
electric potential from those surrounding A'. Let eV be >eV, and surround A with a screen which is permeable to the electrons and maintained at the potential This may be imagined as
V
.
a wire gauze of indefinite fineness which is connected to one end of a battery, the other end being connected to A. The electromotive force of the battery
done by the battery, but even
if
is
V
V.
there were
There it
is
no work
would not
affect
the argument. The nett effect of this arrangement is that the Electrons in the enclosure outside the potential filter have the temperature T and the potential V. Their pressure p Q is different
from the pressure^ outside A, the relation being evidently
log^ = logjp-
e
(V'-V)
RT
i
.(15).
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS With the arrangement shown in the figure the electrons made to perform a reversible thermodynamic cycle between temperatures
T
and
T
in the following
449 are
the
manner:
N
By means of a' suitable piston and cylinder arrangement electrons are taken out of the enclosure bounding A. This operation
is
conducted at constant pressure
and the potential V. absorbed from
A
The
external
_p
,
the temperature being T is and the heat
NET
work done
N{w + e(V-
N
7) + RT}.
The electrons are then caused to expand adiabatically to the temperature T'. The work done during this adiabatic transformation is
where y
is
is
the ratio of the specific heats of the electrons at
constant pressure and at constant volume. The heat absorbed nil and the pressure is changed from p Q to
is
The next step consists in expanding to the pressure p' at the conf The work done here is stant temperature T .
and an equal amount of heat temperature T' and potential
is
absorbed.
V are
The
electrons at the
then allowed to flow into A'
under the uniform pressure p. The work done during this stage - N(w'+RT'\ where w' is the is - NRT' and the heat absorbed is internal latent heat of evaporation of one electron at T Finally the electrons are allowed to flow down the connecting conductor back f
.
to
we
A, when the whole system is in precisely the condition at which There is no work done in the last operation and started. rT
the heat absorbed
is
Ne
SdT, where
S
is
the heat absorbed
J T'
when
unit
quantity
of
electricity
flows
against
unit rise
of
temperature.
Since the heat absorbed during the flow down the conductor is taken in at a series of continuously varying temperatures we have
EQUlLl.BKl.Um T.H..KUKK Ui
THE
to apply the second
UUJNJJUUTOKS
WL,JBiUJ..UiNl(J
law of thermodynamics in the integral form
/f
= 0.
This gives
/T\ ~y w' w+RT' + RT ^ LOg^ po/T^y +^ -(rim)
rn~
T
f
l
[ [
7rt
1
PQ\JL J
JL
J
S
7r,dT'= T 1
0,
'
whence, by (15),
|
-
^+R
(logj>
- log/} -
^
^ 1 (log 7
log T')
+
<>Jl%dT= T ....
/T
1
n/
or
^
^yj
/
'...(16),
CY
iS
7 -l
and
r JZ;,T--
vofif-jp-^r-x-ie
-4,e BT
or
J
VT-I
^7
( 18))
2'
(19).
In these equations A, A l and JL 2 are constants which are of the substance and are independent of T. e is
characteristic
used
for
the base of the natural logarithms to distinguish
the electronic charge
A
certain
it
from
e.
amount of caution
is
necessary in the interpretation
In measuring the specific heat of electricity cr, a continuous current is driven along by means of an impressed If we are to be certain that S and cr are the electromotive force. of these equations.
same thing
it is
desirable that the flow
be a continuous process.
down the conductor should
This can be realized to any desired degree
by making the cylinders sufficiently small and numerous and working them rapidly enough. As we have not
of approximation
proved that the thermal effects arising from a current inspired by diffusion are identical with those due to a current driven by an impressed electromotive force, it is desirable that the current in the unequally heated conductor should be driven in this way. Our cyclic process
may be adapted
in the wire, which
A
A
is
made
to this
end by introducing a battery
indefinitely thin, whilst the conductors
and are indefinitely large. The transference in the wire is thus electrically driven and outside of and it is effected mechanically, the whole process beinsr continuous.
A
A
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
Under
these circumstances the
necessarily the
same thing
as the
451
w of equations (16) (19) is not w of equation (14). The latter
refers to a virtual displacement of electrons across the boundary to the subject equilibrium conditions, whereas the former refers to
a continuous stream such as actually occurs under an impressed force. Since the electric transference produces no permanent
change in the materials, the difference between these two quantities will be equal to the difference in the quantities of energy which
accompany the transference of unit quantity of electricity in the respective media on the two sides of the boundary, if such a If we keep the symbol w for the w of equations the true latent heat of evaporation the w denote and (16) (19) we have of equation (14) by
difference exists.
where XT
is the energy transferred by unit electric current in the metal in unit time and X is the corresponding quantity for the
current outside the metal.
Thus instead
of (19)
we get ...... (20).
We (20) for
have now arrived at two different expressions (14) and VQ in terms of 0, cr and T. Since, if and a- could be
would have to become expressed in terms of T, these expressions identically equal,
follows that
it
X.-*
T [ ]
for all values of T.
tions
:
^^-T-l-^JT a W-l R)T~l RTdI f
1
Hence
will
be given by each of the equa-
THE EQUIL1J5K1UM
UUJNUUUTOKS
TJU1XJKX
Approximation
to
VQ as
a function of T.
If the electrons are treated as point charges, they have three If the collisions of the electrons of freedom and 7 $.
=
degrees occur with centres of force varying as a power of the mutual distance, it follows from the kinetic theory calculations in the last that chapter ^
1
-r ( aJL \
m
=
so
that,
under these conditions,
JL
s (25).
The value
of
a
is
different for different metals
and may be
either
In general it also varies with the temperapositive or negative. ture, often in a complicated manner. If the thermoelectric powers of different pairs of metals were accurately linear functions of the
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
453
temperature, then the difference in the values of cr for any two metals would be proportional to the absolute temperature (equa-
This is not borne out by the preceding values of a which are taken from a table compiled experimental
tions (51), p. 426).
by Baedeker*. value
is
reliability of the
all
cases
it
negative and
is
It
is
value of
we
where
e/>
in (14)
-j^.
numerical
its
questionable, however,
~
experimental determinations of
has been found that
than | R, so that f R
(j>
cr
warrants us in rejecting equation (50),
cr
In
of
iron
for
greatest at just over 200 C.
whether the
and
Thus
ecr is
p.
426, Chap. XVII.
a good deal smaller
taken as a first approximation to the
may be
Thus neglecting the second and higher
derivatives
have, approximately,
is
we
independent of
Substituting this value of
T.
c/>
for
w
obtain v
= A T^e~"^
(27).
1
are independent of T to the extent to this equation A l and which the assumed approximations are valid. Now there is a simple relation between v and the number
In
<
of electrons emitted
by unit area of the conductor in unit time,
of the conneglect the reflexion, at the surface It is the exterior. to it from return which ductor, of the electrons is reflexion such that results appreciable, known from
provided we can
experimental
but we
shall disregard it for the present,
and consider how
it
may
be allowed for later. In that case, since VQ represents the number of electrons per cubic centimetre of the space in the steady state, be the number emitted by the surface in unit time will evidently returned from the exterior in the same interval. equal to the number This is CLU
I
J-oo
Cuv
QjW
I
J -
li
\
1
'
fc
\ vr /
oo 1
yi n Metallischen Leitern,
(28).
p. 76.
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
454
we apply a
If
difference
sufficiently great potential
all
N
the
by the conductor may be collected on a neighThe ensuing saturation current is, per unit electrode. bouring area of the emitting body, electrons emitted
where
A
2
and
<J>
are constants characteristic of the substance but
independent of T. Had we not made the approximations above the index of T would be slightly different from 2 and we should e ap
have had exponential factors of the type
where ap
,
an
is
extremely small constant. The values of these factors cannot differ should therefore expect (29) to give a fair much from unity. the mode of variation with temperature of the of representation
We
electrons emitted
by hot
bodies.
The number of substances which now very considerable and for all
have been tested in this way is of them the emission has been found to be consistent with an
- AT- e~ b/T As, however, the variation equation of the form Ne is almost all in the exponential term the results can be fitted as .
by Ne
well
= AT
2
6~ b/T
,
by taking a slightly different value of the
constant 6 in the exponential index.
We
shall
now
consider the bearing of these effects on the
nature of contact electromotive
force.
Contact Difference of Potential.
Imagine an enclosure limited by an insulating boundary, maintained at temperature T. Suppose the enclosure to contain q material bodies, arranged in any manner, and that the whole system has come to a state of thermal equilibrium. The temperature will then be uniform and In equal to T throughout the system. general the surfaces of the bodies will assume different potentials
V
l9
Vt,...Vg
where Tf m
mth
Clearly
.
is
the potential energy of an electron just outside the show that Qm are uniquely determined
W
If we can
Wf
body. by the constitutive equations of the system IT"
_
T7-
^>V^
A
J-
4-,
rt
:~.L -
,1:^
.
.
it ,-
follow that / i-i T
will ,
,
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
455
etc. represent the concentrations of the electrons at points just outside the different bodies. Since there is equilibrium between
the electrons which are just outside each pair of bodies we evidently have q - 1 equations of the type
From m v* =
(14) and (12)
we
see that there are q equations of the type From (7) there are also 3 equations of T).
m Jm A m
4>m(W Jm = ^m (T) and evidently there are q equations of. the = A type m constant. Altogether there are 4g 1 equations between the Sq variables J and differences TF m and T. Q
the type
i/
the
latter,
say
T
t
1 is
Wm W
differences
^
A,theq-l
,
Thus there are 4# Q
Q
equations and 4g variables, so that if one of 1 given all the variables including the q
P are
determined.
These equations involve
shape nor relative orientation of the bodies, so that the differences of potential m Vp are characteristic of the
neither the
size,
V
substances under consideration.
They are
clearly the
same whether
the bodies are in contact or not.
Since from
We Wm
shall see
wp
\
potential
(13)w = TF
below
(p.
/it
457) that
follows that
Jm
Jp
is
small compared with
so that according to this view the contact difference of may be estimated from the internal latent heat of
determinations evaporation of the electrons. The experimental of w for different substances are not yet sufficiently trustworthy to furnish a satisfactory test of this relation; although such indications
as
there
are
tend to show that
w
is
smaller for
for electronegative elements. electropositive than
The Peltier
We physical
shall
to obtain expressions for the various
now proceed
quantities
Effect.
which
are
grouped
under
thermoelectric
consider the Peltier coefficient P.
To
phenomena. Let us first do this we need only to consider a reversible isothermal cycle The arrangement and operainvolving two different conductors. that tion is in fact much the same as that shown in Fig. 50 except 4-u^ ^/vn/iii/vfrkvo A and A' arp nf rlifpp.rp.nt, materials and are at the
THJi JijyUlJUiJJXlUM
same temperature
Let the
T.
UJ?
TJHJJiUJK, X
J&.UJDj\jJ.xt,uiN JLU
suffixes 1
and 2 be used
to denote
the various physical quantities which relate to the separate materials, Let eVz be Surthe notation being otherwise as before.
>eV
l
at
F
2
and
.
by a potential filter maintained This will reduce the pressure of the electrons from p l to p-f
round the .
will
conductor, as before,
first
change their potential energy from
eV
eK
to
l
but
will not
temperature T. The cycle commences by removing JV" electrons from the enclosure surrounding A l under the uniform and the heat absorbed is, by The work done is pressure pf.
affect their
NRT
N [w, + F - V,) + RT].
Next expand at T from 2 ( are each equal to and heat absorbed work The done p\ p2 r Q and F in the second body. NR Tlog(p /p2). Next condense atp2 2 The work done is NRT and the heat absorbed is - N[w* + RT]. The electrons are then allowed to flow along the connecting convirtue of (13), to
e
.
l
A
ductor to
ly
thus completing the reversible cycle. During this is done, but there is an absorption of heat NQe
operation no work
The
at the junction.
total
amount Q
of
work
in this isothermal
Since the cycle
is isothermal NRTlog(p /p2 = this must vanish, so that p The total absorption of heat is p N[w, + e(V2 -V ) + RT + RT log^/p,) - wa - RT -f Qe], and since
reversible cycle is
l
).
Q
1
2
.
l
this also
must vanish we have Qe =
If
Q
is
to be the
w^w -e(V^V
(31).
l)
l
same thing as the
coefficient
P which is measured
on the Peltier effect, it is necessary that the cycle should be operated under the same conditions as when a current flows continuously under an impressed electromotive force. Hence,
in experiments 1
w
as in dealing with the Thomson effect, w 2 and will be the values l of the latent heats which to a correspond steady flow and not to conditions. the former and the latter equilibrium Denoting
by
by
as before,
<
we
w
therefore have the equations
ftP-ttfe-Mx-^Fa-FO
(32),
= ^-^ -(X -X ) + (Fa -F ) = / -/ ~(X -X ) 1
1
since
A
<
series of
=
W
2
s
2
1
l
1
(33),
(34),
J,
P
as well as the values experimental values of power of the same pairs of metals at the
of the thermoelectric
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS same temperatures
457
are given in the following table, taken from
Baedeker*.
In
all
tbese cases the Peltier coefficient
equivalent volts
thermal to the
a
factor,
is
of the order
shows that - (X2
this value of P.
On
X
:)
cannot be
is
when expressed in The ratio of the
volt.
much
the other hand
several volts in extreme cases.
and
10~3
electrical conductivity of metals,
Q
than
10~2
F
2
which contains
*
LOG.
cit.
as
greater numerically
V may
It follows that e
l
(
F
fa are approximately equal to one another. small compared with w m WP as stated on p. 455, <j6 2
A,
2
amount
to
w
2
w^
Thus Jm
Jp
Fj),
p. 73.
t The values in brackets were obtained from specimens of material different from those used in measuring the Peltier coefficient.
The Thomson Effect
A
of expressions for
number
is
desirable to express
J
etc.
some
them
of
in
terms of the
quantities.
which describe the internal structure of the material
instead of
=
<
We
Consider, for example, equation (23).
etc.
and from
(W -J) ~dT
and (14)
(2)
RT IT
RT*
dT
have
~
RT*
RT*
'
Thus (J
so that
This expression is more general than equation ((51) of the lastchapter, and the two become identical only in rather simple cases,
We
have seen that the kinetic theory methods employed in Chapter XVII can only be regarded as strictly accurate when the
linear dimensions of the regions, within
which the forces exerted
on the electrons are appreciable, are small compared with the distance between collisions. In that case the potential energy w of the electrons will only differ by a negligible amount, from the
mean value
Also the
/.
A
of the
-last
chapter 1
is
emial to l
N
\
}
\7rJ if
N
is
the
number log
of free electrons in unit volume.
Since
N=
Thus, from (35),
e
(y
~ w
1
z
3
f^_
d A
\I~\Q]
.
f
5
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS This
is
in
agreement with
since
(61),
under
the
459
conditions
contemplated
These
equations, which
have been deduced
from thermo-
dynamic considerations, are thus in agreement with those given by the kinetic theory calculations when the latter are accurate, In virtue of (34), equation (35) satisfies the condition
o-j-cr^a
P
7
(
postulated by Lord Kelvin's Theory.
Thermo electro motive
Force.
Having found expressions for the Peltier and Thomson cowe can, by means of the energy equation (48) of the last from them the value of the thermoelectromotive deduce chapter,
efficients
For a circuit of two materials with force of the complete circuit. and one finds in this way, for example, at T T' junctions
"
eJT
n
T
In a similar way equations may be obtained containing other combinations of the variables which occur in equations (21) (24)
and (32)
(34).
the assumption of thermodynamic reversibility and and cr from equations (23) and (32) substituting the values of in equation (49), p. 426, we obtain
By making
P
.
V\
T'
and, after integrating by parts,
^^
T Tn (]
or
J
T
rpO
~A T~ -L
'd!F,
TPVV/
JLXlJil
use of (37) the following expressions for be obtained from (36) or the equivalent equations
By making
-
w,- Wl + e(V
The values substitutions.
E
may
:
1
'
T
in terms of
J J
2
l}
,
-V ),dlT a
may be derived by dE
etc.
The thermoelectric power
is
^>
obvious
given by
(41) respectively by the upper limit T'. equations obviously satisfy Lord Kelvin's condition
dif-
These
ferentiating (38)
a#_p
df'T* Alternative Expressions for the Thermoelectric Coefficients. If
we wish
of electrons (v \
tor,
to express these quantities in
=
-r-
)
in unit
drJ
volume
at each Lpoint of the conduc-
instead of in terms of the functions
mean
potential energy,
terms of the number
we may proceed
J
which depend on their
as follows
:
_ TT
Since from (3) v
= Ke RT we ',
~
have
TD/TT
= RTlogK-~
I
ni
~ RT n
i
f
vlogvdr
(42).
dn
J ni
Thus, for example, l
*
and similarly with the other quantities
f
fi
dn \ j
E and ^.
11
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
461
If the distance between collisions is large compared with the distances within which the forces are appreciable, then the potential of the electrons is constant in all but a small fraction energy
of the available space.
values of v and
In this case we can replace the average values without serious error. If, in
W by actual
addition, the law of force during collisions varies as a distance, finally, if
we
see from the last chapter that
the law of force
\m
(
T/T)
= 0-
And,
the same for the different materials
is
\p
^m
power of the
= 0.
Under these circumstances the any pair, which have been found reduce to expressions then, for
Pe^RTlogvt/v!
E[ e
I
l
.,*
Vl )]
-w"r
(44),
(45)
'
D
T7T
8T
T~(log
7
R \
(43),
=
< 46) -
7 log S
These equations are exact between hard elastic spheres.
Comparison
if
the collisions are always like those
with-
Experiment.
There are a number of reasons why formulae (43) (46) are unsatisfactory. In the first place the ratio of thermal and electrical conductivities indicates that the force during collisions does not vary very sharply with the mutual distance. Moreover the computed
never free strength of the centres indicates that the electrons are are conditions theoretical Thus the forces. considerable from very
In the second place the formulae are contradicted by the thermoelectric data themselves. By hypothesis all the electrons are to be treated alike, so that v in far
from being
satisfied.
the number of current-canning equations (43) (46) represents Now the electrical electrons in unit volume of the material. is no reason to expect and there to is z/, proportional conductivity that the other factors which enter into it will not be of the same order of magnitude for different substances.
We
should therefore
THE
EQU1L111K1UJ.YL TJ
Formula with the ratio of their specific electrical conductivities. lead us to expect an exceptionally therefore would large (43) Peltier effect at the junction between a very good and a very bad conductor.
The
Peltier effect in a
number
of cases of this kind
has recently been investigated by Koenigsberger and Weiss*. Although these experiments are difficult to make, they seem to
have established that the Peltier effect in such cases
is
not, in
In some cases it was found general, of exceptional magnitude. to be in the opposite direction to that given by equation (43).
The more complete theory involves two considerations which In the
are
place, the law of force neglected in equation (43). to another. It is, during collisions may vary from one material first
\
X 2 which arises however, extremely improbable that the term from this circumstance, can be comparable with the term involving log
We
i/x/z/a,
when
the ratio of the two conductivities
,
is
very great.
are thus driven to the conclusion that the difference in the
potential energy of the electrons
is
not, in general, measured, even
approximately, by RTlogvJv^ where v l and z/2 are the volume This objection cannot be concentrations of the free electrons. in the case of the formulae given by the more complete theory, which make the Peltier effect depend on the mean poten-
made
energy of all the electrons which may, from time to time, free under the dynamical actions actually occurring, and not on the actual or average number of those free instantaneously. tial
become
The number of the former may be quite considerable, although, on account of the intensity of the attraction of the rest of the atom, very few of them are able to get free enough to take part in carrying the current at any instant.
A second
point which is of interest in this connection is that the by phenomena exhibited at the melting- point. For all the metals which have been tried, except antimony and bismuth,
raised
the specific conductivity of the solid at the melting-point is greater than that of the liquid. The changes are quite considerable, the ratio of the two specific conductivities varying from T34 in the case of
sodium
to 4-1
case of antimony the ratio All the metals except 0*46.
they
solidify.
is
in the case of mercury. 0'70 and in the case of
antimony
arid
In the
bismuth
bismuth contract when
In accordance with (43) and (46) we should expect *
Ann. der Phwik*
vol. xxxv. D. 1 (191 1L
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
463
these changes in the electrical conductivity to be accompanied by very considerable changes in the Peltier effect and the thermoelectric power. Such effects have been looked for most carefully .and with negative results. It is clear that the change in the
thermoelectric power, if there is any, is much smaller than we .should expect from (46). However, it cannot be said that the absence of Peltier effect at the melting-point is properly under-
stood on the more general theory. in a
happen
number
of ways which
It is evident that it it
might would take too long to
discuss here.
In the case of the iodiferous cuprous has already been
made
iodide, to
which reference
in connection with the Hall effect,
Baedeker*
finds equation (46) to be accurately verified if the relative values of v are measured by the relative conductivities. It seems as affecting the motion of the electrons are the case of this substance than in the case of the in simpler metals.
though the conditions
much
The Conducting Electrons.
The expressions which have been obtained for the various thermoelectric quantities involve integrals which are extended over all the electrons which are able at any time to take part in thermoelectric phenomena. In general we should expect only part of these electrons to be capable of engaging in the transportation
Many of them will be so strongly attracted atom nearest to them that they will only rarely be able to escape from its immediate neighbourhood. Such electrons will
of the electric current. to the
only be slightly displaced under the influence of an external field will not participate in the conveyance of the conduction
and
current.
seems natural
It
carried
by
to suppose that the conduction current is
those electrons of the class contemplated which are
An orbits. instantaneously executing open, as opposed to closed, of electrons which are executing for the number expression
N
orbits
be obtained on the assumption, which accords
may open with the results of the discussion in the -electrons are attracted
by
last chapter, that the centres of force varying inversely as the
R
be the instantaneous radius cube of the mutual distance. Let makes with the direction of this the 6 and centre the to angle
R
when =0. of velocity.
R
t
the radial component be the tangential and from Let Vl be the velocity infinity, then
Let
m Vf = It is
shown
that p. 231,
in Routh's all
Dynamics of a Particle (Cambridge, 1898),
V? < T*
the orbits for which
The equation
W.
to the others
is
are open.
given by
K where
T
and
T = h (A + JJ), R = nh (B - A), T
R
when
t
are zero, n2
and
R
and
Q
G
V*
= -^-
1
are the values of
and h = RT.
Evi-
2
J.Q
dently Since
R
can only become infinite
if
A
and
B
have opposite signs.
RQ the condition that
jB
A
and
7
7
a
should have the same sign
F
2
1
""
T*
2
I;
i.e.
T
2
Rf > F^ tf +
^o
v.
.*
is
The complete condition
Bf.
an open orbit
for
is
therefore that the
energy should exceed the kinetic energy due to the attractive forces supposed to act from infinity to the point con-
initial kinetic
sidered.
If
we put
=
W=^mV^
}
we evidently have )
dudvdw
TT
= Sn
3>/]pm)\3>/*o *
TT
J
Vi
TTT
r2-rr
d0
dr J
d Jo r 00
A^ ^
(>-&(]
"j
\
L
......
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
465
where Sn denotes the number of electrons of all sorts at a point where their potential energy is at any instant, and SN denotes the number of them which will execute open orbits. This calculation assumes that the distribution of kinetic energy among the particles is independent of the potential energy, a conclusion which we have already established (see p. 405).
W
Now
the
of volume dr
number dn is
equal to
of electrons of all speeds in the element
vdr where, from ne -WIRT
whence the total number volume T is
N
(3),
of current-carrying electrons in the
(48).
It is impossible to predict anything very precise about the behaviour of integrals like those in (48) but it will be observed ;
for example, and neglecting by comparing with formula (39), electrons in a of number the both that current-carrying "Xa-Xi,
functions of cubic centimetre and the thermoelectric power are
the distribution of
^
Jtil
in the space inside the conductor.
.should therefore expect that
We
a continuous any cause which produced
substance at a given .alteration of the electric conductivity of the
a corresponding alteration in the temperature would produce The changes in the one quantity would thermoelectric power. in a correbe expected to follow changes of the other quantity
sponding manner. in the case of Evidence of a correspondence of this kind exists to found is depend very largely Their electrical behaviour alloys. this point of view From on the intimate structure of the alloy. classes: (1) those main two be regarded as falling into alloys may
constituents and which consist of mixtures of crystals of the are mutually /2^ those formed of crystals whose constituents
466
THE EQUILIBRIUM THEORY OF JELUUTKOMU CONDUCTORS
In some cases the constituents are only able to dissolve in one another to a definite limited extent. In other cases chemical considerable complication. compounds are formed which give rise to The behaviour in such cases, however, is determined by the mutual
soluble.
solubility of the
compounds and the independent constituents
that the effect of chemical combination
the
number
of substances
their behaviour
is
is
;
so
practically to increase
which have to be considered. Otherwise where no chemical com-
similar to that of cases
bination appears to occur.
Let us consider the two simple cases of complete immiscibility and of complete mutual solubility, from which all the others may
In the case where the constituents are entirely be developed. immiscible the specific electrical conductivity, expressed as a function of the composition, changes linearly from the value characteristic of one of the pure substances to that characteristic
of the other.
-Precisely the
alloys
same statement
is
referred to a standard metal.
electric
power whose constituents are mutually soluble
true of the thermo-
The behaviour is
of
entirely different
The addition
of a small quantity of either constituent to the other pure metal produces a large diminution in the specific The diminution produced by the adelectrical conductivity.
in character.
dition of a given quantity of the foreign substance diminishes Thus the curve progressively as further amounts are added.
which expresses the conductivity as a function of the percentage composition drops sharply from the value corresponding to either pure metal and has a fiat minimum in between. The curve which expresses the thermoelectric power as a function of the composition is entirely similar in character. The reader who is interested in
the electrical properties of alloys will find a very good account of the recent developments in Die elektrisdien Erscheinungen in metallischen Leitern
We
by K. Baedeker (Braunschweig, 1911).
have seen that a considerable change in
electrical
con-
ductivity is unaccompanied by corresponding changes in the thermoelectric quantities in the case of metals at the melting-
pure
It seems probable that in this case there is something in the conditions of equilibrium which makes + \ take the same
point.
J
value for the
electrons
necessarily involve
a
in the
two phases.
This
would not
corresponding equality in the fraction in In addition there is thp. nnsftiViilitxr that. limipfa.r-.t,inn
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
467
a considerable change, without altering J, in the centres which determine what corresponds to the mean free path.
worth while to consider the form taken by (48) in two In the first let us simple that the interior of the suppose conductor may be divided into two classes of for the first of It is
cases.
regions,
which,
T!,
$ (=-
W)
has a large value which
For the second r2 suppose that
<
= 0.
Then
we can denote by
<
.
(48) becomes
This formula makes the conductivity increase rapidly with inThe creasing temperature on account of the factor e~^ IMT .
experiments made by Horton, Koenigsberger and others on the conductivity of comparatively poor conductors show that the temperature variation can be represented satisfactorily over a considerable range by a formula for the conductivity developed from an expression of type (49) for the number of tMirrent-carrying electrons.
In the second place suppose to be 'small. -^p lect
Then
if
we neg-
terms involving (
where
^
is
the
of the conductor.
mean
We
2
taken throughout the volume should expect this type of formula to apply, value of
c/>
for the variation of qualitatively at least, to the best conductors ; conductivity with temperature in such cases is comparatively
small.
According to (50)
N
will
always be less than
n,
which
it
N
zero. Thus will approaches asymptotically as $JRT approaches order In to the with increase explain rising temperature. always
decrease in the conductivity of the pure metals with rising temit is
necessary, increase either in the
perature
volume
if this
theory is to hold, to fall back on an of the centres of force per unit
number
or in their strengths.
Some
of the rather
bad conductors investigated by Koenigs-
VkAvnviv ^v>n"hif, \rar\7 inf.prAst.incr r>hft"nomfvna,.
Thus
in the case of
increases rapidly with increasing temmagnetite the conductivity low at temperatures, to a maximum at about 240 C. perature oft* as the temperature rises in a manner which falls it that After
resembles the behaviour of the metals. The rapid increase at low to the large increase of with T temperatures is evidently due The subsequent decrease may tentatively is large. when
N
/RT
in the number of the repelling centres or to a change in the magnetic structure of the substance. account of the temperature variation of the electrical
be attributed to an increase
A
comprehensive
properties of the comparatively poor conductors, by Koenigsberger, will be found in the Jahrbuch der JRadioaktivitdt und Electronik, vol. iv. p.
158 (1907).
The Reflexion of Electrons at the Surface of Conductors. According to the theory on p. 444 et seq., the concentration r of the electrons at a point close to a conductor in an enclosure at a constant temperature T is determined by T and the intrinsic
()
V
m of the conductor. Thus the equilibrium pressure value and in accordance with equation (14) is a definite has p
potential
given by
p=
v9
RT = ATe^'*
.................. (51),
the internal latent heat of evaporation reckoned per electron and A is a constant characteristic of the material but
where
is
<jE>
This equation can be established in a very general manner and is true even if the electrons are emitted wholly or in part by the photoelectric action of the complete
independent of T.
Now
aetherial (black body) radiation*. in accordance with (28) the number of electrons which reach unit area of the surface of the
conductor from outside in unit time
is
./
...... (52). J ^
In the steady state this quantity must be equal to the number emitted by unit area of the substance in unit time provided all the incident electrons are absorbed by the conductor. However, the experiments of von Baeyerf and the writer J have shown that *
See 0.
W. Kichardson,
Phil.
Mag.
vol.
xxin. p. 619 (1912).
f Ber. derDeutsch. Physik. Ges. Jahrg. 10, p. 96 (1908). Phil.
Mag.
p. 557 (1909).
vol. xvi. p.
898 (1908),
vol.
xvm.
p.
091 (1909)
;
Phijs. Rev. vol. xxix.
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
469
a very considerable proportion even of the very slow moving electrons which are emitted by photoelectric and thermionic action is
reflected
by metals,
so that
it
necessary for us to
is
take
reflexion into account.
If the
body be
electrons will
reflected
Then the number (1
- r) Ndt.
dt be eNdt.
not a perfect absorber then some of the by it. Let the proportion reflected be
is
N r.
actually absorbed
by the conductor in time dt is Let the number emitted by the same surface in time Then the equilibrium condition gives
=1-?"
(53).
Thus the emissivity (compared with a reflecting power are complementary. This result
is
radiation law.
somewhat analogous
perfect absorber) and the
to
Stewart and Kirchhoff's
introducing kinetic theory considerations it can be shown that the equality holds for each group of velocities
By
*
u, v, w, du, dv,
dw*.
of slow electrons
by
A number of different
measurements of the reflexion
metals have recently been
made
by A. Gehrtsf. Photoelectric Action.
As
is
known, the fact, discovered by Hertz, that a spark easily between two terminals when that which is
well
passes more
negatively charged is illuminated by ultra-violet light, led to the discovery that the incidence of light of sufficiently high frequency
caused the emission of negative electrons from conductors. This phenomenon, which is called the photoelectric effect, is certainly very general and appears to be a universal property of matter. There is no doubt about the universality of this effect when the
term light is understood to include some experimenters have recently
manence of the
effects exhibited
X
rays and
7 rays, although in question the permetals and ordinary ultra-
called
by
We
shall now consider what conclusions may be violet light J. drawn as to the nature of photoelectric action, by the application *
W. Richardson, Phil. Mag. vol. xxm. p. 606 (1912). Ann. der Physik, vol. xxxvi. p. 995 (1911). d. D. Physik. Ges. J G. Wiedmann and W. Hallwachs, Ver. 0.
t
vol. xvi. p.
107
of the principles of
thermodynamics and the kinetic theory
of
matter.
T
Consider an enclosure maintained at the constant temperature which is photoelectrically active but which containing a material
has negligible thermionic emission. but this will not vitiate the results
No if
such material may exist; thermionic and photoelectric
The actions are independent, a hypothesis which we shall adopt. of the electrons near the photoelectric concentration equilibrium material will be determined by equation (51), as a piston which considering the work done against
we may show by is
transparent to
The number which return radiation but impervious to electrons. from the enclosure per unit area of the surface of the material in by (52). If a is the proportion of not reflected, it is necessary, in order absorbed, that the conditions should be steady, that the number emitted in
unit time
is
these which
therefore given i.e.
is
unit time should be equal to
But the number emitted
is
a function of the intensity or
density of the surrounding radiation. that for monochromatic radiation the
Experiments have shown
number
i,s
almost,
if
not
exactly, proportional to the intensity of the illumination and varies little if at all with the temperature, when the other conditions are
We may therefore assume the photoelectric emission to constant. be proportional either to the density of the. aetherial radiation
We
shall take present or to its rate of emission or absorption. the latter as being the more general. Our results can then be adapted to the former hypothesis by making the cmissivity e equal to unity. If the steady energy density in the vibrations between v and v 4- dv is L (v) dv the energy belonging to these frequencies which is incident on area dS in unit time is
T---
where
L (v) dvdcodScos 6,
= 27rsin Odd
and the limits are from to 7r/2. Thus, if the emissivity of the material, the amount of of these energy frequencies which is emitted from unit area in unit time is da>
e is
^ceL(v)dv. Let us suppose that the emission or absorption of unit quantity of radiant enerev of
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
471
causes the liberation ofF(v, T) electrons. Then the total number of electrons emitted in unit time by the complete radiation is
JV =
-r
/
eF(v. T) '
4Jo
L (v)dv
(55).
For equilibrium (54) and (55) must be identically equal for all values of T. Thus if we can L express (v) as a function of v and T, and <j> as a function of T, we shall have an integral equation of which eF(v, T) is a solution.
We
see from equation (26) that
+ f RT,
<
where
<
is
<
is
independent of T.
approximately equal to Over the part of the
spectrum we are dealing with, L(v)dv can be represented with great accuracy by equation (47) of Chap. XT, or
L (v) dv == Sirh
e
dv.
(/
Thus from
(54) and (55) _
uAc-
,
where
We
shall
assume that
been proved, to 00
-TQ
+ f-KT
A
l
,
independent of T.
is
MWX
This has not
strictly speaking, as is only approximately equal and a will also involve T. All that we shall claim
our results, therefore, is that they will represent the way F(v, T) varies with the parameter ^> when the matter is supposed to satisfy certain ideal conditions which may be only approximately
for
realized in practice. However, solutions could be found for other 2 cases in which T' on the right of (56) is replaced by certain other
functions of T,
When A!
is
is
constant, (56)
6jP(i;,r)=0 when
and
e
F(v,T) =
:
l
-
solved by
0
..................... ( 58 )>
^<^<
-C 59
)-
This solution makes the photoelectric action of monochromatic of 1\ and the emission zero when the frequency light independent These results are both in agreeis below a certain critical value.
E
Now
consider the average kinetic energy which are emitted by light of frequency v.
v
of the electrons
The total energy emitted under the influence of the complete radiation at T
E is
clearly
E = 4~T E
v
Jo
eF(v}L(v,T)dv ............... (60).
This must be balanced by the kinetic energy returned to the metal by the stream of electrons, per unit time, from outside.
N
n
If
the
is
number
volume of the space
stream of energy which reaches unit area in unit
outside, the
time
of electrons per unit
is
,n w
/
i-m \% \ -
/*
7T J
Jo
r
r
u^
fKm_
% \
and since k =
J
-ooJ -oo
2RT~
N=
and
l
(61).
Of
this energy let the proportion
the proportion which
is
1
be reflected; then
yS
Thus
absorbed.
for
/3
is
equilibrium
E = ]3.2NRT ........................ (62). we neglect the effects of reflexion, by putting a = # = 1, and substitute the value of given by (47), Chap, xv, making the If
N
same approximations as
we obtain from
before,
(60)
and
(62),
(63).
/' Jo Subject to
and
ejF(z>)
=
eF(v)
= -~^(l
from v
the solution of this /?
Q
to
/hv)
i>
=
from
<j6 ()
v
/A,
= $ //i
to
z/
=
oo
,
is
= hv
For values of hv which as the
=Q
lie
when
<jfr
between
<
Ai/
and
<
< ,
DO
E
(64). v
has no meaning,
corresponding electrons have no external existence.
The reflexion
solution (64) is dependent on the assumption that the of electrons can be This is equivalent to neglected.
assuming that effects which electrons may be disregarded.
arise
But
from the if
of
collisions
we consider
this
mm
fr.
the
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
473
point of view of the electrons emitted under the influence of the light, we see that neglecting the effect of collisions is tantamount to
assuming that the only energy
lost
by the
electrons
is
used up
in overcoming the work of the forces which tend to retain them in the interior of the substance. Under these circumstances the
kinetic energy of the escaped electrons will be equal to that which they acquire by the action of the light (not necessarily from the light directly) minus the work which they have to do to escape. It is clear that the energy which they acquire under the influence
of the light
is
hv,
the same value for
where A
is
Planck's constant.
It evidently has
the electrons liberated by light of the same difference in the energy of the electrons emitted frequency ; any by monochromatic light must therefore be attributed to the effect of collisions
all
of the
escaping electrons in the interior of the
substance.
We
can take account of the reflexion of electrons tentatively
by putting
/3
= SOL.
E
v
Then
= s(hv
instead of (64) $Q)
when
<
we get
(65).
For small velocities of incident electrons, such as those with which we are concerned, the proportion reflected increases with It follows that out of a mixed aggregate of increasing energy. will be incident electrons a proportion of the slow ones greater
absorbed than of the fast ones, and that the proportion of incident electrons which is absorbed will be greater than the proportion of their incident energy which is absorbed. Thus s will be a positive quantity which is less than unity.
no conclusive reason for denying the applicability of the type of argument given above to the emission of material whether charged or uncharged and of particles of all kinds, whatever chemical nature, under the influence of every type of
There
is
aetherial radiation, provided the chemical actions are of a reverof the sible character so that an equilibrium in the material part
system can occur.
The foregoing treatment
of this subject is taken from papers
An equation resembling (64) was first given by writer*. Einstein f as a consequence of the view that the energy of light
by the
*
Phys.
Rv.
vol. xxxiv. p.
146 (1912); Phil. Mag. vol. xxiv. p. 570 (1912).
waves was distributed in discrete quanta. For further developments of the subject the reader may consult the following papers: Einstein,
Ann. der Physik,
vol.
xxxvn.
p.
832 (1912)
;
Journal de
d. k. Preuss. Akad. d. Wiss., Physique, 1913; Planck, Sitzungsber. 350 xvm. vol. Kl Math. Phys. (1913); 0. W. Richardson, Phil. p.
Mag. vol. xxvu. p. 476 (1914). Measurements of the kinetic energy of the electrons emitted by various metals under the influence of light of different been made by the writer and frequencies, which have recently
Dr K.
Compton*, afford considerable support to the above theory. Denoting the maximum observed value of Ev by Em and the mean value by E v these quantities were found to satisfy the T.
,
,
relations
Em = km (v E = k (v V
i;
and
v
v
)
Q)
corresponding to (64) and (65). The values of the constants found are given in the accompanying table :
The
units are:
and k v
,
for
10~ 27 erg
z/
,
10 KJ sec.~
1 :
for
X
,
10~ cm.: and
for
km
sec.
According to these results metals investigated. the radiation value h
The
$ in (65) is very close to for all the values of k m are all somewhat less than
= 6'55
of minor causes which
x 10~ 27 erg
might give
sec.,
but there are a number
rise to this discrepancy.
There
another way in which h may be estimated from these observations. VQ, the least frequency which will give rise to any
is
photoelectric If we may assume that has the equal to //i. same value as in the thermionic emission and that it is correctly
emission,
is
<
(jt>
*
Phil.
Man.
vol. xxiv. n.
575 HQ19A
THE EQUILIBRIUM THEORY OF ELECTRONIC CONDUCTORS
475
given by the measurements of the latter effect which have been made, we can evaluate h from and zv A number of observers have found values of for the thermionic emission from platinum <
<
in the neighbourhood of 8*32 h
=
(
/z/
x 10~ 12
= 8-07
Whence
erg.
x 10- 27 erg
sec.
Thus this method of evaluating h leads to numbers which are about as much above the radiation value as the others are below
From
the experimental values of z/ the corresponding values can be calculated. The differences of for different metals are found to agree fairly closely with e times the corresponding
it.
of
<
<j6
contact differences of potential. This is in agreement with the theory of contact electromotive force which has already been given.
The
linear relation
between
Em
and
v
has been verified inde-
pendently by A. LI. Hughes* whose values of km exhibit a failconcordance with most of those given in the preceding table. The first measurements of the energy of the electrons emitted under the influence of ultra-violet light were made by Lenardf.
These results must be considered as
confirmatory of
validity of Planck's radiation formula. *
Phil. Trans. A, vol. ccxn. p. 205 (1912). t Ann. der Physik, vol. vm. p. 169 (1902).
the
CHAPTER XIX TYPES OF RADIATION IN recent years the number of different kinds of radiation with which we have become familiar has been greatly extended. This especially the case if the expression radiation is made bo include any invisible form of energy which originates at a material source
is
capable of travelling through empty space at a very high We shall adopt this extension of the term for the present, speed. as it is convenient for the immediate purpose of our discussion.
and
is
To
classify the different radiations it is desirable to place
in the following groups
them
:
(A)
Material and electrically charged.
(B)
Material and uncharged.
(C)
Aetherial.
To determine whether a given radiation
falls in
group (A) or
The rays are bundles usually a comparatively simple matter. of electrified particles in rapid motion. therefore behave will They
not
is
like
an
electric current flowing in
a flexible conductor and thus be
deflected in a magnetic field. The direction of the deflexion enables the sign of the charge which the particles carry to be determined.
Another method
is
to receive a
beam
of the radiation in an elec-
trically shielded and insulated conducting cylinder.
The
cylinder
then charge up with electricity of the same sign as that carried the This method, by particles which constitute the radiation. more is often less of direct, though easy practical application than
will
the method which makes use of the magnetic deflexion.
To
distinguish between groups (B) and (C) may be extremely In the case of radiations whose frequencies lie within
difficult.
TYPES OF RADIATION
477
or close to the range covered by the visible spectrum the occurrence of interference, diffraction, refraction and dispersion is taken to indicate that they are aetherial ; since it is only on the theory of waves propagated in a continuous medium that the phenomena
enumerated have received an adequate explanation. On the other hand if the frequency of the waves were very great it might be extremely
difficult, if
not impossible, to detect these effects
;
so
that this criterion would not necessarily be available.
Rather recently W. H. Bragg* has suggested a different method of distinguishing between aetherial and uncharged material rays. The distinction depends upon the geometrical distribution of the secondary rays which arise when the primary rays fall on matter. Recent investigations have shown that these secondary rays may differ greatly in character according to circumstances.
Generally
speaking, their nature is determined partly by that of the primary radiation and partly by that of the matter on which it impinges.
happens that the impact of a simple primary radiation on a chemically simple substance will cause the simultaneous emission The different types of more than one type of secondary radiation. of radiation which may thus arise will be considered more fully It often
below.
For the present
it is
only necessary to realize that the
secondary rays in general consist partly of negatively charged particles (electronic or ft type) and partly of rays which are similar in their properties to
Now
Roentgen rays (X
type).
/3 secondary rays which occurs primary rays impinges normally on a slab of
consider the emission of
when a beam
of
X
absorbing matter of indefinite thinness. The number of j8 rays emitted from the side of the slab on which the primary rays are incident would be expected to be smaller than the number emitted
X
side, if the rays are uncharged material particles. electrons will be either those which emitted the in this case, For, were originally present in the slab and which are knocked out of of those it by the moving uncharged particles or they will consist of the moving uncharged material particles which have lost a
on the emergent
;
collision
with the atoms of the
positively charged constituent by In either event the average value of the slab.
component of
the normal to the surface velocity of the emitted electrons along will not be zero and will be in the direction of the incident radiation. *
Nature, vol. LXXVII. p. 270, Jan. 23 (1908)
;
Phil.
Mag.
vol. xvi. p.
918 (1908).
will therefore be a greater number of /3 rays emitted from the emergent than from the incident side of the slab,
There
At found
sight it appears that this difference ought not to be the incident radiation consists of aetherial pulses similar
first if
to those which, as
a charged
body
is
we have seen accelerated.
in
Chap, xn, are emitted when
Under
natural view of the emission would
these circumstances the
seem
to
be that the
electrons
slab receive an impulse from the electric intensity present in the the electric intensity is always normal in the passing pulse. to the direction of propagation of the pulse and thus lies in the
Now
From this point of view there is nothing to plane of the slab. favour one side of the slab rather than the other, so that the emission of electrons should be the
same on the
incident as on
the emergent side.
The distribution of the secondary /? emission produced by the and "7 Roentgen rays has been examined experimentally by Bragg and others. In every case it has been found that there, is a larger emission from the emergent side of a thin plate than from the incident side.
These results have led Bragg to maintain that
the 7 and Roentgen rays consist of uncharged material particles and are not "aetherial" pulses. It was shown first of all by 0. Stnhlrnann *, and about
the
same time, independently, by symmetry occurs in the when illuminated by the plates
R. D. Kleernanf, that a similar lack of
emission of electrons from thin ultra-violet light from criterion leads to the
the
arc.
view that
It
thus appears that Bragg's well as
light, as
the radiations
previously enumerated, consists of neutral material particles.
Under the circumstances position.
One
it
is
desirable
of the chief difficulties
lies
in
to
reconsider the
the fact that we
have no adequate theory of the mechanism of the absorption of light and similar radiations leading to the emission of electrons.
The view which imagines the kinetic energy of the electrons to be derived from the work done by the electric intensity in the pulse, as it passes over
which are
far smaller
them, leads to values of the kinetic energy than those observed experimentally. Thus
the occurrence of asymmetrical emission *
Nature,
May 12
(1010); Phil.
May.
vol. xx. p.
is
not the only difficulty
331 (1910), vol. xxn.
(1911).
t Nature,
May
19 (1910); Hoy. Hoc. Proc. A, vol. LXXXIV. (1910).
p.
854
TYPES OF RADIATION
479
experienced by this simple theory. In this connection it is worth while to point out that the simple theory does lead to a slight excess of emergent emission owing to the deflexion of the electrons caused
moving
by the magnetic
force in the aetherial This pulse. deflexion, however, is far too small to produce the observed effects if the force in the is to be distributed
pulse supposed uniformly, On the other hand the application of thermodynamic and statistical principles to the study of thermal radiation and photoelectric action led us to the view that when radiant
in the usual way.
energy causes the disruption of an electron from a material system, the electron acquires an amount hv of 27 energy, where h = 6*55 x lO"
and v is the frequency of the radiation. This value is in accord with experiment, although for good any moderate value of v the amount of energy hv is much greater than that which we erg
sec.
should expect the electron to acquire, on the simple view discussed above, from the direct action of the It seems pulses. fairly clear either that the (electromagnetic) constitution which
we have assumed of the
process
for the radiation is at fault or the
of absorption
is
different
mechanism
from what we have
supposed.
In view of the
latter possibility it is very desirable to see if
we cannot find out anything about the magnitude of and Stuhlmann effects to be expected from aetherial
the Bragg radiations,
without making any definite assumption about the way in which the radiation is absorbed, but keeping to the value hv of the energy acquired by the disrupted electrons, which, as we have seen, is confirmed by experiments on photoelectric action and by the theory of heat radiation. Consider again the case of aetherial radiation incident normally on a thin slab of absorbing material.
In general, absorption may occur through the operation of processes of very different nature, for example, conduction as opposed to accumulation by relatively fixed and stable arrangements but we shall suppose that the only type of absorption which we need to ;
that which results finally in the disruption We shall fix our attention on of electrons from material systems.
consider in our slab
is
the state of things which exists after the slab has been illuminated for a sufficiently long time so that there is no further accumulation, in the slab, of energy abstracted from the incident beam.
Under
these circumstances the energy absorbed from the incident radiation
will appear, at
any rate in the
first
instance, as the kinetic energy
hv which the disrupted electrons possess at the instant of disruption. of them disrupted in unit time the Thus if there are energy
N
absorbed from the radiation is Nhv. It is important to observe that the energy of the electrons which we have to consider here is
that which they possess before, not after, they are emitted from
the
slab.
not the only physical quantity of which the We have seen in Chap, x, p. 211, suffers depletion.
But energy
is
beam when a material system similar to that under consideration absorbs an amount E of aetherial energy an amount of momentum incident
that
case this momentum E/c disappears from the aether. In the present and the electrons it contains, the slab to communicated must be Now and reaction. of action law in order to satisfy the according to the electron theory the action of the radiation so that this momentum is communicated to instance,
and such of
matter only reaches nature of collisions.
it
is
on the electrons;
them
in the
first
by the slab of indirectly through dynamical actions of the shall therefore assume that the electrons
it
as is ultimately received
We
momentum
as well as energy from the incident radiation receive previous to the occurrence of disruption ; although we do not know the precise nature of any process which will communicate an appreciable
amount
either of
momentum
aetherial disturbance to them.
or energy from a periodic
Now consider
the accumulation, in
any small interval of time, of momentum by the electrons in the slab. The increase of momentum of the slab and contained electrons is
due
to (1) the
momentum
of electrons which
come
into the
momentum, reckoned negatively, of the disrupted and electrons, (3) the momentum accumulated during the interval electrons the by present in it. Since the state of the electrons system, (2) the
instantaneously present in the slab is steady, by hypothesis, it follows that the difference of (1) and (2) is equal to (3). When (1) is zero, (3) is the momentum derived from the radiation, It follows that the
momentum which
from the radiation,
is
acquired by all the absorbing electrons exactly equal to the momentum of the disis
rupted electrons at the moment of disruption. But since the energy absorbed is Nhv the value of the former amount of momentum is If u
the average component of velocity of the disrupted electrons, in the direction of incidence of the radiation, an Nhv/c.
is
TY.P.ES
OF RADIATION
momentum
alternative expression for the
481 of the
N
electrons
--A^-Z me if v
2
is
Thus
Nmii.
is
^
2c
"
the average value of the square of their velocity at the
instant of disruption. i
i
As (V)~ approaches
c,
u approaches \ (v2 )
,
so that for those
radiations which give rise to the emission of very high speed /3 rays there will be a very marked preponderance in the emergent direction.
In the case of ultra-violet
light,
f
15 2 equal to about 10 the ratio of u to (v in this region there is a frequency i/ ,
however, for which v
only about 1 500. such that if v is <
is
is
But
:
z>
no
electrons are able to escape so that the number emitted is very sensitive to very small changes in the energy communicated to the ;
electrons.
fraction
u
:
Thus a very small difference in the value 2 (v )^ may make a very considerable difference
emission of
emergent It
/3
of the in the
secondary radiations as between the incident and
directions.
thus appears that the results obtained by Bragg and for without supposing the primary to be of a material nature. There
Stuhlmann may be accounted radiations which exhibit them
does not therefore appear to be any simple criterion which will and neutral invariably enable us to distinguish between aetherial material
radiations.
In fact there does not seem to be any
radiations which convincing evidence of the existence of any it seems a as that so to working hypothesis belong group (B),
most reasonable to
all those radiations classify as aetherial
which
do not belong to group (A). This conclusion is fully substantiated which have shown that X rays by recent experimental discoveries
under suitable conditions can exhibit the phenomena of diffraction and interference (cf. p. 507).
currently employed in describing based on any definite confusing, as it is not
The nomenclature which these radiations
system.
Thus
is
very
reflexion,
is
in different contexts radiations
may be
differently
mode
of origin or according to named, either according to their what is believed to be their nature, or according to the effects v,vhich
often happens that radiations, have every reason for believing to be identical in nature,
they produce.
which we
In this way
it
have entirely different names depending on more or less fortuitous Thus a high speed negative electron is called circumstances. it if a /3 ray originates in a radioactive substance, a cathode or Lenard ray if it is produced in a vacuum tube, a secondary
Roentgen ray if it is produced by the impact of the Roentgen on. As it is not our purpose to rays on a solid obstacle, and so describe the properties of all the different kinds of radiation in detail we shall extend the scope of the term /3 ray so as to cover
any negatively charged material ray. In the same way we should call a positively charged material ray an a ray, although this term also is usually applied only to those rays which originate with radioactive substances.
The uncharged, and presumably
X rays. We
radiations will be referred to as
aetherial,
shall use the
term
when they originate from radioactive substances 7 and Roentgen rays when they are produced in vacuum tubes. rays for these
For a detailed account of the properties of these radiations the reader maybe referred to the following authorities J. J. Thomson, Conduction of Electricity through Gases, Chaps. XI, XII, XIX and xx; :
Rutherford, Radioactivity, passim.
The
difference
between a and
/3
rays
is
not merely one of sign.
cases the specific charge (efni) has been found to be of a different order of magnitude. For the /3 rays e/m always has the to value which electrons, whereas for a rays e/m is corresponds large
In
all
always of the same order as the corresponding quantity in electroAs all the evidence points to the charge e being either lysis. equal to, or a small multiple of, the elementary electronic charge in all these cases, it follows that the a rays consist of
atoms or
molecules which have lost one or more negative electrons. Their not do furnish we have succeeded evidence that properties any in isolating any fundamental positively charged electrical atom which would correspond to the negative electron, unless it be the
hydrogen atom which has
The most convenient
lost a
negative electron.
test for the presence of the various types
of radiation under consideration
is that furnished by the production of electrical decomposition (ionization). Thus the passage of the rays through an insulating gas imparts to it the property of electrical conductivity. The amount of this ionization is often
taken to be a quantitative measure of the energy absorbed from the rays. In the case of the X radiations this has proved to be
TYPES OF RADIATION
483
the only method of estimating the energy which is available in rare cases, although with the a and /3 radiations more except direct methods can usually be The other tests for employed. radiations, in addition to the production of ionization, such as the
excitation of fluorescence, photo-chemical action,
etc., are probably secondary and depend on electrical decomposition, or at any rate disruption, in the first instance.
The Aether Pulse Theory of
the
Roentgen Rays.
The suggestion that the Roentgen rays are pulses in the aether, which are produced when the cathode rays are stopped by the walls of the tube or other material obstacle, was first made Wiechert* and Sir G. G. Stokes -f.
A
view, which is really are transverse aetherial vibrathat these to this, rays equivalent had short of tions previously been put wave-length, exceedingly
by
E.
forward by Schuster J and others. The consequences of the aether pulse theory have been worked out by various physicists, including
Thomson
J. J.
We
,
Abraham and Sommerfeld^j. ||
have already seen in Chap. XII that when an
charged body
or electron is accelerated or retarded,
electrically
an
electro-
with the velocity magnetic pulse spreads out in all directions In accordance with equation (11) of Chap. XII the total of light.
of energy in the whole pulse remains constant as it spreads outwards, so that the amount falling on unit area varies inversely
amount
We
also saw that as the square of the distance from the source. are intensities and at any point of the pulse the electric magnetic
are also at right angles equal and mutually perpendicular. They The thickness of a pulse proto the direction of propagation. duced by stopping a particle which moves with a given velocity the acceleration, whilst, at the same time, is greater the smaller the pulse is less. On this view the main in the energy present radiation or white the between difference rays and thermal
X
*
Ann. der Phys.
vol. LIX. p.
321 (1896).
t Nature, p. 427, Sept. 3 (1896);
Mem. Manchester
Lit.
and Phil Soc.
vol. XLI.
(1896).
Nature, p. 268, Jan. 23 (1896). Phil. ||
r
Mag.
vol. XLV. (1898).
Theorie der Elektrizitiit, n. Chap, n., Leipzig (1905). as** in n*i v /?,*. WKn "Rfi.iif.ri.Kfih. Akad.. Math. Phvs. KL, Jahrg. 1911, p.
1.
the fact that the X rays originate from particles light is due to which move with much greater speeds. These high speeds enable the electrons to penetrate right into the interior of the atoms,
where they are subject to fields of force much more intense than Thus in those to which the slow moving electrons are exposed. and intense also more the X ray pulses the forces are both change
more sharply than those in the light waves. cases,
Since, in
the aetherial disturbances are quite irregular, the
X
both rays,
on this view, evidently correspond to light waves of high average frequency.
X
between rays and light are at once The absence of diffraction these considerations. by and interference under ordinary circumstances is due to the extreme
The
salient differences
accounted
for
shortness of the equivalent wave-lengths. The absence of refraction is probably to be attributed to the very great difference between the equivalent frequency of the rays and the natural frequency of the bodies traversed
although the fact that the equivalent
;
wave-length' appears to be comparable with atomic dimensions may also be an important factor. Either of these causes would
probably be competent to explain the absence of refraction, so that there seems to be no difficulty here (c, however, p. 507).
There
is
one peculiarity of the aether pulse theory which
By means
deserves further consideration.
and the expressions Chap. XII in
we can
different
of Pointing's theorem and magnetic vectors found in
for the electric
write clown the density of the stream of energy the accelerated electron. In the
directions from
F is in the same straight the v(= /3c), energy radiated across unit area at a point distant r from the accelerated particle, where r makes an angle X with the direction of v, is
simplest case*, line as
when the
acceleration
the velocity
167T2 cV 2 (1
In this expression the instant
=
or
TT,
............... {(9)}
'
5
t
=
/3 are the values of those quantities at radiation left the accelerated particle, and r/c previous to that at which it reaches the
point under consideration.
\
cos X)'
T and
when the
therefore at a time
-
so that there Cf.
We is
no
see from (2) that E' = when radiation along the axis of
Sommerfeld,
X
loc. cit.
TYPES OF RADIATION motion,
If
that the
maximum
maximum
small the
/3 is
radiation
no longer the case when
is
of
E
485 f
is
when X =
in the equatorial plane.
7r/2
so
5
This
is
becomes comparable with unity, i.e. when the velocity of the electron becomes comparable with that f of light. In fact as approaches 1 the value of X for which E is
in
a
/3
approaches zero, so that the resulting X rays all lie infinitely narrow hollow cone about the original axis of
maximum
an
motion.
Some
X
(or 7) Gray* on the are /3 rays stopped by matter, that the intensity of the resulting
recent experiments by
A.
J.
rays which are produced when
show
X
in a convincing
radiation
is
manner
much
@
direction of the
greater in the forward than in the backward Similar phenomena in the production rays.
X
rays by the stoppage of cathode rays had previously been observed by Kayef. Gray concludes that the asymmetry which
of
is too large to be accounted for of the rays. theory production of
he has observed
by the aether pulse
X
The Scattering of
A
X
Rays.
X rays
by matter has been is traversed by an given by Sir J. J. Thomson J. will be accelerated by the aether pulse, the electrons in the former electric intensity in the latter. Consider the case in which matter simple theory of the scattering of
When
matter
traversed by a plane pulse in which the electric intensity is direction of imagined to be parallel to the axis of x. Let the is
"
"
be along the axis of z, and propagation of this primary pulse denote the value of the electric intensity in it. When the let will be subject to primary pulse passes over an electron the latter
X
an acceleration F
=
m
.
If
we suppose the
initial velocity of
the
electron to be negligible compared with that communicated to
we can put
sin
by the pulse, Thus the energy radiated by primary pulse
is
passing over
in equation (11) of Chap. xn. the electron in time dt whilst the
it is
dt
This result will only be true provided *
Roy. Soc. Proc.
it
e= e =
vol. LXXXVI. p.
Xe
,QX
is
large
compared with
513 (1912).
t Canib. Phil. Proc. vol. xv. p. 269 (1909).
t C.nn&iMtian of Electricity throuah G-ases, Second Edition, p. 321.
by the resulting displacement of the The limitations electron from its original position of equilibrium. below. The thus introduced will be considered energy radiated the forces called into play
by the electron during the complete passage of the pulse evidently
is
'
where the integrals are extended over the time of passage.
It is
of very penetrating radiations, probable that, except in the case 2 if we neglect /3 error led into serious we shall not be compared
with unity.
Thus
W
d
X*dt
= ^ (X* 4- .H"2 ) = X*
is the energy in unit volume of the is the number of If is its thickness. and d primary pulse if and the primary pulse electrons in unit volume of the matter, which is lost by 8 only loses energy in this way, the energy
where
JE
N
W
the primary pulse in travelling a distance &z
Thus the
relation
and the distance
if
W
Q
is
On
is
given by
W
between the energy of unit area of the pulse matter is
z traversed in the
the value of
W when z =
the theory that
0.
we
are considering, the thickness of a primary pulse clearly equal to the distance traversed by an in the time during which the acceledisturbance electromagnetic ration of the emitting electron is appreciable. This time will be is
determined by two factors
(1) the velocity of the moving electron relative to the retarding atom, and (2) the geometrical distribution of the field of force, inside the atom, which the accelera:
produces
tion.
The
velocity of the moving electrons is always large has the value c of the radiation velocity as an upper limit.
linear dimensions of the fields of force will
and
The
depend upon the
constitution of the atom, but will be comparable with the distance between the electrons which are present in the latter. As we
TYPES OF RADIATION shall see later, there are
good reasons
for
487
believing that the total
number of electrons present in the atom is
nearly proportional to and comparable with the atomic weight so that the distance between them will be comparable with, but somewhat smaller than, the ;
We should therefore expect that the very penetrating pulses which are produced by the stoppage of very rapidly moving electrons by heavy atoms would have a thickness of the order of 10~9 cm. If the stoppage (or acceleration) is
diameter of an atom.
produced by light atoms the thickness of the pulses might be ten times as great with electrons moving at the same speed.
With slow moving
electrons the pulses in each case will be correspondingly thicker, since, other factors being equal, the thickness varies inversely as the speed of the electron. These conclusions
would not be materially affected
if the pulses were supposed to the of from acceleration electrons in the anticathode by originate the field of force of the moving electrons passing near them, instead
of arising, as we have hitherto assumed, from the acceleration of the moving electrons themselves.
X
When
radiations fall on light atoms there very penetrating believe that the pulses are so thin and the to reason every electric intensity in them so large that the forces called into play by the resulting displacement of the constitutive electrons of the is
matter are relatively small. We should therefore expect the theory which leads to equation (7) to hold in such cases. Under the *
same circumstances the thickness of the secondary pulses will be almost the same as that of the primary pulses. For the thickness of the secondary pulse is equal to the distance travelled by radiation during the time taken by the primary pulse to pass over an electron. But the diameter of an electron is only about-
10~13 cm. and is therefore practically negligible compared with the thickness even of a very penetrating pulse. It follows that the thicknesses of both the primary and secondary pulses are the same under these conditions and we should therefore expect them to In confirmation of this result exhibit very similar properties. Barkla has found that the
"
scattered
"
secondary Roentgen rays, which are produced when penetrating primary Roentgen rays pass over substances made up of light atoms, have the same absorption nature of primary rays, whatever the chemical the matter in which they originate.
coefficient as the
For such
rays, according
to equation
m
Since
the coefficient of
(7),
and
c
are independent of
e, absorption is Ne*/67rm*c\ the nature of the matter, the coefficient of absorption for different made up only of light atoms should be protypes of matter the number of electrons in unit volume. In the portional to N, case of air Barkla found that the energy, measured by the ionization produced, of the scattered radiation from 1 cubic centimetre is equal to 0'00025 of that of the primary radiation passing through it. Thus, for this substance,
Since c
=3
x 10 10
But
V^ = 4-81 Ne = 2 VTTC x
e/2 ,
n
if
the
is
x 10~ 10
,
eft
vWc =
1-77 x 10 7
number
of molecules in 1
c.c.
of
the results
air,
ne = 2 VVc
Whence number
about half the molecular weight.
Since the absorp-
x O4327. of electrolytic experiments show that of divided electrons number the Thus 14. by the JV/7i
=
of molecules
is
and
5'95.
X
tion of penetrating rays by light atoms depends only on the and not on its chemical nature, we traversed of matter quantity
are led to the further conclusion that the atomic weights of the different elements are proportional to the number of electrons their
atoms contain.
that the
elements
Combining the two
inferences,
it
follows
number is
conclusion
of electrons present in the atoms of different a common submultiple of their atomic weights. This
we
shall find to
be supported by other lines of reasoning
490496).
(see pp.
Polarization of
X
Rays.
X
use of the properties of the scattered radiation show that Roentgen rays possess features analogous to the polarization exhibited by light and other electromagnetic
By making
it is
possible to
The existence of this polarization was first demonstrated Barkla* and his results have since been confirmed by a number by waves.
of other follows
:
The principle of these experiments is as be the direction of the cathode stream in the
observers.
Let
AB
tube in which the Roentgen rays originate. The cathode rays are stopped by the anticathode at -B. Let the Roentgen rays which travel in the direction *
BC, which
is
normal to AJB, pass through
Phil. Trans. A, vol. cciv. p. 467 (1905^.
TYPES OF RADIATION matter consisting of light atoms at
489
C and
thus give rise to rays. probable that the cathode rays are not stopped by a single impact at so that their motion will become irregular before they cease to cause the emission of "
scattered
"
X
Now
it is
B
We
should, however, expect the most penetrating Roentgen rays. rays to arise when the cathode rays are moving at their greatest speed, that is to say, before they have been deflected by many
encounters.
The main
features of the penetrating rays will there-
be determined by the properties of pulses emitted during the acceleration of electrons moving in the direction AB. Let GE be a line parallel to AB in the plane of ABG and let CD be perfore
pendicular to EC,
OB
and BA.
The
acceleration at
B
of an
AB
electron whose velocity is along will give rise to an element of pulse travelling along in which the electric intensity is and the magnetic intensity is parallel to CD. If parallel to
BC
GE
this pulse falls
on an electron at
acceleration along GE, initially,
axis
CE
through
If
(7,
the latter will be subject to an the electron to be at rest
we assume
the secondary pulse will have zero intensity along the and its maximum intensity in the equatorial plane
C which
stopped by the
contains
first
CD
and BC.
If the cathode ra}-s were
B the secondary rays would have axis CE and a maximum intensity
encounter at
zero intensity along the polar at points in the equatorial plane through C.
If the primary tube
AB
became parallel to CD, the secondary were rotated so that would rays would fall to zero in the direction of CD, and become one of the directions of maximum intensity. Since the
CE
electrons are believed to be only partially stopped by a single collision the intensity of the secondary rays will never drop to zero. shall only be able to observe a minimum value in the directions with the maximum value in the perpenindicated as
We
compared
dicular plane.
c
\B
A
E Fig. 51.
observations alluded to are in satisfactory accordance with fV^vpw in ot. mi t,li npri The maximum difference of intensity
The
in the directions
CE
and
CD
observed by Barkla
is
about 20
/o
.
The
occurrence of polarization phenomena is not easy to account for on the material theory of rays, although such a position has.
X
been defended by Bragg*. The Scattering of a and
/3
Rays by Matter.
When seems
these very rapidly moving particles traverse matter it that they pass through the interior of the fairly certain
atoms and not simply through the spaces between them. For instance, it is found that when a pencil of a rays is made to pass through sufficiently thin sheets of solid substances only a small fraction of them are either deflected through appreciable angles or stopped, whereas a calculation based on the fairly concordant values of the dimensions of molecules given by various
methods shows that the chance of a particle passing through such a sheet without colliding with a molecule is excessively slight, In
fact,
particles
we have
since Rutherford's experiments
have shown that the a
from radioactive substances are charged atoms of helium, every reason to expect that they are of atomic dimensions
it is probable that they would be geometrically unable to pass through a sheet of solid of the kind contemplated without intersecting the atoms of the latter.
themselves, and
Admitting that the moving charged particles pass through the many cases without sensible deflexion, it
interior of the atoms, in
follows that such deflexions as occur
must
arise
from encounters
with systems forming part of the atom rather than with the atom as a whole. We should therefore expect that a study of the of scattering pencils of these particles which results from their transmission through matter would afford valuable indications toward a knowledge of the structure of the, atoms of the latter; it is possible that the effects to be expected might not be very dissimilar even if widely different views of the structure of the atoms were The problem has been considered adopted.
although
theoretically
by both
Thomson f and
Rutherford.!,
who
have
adopted rather divergent views both as to atomic structure and as to the nature of the deflexions. Let us consider Thomson's treat-
ment
first.
*
Phil.
Mag.
vol. xiv. p.
429 (1907).
t Canib. Phil. Proc. vol. xv. p. 405 (1910). Phil. Mag. vol. xxi. p. 669 (1911).
TYPES OF RADIATION
Thomson
491
atoms through which the charged be made up of Q negative electrons accompanied by an equal quantity of positive electricity. The deflexion of a moving negative electron, for example, will then arise from two considers the
N
particles pass to
causes
(1) the repulsion of the electrons distributed through the atom, and (2) the attraction of the positive electricity. In regard to (2) two cases are considered: (1) the positive electricity is :
distributed with uniform
volume density throughout a sphere equal to that of the atom (see Chap, xxi), and divided into separate equal units, as to which it is con-
whose volume (2) it is
is
much
sidered probable that they occupy a
larger
volume than that
occupied by an electron.
The
deflexion of an individual
due
electron
to its passage
through a single atom will depend on the manner in which the atom is approached, but the average value of this deflexion taken over a large number of encounters will be a definite quantity 0. If we consider a sufficiently large number of consecutive deflexions it is
probable that no serious error will be introduced
treated as equal to the mean value d. It that the direction of any individual deflexion all
and
its
magnitude
is
if
they are
be remembered a matter of chance to
considering the resolution of n of such deflexions 6 along any two perpen-
is
any large number
is
very small.
By
dicular axes in a plane at right angles to the original motion, it evident that the problem of finding the average effect of the
is
n deflexions
is
the same as that of finding the average amplitude n vibrations, the amplitude of each of which
of the resultant of
and the phase of each of which is entirely fortuitous. This problem has been solved by Lord Rayleigh* who has shown that
is
the average resultant amplitude
is
Vn0
(8).
It follows that if the rays pass through a plate of thickness t, atoms, of radius 6, per unit volume, the average containing 2 The problem is deflexion which they experience is 8 \/iW& t
N
thus reduced to the determination of the value of
The
part 6 1 of 8 which arises negative electrons can be found by
moving under
6.
from deflexions caused by the of the theory of particles
means
central forces which vary inversely as the square of
* Thfnrti of Sound. Second Edition, vol.
i.
p. 35.
When
the distance*.
the velocity
V
of the
/3
particles
is
so
large
that the resulting deflexion is small, the deflexion produced by a 2e2 1 - where x is the is perpendicular distance single encounter =^ ,
between the electron encountered and the undetected path. The all the electrons which lie within a average value of this for 4e~
distance a of the line of motion
is
y^
1
-.
If
I
is
the average
the moving particle which lies inside the length of the path of are uniformly distributed in the atom and if atom, if the electrons there are v of collisions
If
them per unit volume
with electrons which
we suppose
for the
moment
lie
of the latter, the
number
within a distance a
is
of
vn-aH.
that the negative electrons are alone cr of encounters
number operative and consider any very large be deviation would total the with atoms average 408
-
~~
mV
2
mV
i?:> -
a
Since this must be Vcr times the effect of a single encounter the effect of the latter is
16
substituting VZ
= $V2&,
where
e
b is the radius of
an atom.
the average deflexion due to the sphere* of positive electrification the theory of central forces shows that this is If
c/>!
is
given by
provided fa
is
mean
16
^= T where r
is
When
small.
definite units the
^ e
2
the positive electricity is made up of due to these is given by ^> 2
deflexion 1
V/3#; TV/""/ "I X
6
TT
1
~8
volume occupied by the positive volume of the atom.
the ratio of the
electricity to the
Thus, according as we adopt the
first
or second hypothesis as
to the geometrical configuration of the positive electricity, the value of 6 will be given by the equations or *
-
Routh, Dynamics of a Particle, Cambridge, 1898, Chap.
vi.
TYPES OF RADIATION
493
and the average deflexion i)r m in passing through a thin plate of thickness t will be given by either
(384 -
or
As the only unknown
quantities on the right-hand sides of (13) the equations are supported by experiment they ought to enable us to determine the value of the number of electrons in the atom.
and (14) are JV and
T, if
N
The extent to which these equations are in accordance with the observed scattering of /3 rays when they pass through thin sheets of various solids has been examined by Crowther*. If ty any particular angle of deflexion it follows from the theory of errors that the probability of a deflexion greater than i/r is
is
-Wm
2
e
is
-|
Thus from
.
will
(8) the thickness
t
for
which this probability
be given by 2
e-*W =i ........................ (15), i ^/ = 0V(clog2) ..................... (16),
or
where
c is
a constant
that the deflexion
is
any particular substance. The probability e~^cf& \ that is to less than ty is equal to 1 for
I-e-W where k
is
constant.
........................... (17),
constant for any particular substance,
From
(8)
fAo^W^x^bp and from either (13)
= const x
where the absorbing medium is
-^ is kept
.................. (18),
or (14) -tylt$
the incident rays
if
and (16)
is
=2
.................. (19),
kept the same and the velocity of
When ty is also kept constant m V*/t
varied.
of investigation adopted by Crowther was to the proportion of the incident @ rays whose deflexions
The method measure were
less
when they passed through abmaterials and when the thickness of
than a fixed value
sorbing sheets of different *
Row. Soc. Proc. A,
^
vol.
LXXXIV. p. 226 (1910).
In this way the sheets and the velocity of the rays were varied. confirm and to able he has been equations (16), (17) (20) using .aluminium sheets, and equation (1C) with platinum sheets
N
also.
be calculated from (13) and (14). The value of yfr/t^ enables Q In reducing (14) r is assumed equal to zero, but it is shown that to
If (13) the value assumed for r cannot affect the results vitally. be proportional to the atomic weight of is taken, Q is found to
W
N
and Pt, the value of
the elements C, Al, Cu,
Ag N^fW varying only between 2*87 and 3*32. This variation is comparable with the error of measurement. If (14) is taken, ranges from 3*7 for carbon to 33*5 for platinum. Since the scattering of rays by
N/W
X
matter leads to the conclusion that the number of electrons per atom is proportional to the atomic weight, the hypothesis underlying (14)
is
discarded and the experiments are taken to establish
the view that the positive electricity is uniformly distributed. Subject to this hypothesis the results show that the number of electrons present in
any atom
is
equal to three times the atomic
Alweight, within the limits of accurac}' of the measurements. this is than estimate that though considerably higher given by
the scattering of Roentgen rays by light atoms note that it is of the same order of magnitude.
in
it is
satisfactory to
Rutherford's treatment of the scattering differs from Thomson's He attributes the main features of
two important particulars.
the scattering of the particles when they pass through thin layers of matter to the effects of single encounters (single scattering) .and not to the chance combination of a
minute deflexions.
This
is
multitude of excessively the most radical difference. It is
*" arid others on the supported by the experiments of Geiger
produced by thin sheets of matter. These have shown that the proportion of a particles which experiments are scattered through -large angles is very much larger than that which would be expected as the cumulative effect of a multitude
.scattering of a rays
of small deflexions.
constitution of the
The other atom which
difference is
adopted.
N
is
in the hypothetical is imagined to
This
consist of a central point charge e surrounded by a sphere of uniformly distributed electrification equal altogether to + Q e.
N
*
Geiger and Marsden, Roy. Soc. Proc. A, Roy. Soc. Proc. A, vol. Lxxxm.p. 492 (1910).
vol. LXXXII. p.
495 (1909); Geiger,
TYPES OF RADIATION
495
The assumption
of a large central charge is shown to be required to account for the large individual deflexions.
The is
angle
probability of a single deflexion greater than a given clearly determined, making use of the theory of central
by the chance that the undeflected path of a particle, moving with given velocity, will pass within an assigned distance of the fixed centre. In this way it is shown that the fraction p of
forces,
the particles which are deflected through angles lying between $!
and
is ............ (21),
,
E
The
m the
,^.
Q
7
being the charge,
ticle.
e ~m V-
6=
where
mass and
........................ (22),
V
the velocity of the a par-
shown
to
be in satisfactory
measurements of the proportion of a
particles scattered
results of this theory are
accordance with (1) the
through large angles;
is
(2) the scattering of a particles through all angles, if allowance for the effects arising from the cumulative effects of small
made
deflexions considered
by Thomson
;
(3) the proportion of a particles scattered through large angles sheets of different elements, if the central charge is assumed to by be proportional to the atomic weight.
It appears that Rutherford's calculations, as well as Thomson's, and mV-jt^, for the of the fractions lead to the
^rjtf constancy conditions the under the of investigated by /3 rays .scattering Crowther. The values of the constants, however, are different in Q the two cases and so are the estimated values of Using
N
.
T
Crowther's data Rutherford calculates the following values for i\ the number of electrons which would have a total charge equal to ,
the hypothetical central charge
No =
^
:
for Al,
N
Q
= 22
:
= 138.
for
Cu,
N = 42
:
These values are roughly 78 for Ag, Platinum and gold in the proportion of the atomic weights. data relating suitable which for appear to be the only materials are availto the scattering of the a particles through large angles in substantial is Q able but in that case the resulting estimate of the /3 rays. agreement with that deduced from the behaviour of ;
and
for Pt,
N
;
the theory of single scattering the number of particles deflected through an angle greater than a given angle will be if the thickness t is small. Thus, equal to k't, where k' is a constant, instead of (17), the proportion J// of the rays for which the
On
deflexion is less than an assigned
amount
will
be given by
///=! -#*
(23),
Since Crowther found (17) to be provided confirmed by his experiments this test is in favour of Thomson's This divergence might be reconciled by supposing that theory. t
is
small enough.
the c.umulativo effect of the deflexions of
ft rays by electrons is deflexion of the more than the compound important relatively This seems reasonable, as a moving particle will come a rays. The moving within range of many more electrons than atoms.
electrons will be deviated in varying degree both by the electrons and the large central charge, whereas the a particles will only be .affected appreciably by the central charge, on account of the small
mass of the
electrons.
It is worth while pointing out that Rutherford's estimates of
NQ agree better with those of the number of electrons per atom given by the scattering of Roentgen rays than do the estimates of
N
Q
deduced from Thomson's theory. Secondary Rays.
The secondary rays which are emitted when Roentgen rays are absorbed by matter possess many interesting properties. The scattered radiation, which
is
similar to the primary radiation, has In general two additional types of
already been considered. secondary radiation are found to be emitted. of the "
X
type and
characteristic
"
One
of these
is
reasons which appear below, radiation; and the other is of the /3 type.
Let us consider
is
called,
for a
for
moment
the means which are available
recognizing and classifying what we may provisionally call pure radiations of the type. Suppose that we are dealing with
for
X
primary radiations whose penetrating power is comprised within the limits of that of the rays given off in quantity by an ordinary Roentgen ray bulb. When such rays are allowed to fall on matter found that characteristic radiations are not excited, provided the matter consists entirely of elements whose atomic weights are,
it is
TYPES OF RADIATION all
below 40.
497
If the atomic weights exceed 40, characteristic
radiations are found to be emitted.
In
all
of the occurrence of scattered radiation.
cases there
These
facts
is
evidence
would lead
us to conclude that absorption of the rays by light elements
is
a
complex phenomenon than absorption by matter containing the heavier atoms. Let us take as our criterion of a pure radiation
less
the condition that
its law of absorption, by matter consisting only of light elements, should be an exponential one. That is to say, if the incident i is the after i, traversing intensity intensity
normally a thickness x of the matter, will be i
where k
is
independent of
= i,e-
k:K
(24),
This law results, of course,
#.
if
successive equal thicknesses of the absorbing matter absorb equal The intensity i is fractions of the radiation entering them. taken to be measured by the ionization which the rays will
produce in a thin layer of a gas like air, which is composed solely This method is satisfactory of elements of low atomic weight. because the amount of ionization in such cases has been shown
be proportional to the loss of intensity due to absorption. Now suppose that we have a number of different pure radiations
to
and that we determine the value of k for each of them when they Let the are absorbed by sheets of some particular light element. Let the corresponding v etc. resulting values of k be X1? ^, l3
Then it values of k for some other light element be X2 p z i/2 etc. = etc. No such simple relation X etc. *2 is found that 2 ^ /^ ,
\
:
:
:
^
,
,
:
between the values of k is found to hold when absorption by materials with heavier atoms is considered, so that this result favours the comparative simplicity of the of the lighter atoms.
phenomena
in the case
we take any particular element of low atomic of the absorbability k will give us a means of value the weight As it is very readilyradiation. classifying any particular pure as the is aluminium thin in sheets, generally taken obtainable Barkla* used been has of by method This It is clear that if
standard.
and
investigation
his collaborators,
knowledge *
X
of the
Phil. Mag. Mag. vol. xi. p. 812 (1906) Barkla and Sadler, Radioakt. etc. 550 (1908) vol. xvn. p. 739 (1909); Barkla, Jahrb. der 246 Phil. Mag. vol. xxn. p. 396 (1911).
Barkla, Phil.
vol. xvi. p.
vol. in. p.
whose researches have greatly increased our We shall type of secondary Roentgen rays. ;
;
;
below that there is another criterion which may be used A, namely the maximum velocity of the /3 rays emitted
see
instead of
X
when the for
rays
fall
on metals.
This has been shown to have,
each pure radiation, a definite value which
the metal
(cf.
is
independent of
also p. 510).
When
the primary rays are such as are emitted by an ordinary Roentgen ray bulb the difference between the secondary rays emitted by aluminium (atomic weight 27) and an element like In the first place, the copper (atomic weight 63) is very striking. emission the of (scattered radiation) from secondary quantity
aluminium
is
comparatively insignificant.
secondary radiation from the copper two radiations differ also in quality.
is
The amount of the much larger. The
very
We
have seen that the
same penetrating power as the The copper radiation is much more absorb-
aluminium radiation has the primary radiation. able.
It
evidence radiation
is is
also
The current very nearly a pure radiation. almost entirely of a pure
to the effect that it consists
mixed with a trace
of scattered radiation.
The amount
radiation from copper, and other elements of atomic weight above 40, appears to be of the same order as that of the scattered
given out by aluminium.
This pure secondary radiation
is
that which
to as the characteristic secondary radiation.
we have
referred
It is characteristic
in the sense that its absorbability k has a value which is characIn the case of the teristic of the metal from which it is emitted.
elements whose atomic weights lie between about 40 and 100 the absorbability of the characteristic secondary rays is quite independent of the nature of the primary rays which are used to
X
With the elements of still higher atomic weight the phenomena are more complicated and it appears that these
excite them.
elements give off more than one kind of characteristic radiation. It is convenient to allude to the characteristic radiation from
M
a particular element as the M-X rays. Thus the characteristic from are called the Cu-X This notation is due rays copper rays. to Bragg.
The
absorbability of the characteristic rays diminishes, and their penetrating power increases, as the atomic weight of the
parent
-
element increases. rays in
one
The
characteristic rays differ from the scattered important particular in addition to those already
TYPES OF RADIATION
499
They show no trace of polarization. This leads to the view that the manner in which the primary rays cause the formation of the characteristic rays is less direct than that in
mentioned.
which the scattered rays are produced. considered to be established when we
This conclusion recall
may be
that the scattered
are similar in their properties to the primary rays and independent of the matter in which they originate whereas the characteristic rays have their properties determined solely by the
rays
:
matter of origin and not at
We
shall
now turn
all
by the character
to the absorption of pure
A
sheets of different elements.
Barkla and Sadler
is
30
of the primary rays.
series
X
radiations
of results obtained
by by
exhibited in the accompanying diagram
40
50
60
'0
CO
90
<00
:0
IcO
50
I4O
TYPES OF RADIATION
500
The
abscissae
are
to
the absorption
proportional (Fig, 52). coefficient of the various radiations in aluminium.
The
experi-
mental values of k/p in aluminium are indicated by the abscissae dotted lines alongside which corresponding to the various vertical are written the chemical symbols of the different elements by The ordicharacteristic radiations were emitted. which the
pure
nates are the values of k/p, where p is the density of the absorbing element, for the elements C, Mg, Fe, Ni, Cu, Zn, Ag, Sn, Pt and
In the case of the elements of atomic weight below 40, namely C and Mg, the relation between the absorbability of the radiations by the respective elements and that by aluminium is a
Au.
The same linear one, as has already been pointed out on p. 497. to be true in the case of Pt*, observed thing, however, will be and Sn as well. In fact such a linear relation between Au*,
Ag
the absorption coefficients for different pure radiations has been found to hold quite generally, provided the range of radiations tested neither includes that characteristic of the absorbing substance nor lies near the less absorbable side of
The curves all
they them, that
Cu and Zn
for Fe, Ni,
exhibit the
same
it.
are quite different. As we need only consider one of The absorbability by nickel of
features
for nickel for
example. the radiations from Cr, Fe, Co and Ni, which are either more absorbable than, or as absorbable as, that characteristic of nickel, is
same radiations in aluminium. The next most penetrating radiation, as measured on the aluminium
proportional to the absorption of the scale, is
the
Cu-X
radiation,
and
this
shows a small but definite
increase over the absorbability required by the law of proportionAs the absorbability in terms of aluminium of the characality. teristic
rays diminishes
further,
their
absorbability in
nickel
Thus the absorption coefficient of the increases very rapidly. Zn-X rays in nickel is several times that of the Cu-X rays, although the atomic weight of zinc (65) is only 2 units greater than that of copper. Further diminution in the aluminium absorbability of the
characteristic rays is accompanied by an diminution in the absorbability of the proportional approximately in nickel. Thus the curve rays again approximates to a straight line *
which has a different slope from the linear portion on the It
is
necessary to except the penetrating
Ag-X and Se-X rays
in
these
TYPEvS OF
more absorbable
side of the radiation
501
which
is
characteristic of
It will be noticed that in each case the radiation
nickel.
the
RADIATION
having
maximum
absorbability is that which is characteristic of the element next but one to the absorbing element when they are
arranged in the order of increasing atomic weight. Direct experiments have shown that the abnormal
absorption of radiations which are slightly more penetrating than the radiation characteristic of the substance tested, is the cause of the
emission of the characteristic radiation. characteristic radiation emitted
The amount
by an element
is
of the
found to be zero
until the exciting rays become at least as penetrating as the characteristic radiation as the penetration, measured in terms of ;
aluminium or some substance which behaves
similarly
to
alu-
increased beyond this critical value, the amount of characteristic radiation which the exciting radiation will cause to be emitted, compared with the ionization which it produces in air
minium,
is
or any other gas containing only light atoms, increases at first to a maximum value and then diminishes. In fact, the quantity of characteristic radiation emitted appears, for all exciting radiations, to be proportional to the excess of absorption over what would be given by the linear relation which holds for exciting radiations
that are more absorbable than the characteristic radiation. Since these results appear to be quite general, the occurrence of humps, like those shown in the relative absorbability curves for Zn, Cu, Ni and Fe in Fig. 52, affords a simple and delicate method of detecting the presence of unknown characteristic radiations.
Working
in this
way Barkla and Nicol* have succeeded
in
from each of separating two quite distinct characteristic radiations A comprethe elements silver, antimony, iodine and barium. hensive examination of all the radiations thus far discovered shows that,
when they are arranged
in the order of k/p, their comparative
absorbability in aluminium, they fall into
been called by Barkla the
K
and
L
two
series.
These have
series respectively.
In each
series the order of diminishing absorbability is that of increasing
atomic weight of the metal of origin. The data for the radiations which have thus far been isolated are given in the following table
:
*
Nature, Aug. 4 (1910).
OJJ
502
KAJD1AJL1UJN
Reasons have been given by Barkla
for
existence of series other than the series K,
been explored*.
It will
L
be noticed that the
believing in
the
which have already
fact that the charac-
only excited by radiations more penetrating than themselves is analogous to Stokes's law in optics according to which fluorescent light is invariably of lower frequency than the
teristic radiations are
:
which excites it. For this reason the characteristic radiations have sometimes been called fluorescent radiations. It is obvious
light
that each pure characteristic radiation has Phi?
Tl/V,
features which
are
TYPES OF KAD1ATION
503
closely analogous to those exhibited by a monochromatic emission of fluorescent light, and the sum of the radiations characteristic of
any given element may be regarded as a
X
the
fluorescent spectrum in
ray region,
The Relation between
We
have seen that
X
ft
ray and
rays
are
X
ray Emission.
emitted when high-speed
impinge on matter and also that when X rays are absorbed by. matter secondary radiation of the $ type is emitted. Eecent investigations have shown that this transformation of X electrons
ray energy and the inverse change obey a number of fairly simple laws which appear to be of a general character.
ray into
The
ft
any single ft ray is a quite a moving electron, its nature is determined when its speed and direction of motion are given. In general a beam of ft rays consists of a shower of electrons having specification of the nature of
simple matter.
different speeds
Since each ray
and
is
directions of motion.
By
the insertion of
suitable stops the range of the directions of motion can be restricted to any desired extent. By the application, of a magnetic field
the
this
way
for
ft
of spectrum. In rays can also be spread out into a sort homogeneous beams of ft rays can be obtained The ft rays which are given out by purposes.
sufficiently
experimental radioactive substances or which are emitted
when
X
rays
are
absorbed by matter are, as a rule, of a heterogeneous character. This heterogeneity arises in part from the fact that the rays at different depths in the matter and thus lose varying originate
amounts of energy before they emerge. In many cases it has been shown that the maximum energy of the ft rays which thus arise is a definite quantity which exhibits rather simple relationships. In the cases now under consideration it has not always been found possible to measure the energy of these rays by the comdirect methods involving stoppage by an electric or
paratively in the deflexion by a magnetic field, which have been employed bodies hot emitted electrons the of by case of the cathode rays and for this reason The ultra-violet light. bodies illuminated and
by
is
by
by
much affected either that the energy of the rays is too great to be the that rays are not the electric fields at our disposal or else
be available. concentrated enough for the magnetic method to
A
that they are able to travel in a gas like air before they cease This distance has to produce additional ionization of the gas.
been found to be sufficiently definite to be taken as a satisfactory index of the velocity of the quickest rays in any given group.
Using this show that the
criterion Sadler*
and Beattyf have been able
to
maximum
X
velocity of the /? rays emitted when a radiation is allowed to fall on different
given characteristic metals always has the same value. This maximum velocity is therefore characteristic of the X radiation and independent of the greater the greater the atomic weight of the metal of which the X rays are characteristic.
metal
He
;
it is
This question has been studied more fully by WhiddingtonJ. has shown by experiment that if the original velocity of a
group of /3 rays is v their velocity of matter is given by ,
v after traversing a thickness
V-w^ad where a
d
(25),
Since (25) to the velocity
a constant characteristic of the matter.
is
appears to hold with
fair
approximation down
(approximately v = 0) at which ionization ceases, it follows that the velocity v g of the fastest secondary rays is given by v*
= 2 x 10 maximum distance 40
where a
,
for air at
in
= ad
(26),
atmospheric pressure, and d
which ionization
the results of Sadler and Beatty in a this principle it appears that v8
where
k' is
a constant and
w
is
perceptible.
the
Analysing
manner which depends on
= k'w
is
is
(27),
the atomic weight of the radiator
of which the secondary rays are characteristic. This relation has been verified for the Fe-, Cu-, Zn-, As-, Sn-, Mo-, and Ag-X rays.
The value
We
X
of tf
is
8 very close to 10
(
4/
).
have seen that the characteristic
X
rays are only excited
Using an X ray tube with a silver anticathode Whiddington has investigated the relation between the velocity of the cathode rays in the primary
by
rays more penetrating than themselves.
*
Phil. Mag. vol. xix. p. 337 (1910). t Phil. Mag. vol. xx. p. 320 (1910). t Roy. Soc. Proc. A, vol. LXXXVI. pp. 360, 370 (1912). noiix Roy. SOC. PrOC. A. VOL T,YYYIT r> QOQ
TYPES OF RADIATION
505
tube and the amount of secondary characteristic radiation emitted when the resulting primary X rays are allowed to fall on different metals. He finds that no characteristic secondary rays are emitted until the cathode rays have a critical minimum The velocity vp emission begins quite sharply at this point and subsequently varies as the fourth power of the The velocity of the cathode .
value of vp
connected with the atomic weight radiator by the simple relation is
vp
The value
of k
w
rays*. of the secondary
= kiv
(28).
8 very close to 10 and, in limits of error of the experiments.
is
fact,
=
'
within the
Thus primary cathode rays of velocity just greater than vp = lew give rise to primary Roentgen rays, and these if absorbed by a metal of atomic weight w would give rise to secondary Roentgen rays, characteristic of that metal,
maximum
rays whose
which give v s =:k'iu
is
velocity
rise to secondary ft
= kw = vp
The maxi-
.
mum
velocity of the secondary /3 rays is equal to the minimum velocity of the primary cathode or ft rays which are able, indirectly, to excite the characteristic Roentgen rays which caused the emis-
sion of the secondary
ft
rays.
The formulae above only apply without
K
the case of both the
K and L series
modification to Barkla's
According to Whiddingtonf
series of characteristic radiations. is
covered by the more general
formulae (29),
(SO).
For the
K series -4 = 1
For the
L
series
and
A\ and
5 = 0. B = 25.
Thus the secondary cathode
rays, excited
by the
characteristic
L series from an element of atomic weight rays which belong to the maximum a have w, velocity equal to (^ It will be
-
10 8 cm. sec.-
1 .
25)
observed that there
is
a very close resemblance
ultra-violet
illumination.
light
* Gf. J.
J.
Thomson,
Phil.
X
ray and number of the cases both In
between the emission of electrons by matter under Mag.
vol. xrv. p.
217 (1907).
electrons emitted is proportional to the incident intensity, whereas, the maximum energy of the electrons is independent of this in-
These
tensity.
facts receive
a simple and obvious explanation on
X
rays and light consist of showers of material of energy which are localized in space and bundles of or particles distance from the source is increased. as the out do not spread
the view that the
the other hand in the case of light (and also X rays, see 507) it is difficult to account for the phenomena of interference
On p.
and refraction on such a view.
It
seems a
little safer, therefore, of the emitted electrons energy in some way represents a condition which determines the disrupTo agree tion of matter under the stimulus of a given radiation.
to suppose tentatively that the
with the results of the theory of black body radiation and of the experiments on the emission of electrons under the influence of light it is necessary that one part of this condition should be that the energy of the disrupted electrons is either equal to hv, where
h
= 6*55
or
is
this
x 10~ 27 erg
sec. and v is the frequency of the radiation, an integral multiple of this quantity. It also appears that condition must be of a very general character and necessarily
inherent in
all
types of matter.
We have seen that the maximum energy of the electrons emitted under the influence of illumination by light is. given by the equation T.m
where
W
Q
is
= vh~w
........................ (31),
Q
a constant characteristic of the type of matter by is absorbed. If this relation holds for the radia-
which the light
X type as well as light, the results given above enable us to determine the frequency of the characteristic radiations.
tions of the
X
In these cases
w
l}
is
negligible
mk'~
rtwf ~
Thus Itf
for the
,
m=
K
x IQr* and h
According to this estimate their frequency as great as that of visible light,
vh, so
that
(Aw
copper-X rays belonging to the
= 10 A = 1, w = 63, B = 0, 8
compared with
is
series,
putting
- 6'55
x 10-*
about 2000 times
TYPES OF RADIATION
X
Rays and
507
Crystals.
X
Since the foregoing account of the rays was written, an enormous advance in our knowledge of their properties has taken and place owing to a discovery made by Friedrich,
Snipping These experimenters allowed a narrow beam of X rays to traverse a thin plate cut from a crystal of zinc-blende and then to fall on a photographic When the plate was plate. Laue*.
developed, instead of a single black spot being obtained as in the absence of the crystal, it was found that the central circular spot was surrounded by a regular pattern of circular and elliptical spots.
This pattern varied with the direction in which the crystal plate cut, and its symmetry was determined by the relation of the
was
direction of the primary
beam
to the directions of the crystal axes.
Laue, who appears to have suggested the experiment, explained the phenomena somewhat as follows The crystal is to be regarded :
as a geometrically regular arrangement of points (atoms) in space, each of which when traversed by the primary beam of rays
X
becomes a source of secondary waves. Owing to the regular of these the waves will only reinforce arrangement secondary points one another in certain directions. These directions determine the
The position of the secondary spots on the photographic plate. theoretical investigation has been extended by Debyef so as to take account of the influence exerted by the heat motions of the Considering the case in which a system of atoms is arranged in cubical order so that the atoms lie on lines parallel to atoms.
x, y and z at distance a apart, let a parallel beam of The of wave-length X travel along the axis of x. intensity of the radiation at a distant point so, y, z due to a is small parallelepiped of such a substance at temperature
the axes of radiation
T
proportional to
1-1
rin* JT.
^ _1
(
_j
Tra /, 1 sin 2 ( .
X
\
sin*
Ns
i)
^*
sin- JT.
^^
^
--
\
rj
sm-tray^ .
2
X r
sin
2
e
_^^fi^'
7ra 5 X-
r .(33),
where r =
(a? +-
-if
R
-f 2-)^,
is
the molecular gas constant, f
restoring force for unit displacement of the atoms, are the
number
of atoms which
along the
and
is
the
N KN l
2
ll
z edges, respecThe the are of determined by spots bright tively, parallelepiped. the simultaneous vanishing of the three trigonometrical factors in lie
x, y,
the denominator .
,
sin-
TH/Yi
X
#A
1
vra s Hira-y ~ sin- - - X r X r .
sin-
r/
V
In addition to the diffraction pattern there tion proportional to
relative
a uniform illumina-
2RT4
N&ff^l-e" The
is
.
intensities
of
the
f uniform illumination and the
depend upon the temperature T and on the stiffness of the atoms /. The results of the investigation are
diffraction pattern
supported by the fact that in the case of diamond, for which f is very large, the diffraction pattern can be observed for much
Formula (33) is smaller values of xjr than for other substances. deduced on the supposition that the heat motions of the atoms are in accordance with the
Boltzmann-Maxwell
hiw,
and therefore
requires modification in the light of recent work on the specific heats of solids, The necessary modifications are considered by
Debye.
The production of diffraction effects with X rays is not confined to crystals. Thus Friedrich* has observed halos round the central image when rays have been allowed to pass through certain non-crystalline substances.
X
It has been pointed out by W. L. Braggf that the bright spots in the transmission photographs are. found nt places corre-
sponding to regular reflexion from planes in the crystal rich in atoms, subject to the condition that the waves reflected from successive planes reinforce one another. This suggested that would be reflected from the natural faces and cleavage rays
X
planes of crystals, and the phenomenon was at once found J in the case of mica and other Since, for a particular wavecrystals.
length
X
of the *
X rays,
there
is
no appreciable reflected intensity
Phys. Ztits. Jahrg. 14, p. 1079 (1913). vol. xvn. p. 43 (1912).
t Camb. Phil. Proc.
TYPES OF RADIATION
509
except in a particular direction determined by d/\, where d is the distance between the successive layers of atoms, this phenomenon has opened up a new and powerful method of investigating the
X
structure of crystals on the one hand and the properties of rays and atomic structure on the other. These problems have been
attacked with conspicuous success by W. H. and W. L. Bragg*, Moseley and Darwin f, MoseleyJ and others. These investigations
have confirmed in a remarkable way the conclusions as to the nature of X rays which we had already reached from less direct evidence.
X
radiation from an ordinary bulb, for instance one furnished with a platinum anticathode, is observed, it is found to consist of a general radiation similar to white light, If the
which
is
except that the average wave-length is enormously shorter, accomof much greater intensity than panied by a number of sharp lines In an distribution. the neighbouring spectral ray bulb with a that found the H. W. Bragg strongest line platinum anticathode
X
||
8
had a wave-length equal to 1*10 x 10~ cm. and its mass absorption coefficient in aluminium was 23**7. According to Barkla's measurements this absorption coefficient corresponds to an X radiation of an element of atomic weight *74 if it is in the K series or 198 if it is
in the
L
Since the atomic weight of platinum
series.
is
195 we
are evidently dealing with the characteristic radiation belonging to the L series from this element. Admitting that the measured coefficient of absorption is slightly in error series would correspond to an in the
L
and that the true value atomic weight 195 the
K
series is 72%5. in the According to equivalent atomic weight cathode which a will just of ray Whiddington's results the energy
K
excite the
radiation in an element of atomic weight 72*5 is whilst the energy liv which corresponds to a wave-
8
2'06 x 10~ erg, X = 1-10 x 1C- 8 cm.
8 Thus the absolute is 178 x 1Q- erg. length numerical value of the frequency is in satisfactory agreement with * W. H. and W. L. Bragg, Roy. Soc. Proc. A, vol. Lxxxvm. p. 428 (1913) ibid. p. 248 (1913) H. Bragg, ibid. vol. LXXXIX. p. 246 (1913); W. L. Bragg, W. H. and W. L. Bragg, ibid. p. 277 (1913); W. H. Bragg, ibid. p. 430 (1914); ;
W
W.
;
L. Bragg, ibid. p. 468 (1914). vol. xxvi. p. 210 (1913). t Phil. Mag. June 1913; ibid. ibid. vol. xxvn. p. 703 (1914). + Phil Mag. vol. xxvi. p. 1024 (1913) 2 2 Mar. (1914); De Broglie, C. JR., 17 Nov., 22 Dec. (1913), 19 Jan., Feb., ;
Herveg, Ver. -r,,..
d.
o_
Deutsch. Phys. T>
n~
A
l
Gt.s. vol. xvi. p.
T.VVYTV.
-n.
246
73 (1914).
the relations on p. ,506
L
o,
the elements which
examined in the case of t7 (atomic wei.h 107which referreTtoi
S1 lver
lines
uld
L
the case of the elements of
i
J
lines are present in addition examined in the case of
In tho
T
eteL
t tht ^andf
weight 90-6) to
whlch has been 2im)nilim
Id
T?
"*"*'"
in
are
ou
various elements
Phil.
Mag.
vol.
xxvu.
p.
703
fa) 'n
the
TYPES OF RADIATION
511
complete table of the elements arranged in sequence according to the atomic weights. In arranging this table the sequence of atomic weights is departed from when it clashes with the sequence of chemical properties as required by the periodic law. Thus argon is placed before potassium, cobalt before nickel and tellurium
In addition blank spaces for undiscovered elements between molybdenum and ruthenium, between neodymium and samarium, and between tungsten and osmium. is called
before iodine.
are
left
N
the atomic number of the element.
aluminium
N=l%,
for silver JV
Thus and
= 47
for
hydrogen iV= 1, for In accordance
so on.
with the periodic law the chemical properties of the elements are rather than by the atomic
determined by the atomic number
N
weight. Moseley finds that the frequencies v of each sub-series are determined by the equation v
where
A
and
= A(N~b)*
b are constants characteristic of each sub-series.
the lines denoted by a in the
and
(35),
for the lines
denoted by a in the
^=
For
K series
^-^J^
L
series
and
6
= 74
(37).
In (36) and (37) VQ is the fundamental Rydberg frequency in the series formulae for the optical spectra of the elements. For the the agreement is quite exact, but for the L series there is a slight
K
deviation from linearity with the elements of very high atomic It is evident from these results that equation (32) is weight. on the fact that only an approximation, its validity depending the elements are pro speaking, the atomic weights of
roughly
of N. portional to the values
CHAPTER XX PHENOMENA
SPECTROSCOPIC
THE power
of emitting light
at a high temperature.
is
When
a universal property of matter is in a sufficiently
the matter
concentrated condition the energy emitted within a given range of frequency dv does not exhibit sharp variations from one wavelength or frequency to another. This is true of highly compressed The general character of the gases as well as solids and liquids. function which expresses the emitted energy of a given frequency in terms of temperature and frequency is similar to that of the
In accordance corresponding function for black body radiation. with Stewart and KirchhofFs law the difference between the emissivity and that of a black body depends only on the reflecting power; and when the latter is given as a function of v and T the
emissivity can be calculated from that of a black body.
The behaviour of gases which are not highly compressed is Almost the whole of the emitted energy is then quite different. confined to a limited number of quite narrow ranges of frequency. In consequence the gas is said to emit a line or a band spectrum according to the appearance of the light when examined in the The distinction between a line and a band spectrum spectroscope. is
a sufficiently real one, but
in simple terms.
The width
it is
one which
is
difficult to define
of the bright regions in the
bund
greater than in the line spectra and the boundaries are spectra less sharply marked. However, a number of so-called band spectra is
are found, under high resolving power, to consist of a multitude of fine lines very close together and it is possible that this may be
a general feature of band spectra. Nevertheless, these two classes of spectra differentiate themselves quite sharply in other ways, as we shall see in a moment.
One
of the most striking features of spectral lines is the of their It is this fact which constancy position in the spectrum.
renders spectrum analysis so reliable.
The only
physical agencies
which have been found capable of displacing them are intense magnetic fields (Zeeman effect), intense electric fields (Stark effect) and the application of high pressure. In all three cases the observed displacements are quite small. The intensities of different lines may be varied by any desired amount by changing the and the manner and of temperature excitation, without degree in the These frequencies are producing any change frequency. evidently characteristic of systems which remain identical under very varied physical conditions.
One
is
naturally tempted to try to account for spectral lines as
the radiation from negative electrons vibrating about equilibrium configurations in the normal atom. Although there are many facts
which seem
to
of the spectroscopic
support such a view, the enormous complexity phenomena which have already been discovered
In the to harmonize with such a simple hypothesis. of elements, iron for example, there are a number of spectra thousands of bright lines. In the case of iron, and many other
is difficult
has not been possible to find any simple relation between the spectral lines, but in a number of other cases many elements,
it
of the lines have been found to
fall
into series
which exhibit
fairly
between the frequency numbers. simple numerical relationships What follows is the merest outline of the more important facts
which have been discovered in the reader
dynamik, vol.
II.
may
vol.
chap.
II.
chap.
II.,
For further information Atom-
this field.
be referred to
J.
Stark, Prindpien der
and Kayser's Handbuch der Spectroscopie,
vm. Series of Spectral Lines.
this kind was discovered by Baliner* in regularity of 1885 in the case of the line spectrum of hydrogen. The frequencies are found to be given by a Hp, y etc., of the well-known lines
The
first
H
,
H
,
the formula
*
vn. Verb, der Natur. For. Ges. Basel, vol.
vol. xxv. p.
80 (1885).
p.
548 (1885)
;
Ann. der Phys.
where VQ is a constant and m takes in succession the integral values = the lines are 3, 4 5, etc. Starting from H^ for which i> $z> the distance but at scale in the first, frequency widely spaced ,
;
For lines continually diminishes. The all the lines practically coincide with v = z> values of large first 31 lines of this series have now been discovered and they all between each two succeeding
m
.
o
agree with the formula to within 0*6 unit=10- 8 cm. = A).
Balmer's series
is
much
o
m
Angstrom unit
(1
Angstrom
simpler than the series which have
In
been found in the spectra of elements other than hydrogen.
the case of the alkali metals for example the series are made up of pairs of lines very close together (doublets). In the spectra of the alkaline earth and other metals the doublets are replaced by The individual lines in a doublet or triplet are called triplets.
components.
In a large number of cases 3 series and in some been found in the spectrum of a single element.
cases 4 series have
The separate series are also interrelated in an The values of the frequencies in the general
interesting manner. case of 4 series of
doublets are given by the following expressions.
Series.
Sharp Principal First
Component,
,,
= --^- (1
Second Component,
Third Component,
+sy>
Component,
Second Component,
Third Component,
^
^^-^L^ ^ ^ ,
l7 a
=
2
Sharp Subsidiary First
-
(m+pj-
......... (2). ^ '
......... (3).
......... (4) ^ '
.
Series.
^= 2 i/ 2
2
v8
=
-
--^L
^
= l
2
-
-
N (m +
2
......... (6). v J
s)
--^~
......... /v 7 }\
'
SPECTROSCOPIC PHENOMENA
515
Diffuse Principal Series.
First Component,
3
,I
=
^_ _^_
......
-^
...... (9 ).
(8)
.
(
Second Component,
*, =
Third Component,
3
-
^ ^_ (
_^. _A_
...
(10) .
(
Diffuse Subsidiary Series.
(In general this series consists of groups of six lines, divided unequally into 3 sub-groups, which we may term components.)
Component
Second
Component
(""" 1
r
Component
4
,
^
__^o
/
i
v
\
^o
"~(l+^
(
In these formulae
m
-^o
-o+^"(STi^^
Third }
JV
,
2
N
are constants.
2 3)
/x/
(m-fd
2
)
the quantities on the right-hand side except has the same value for all elements. The
all Q
others vary from one element to another. The successive lines the successive integral values 1, 2, 3, 4, are obtained by giving to In the doublet series the frequencies denoted by ^3, 2 ps 3 z>3 otc. and in the single line series the are and
m
,
4 /V",
&"
^
X"
>
missing
^
and pi' in addition. For a critical disfrequencies this cussion of the degree of accuracy with which formulae of character fit the observed facts the reader may be referred to recent W. M. Hicks in the Philosophical Transactions of the papers by Rm/al Society.
2
^2 ,v2) ,
SPECTROSCOPIC PHENOMENA
516
The formulae given above show the
lines all
that, as in the hydrogen series, crowd together as the frequency correspondgradually
m = oo
This frequency, in the case of the first component for example, for each of the four series, has the values ing to
is
approached.
TT^> (if^r (TT^7
(
series
and
(ifb'
have the same termination.
ThuS the tw Subsidiary
This
corresponding pair of components. between the different series.
is true separately of each There are other relations
In the case of the sharp subsidiary series and the diffuse difference in frequency between the components principal series the A similar of a given line is constant throughout a given series.
between the lines whose frequencies are given by and (16) on the one hand and by (12) and (14) on the In the case of the first other in the diffuse subsidiary series. two components the values of the constant frequency differences
relation holds (13), (15)
in the respective series are, in the order of the table above,
...(17).
In the case of the sharp principal series on the other hand the difference in frequency between the components is not constant but decreases as the value of increases. Thus the all
m components crowd together as the end of the series is approached. A similar relation holds between the lines given by equations (11) (13) and and in the diffuse In (14) (15) respectively subsidiary series. these cases the series of individual components terminate at a common frequency; in the former cases there is a separate termination for each
The first
component of a given
series.
series are also interrelated in the following
term (m =
l) of the sharp principal series
manner
and the
first
:
The term
of the sharp subsidiary series have the same frequency; this equal to the difference of frequency between the ends of the
principal
and of the sharp subsidiary diffcvrp.nr.p. hptwApn t.hp p-nrlc nf +.VIA
series
series. rlifFncn
is
sharp Also the TMn'ns.innl
SPECTROSCOPIC PHENOMENA
517
and
the diffuse subsidiary series is equal, for each component, to the frequency of the first corresponding term of the diffuse
series
subsidiary series.
Roughly speaking the intensity
as
of the lines tends to diminish
m
increases, although the relative intensities vary a great deal with the mode of excitation of the lines. In a sharp principal
of doublets the component of higher frequency is more intense than the slower component. The order of variation of intensity of the components with increasing frequency is usually the reverse of this in the case of the lines belonging to either series
of the subsidiary series.
The most important
steps in the establishment of the serial due to the work of Runge*, Rydbergf,
relations just described are
These relations have been Kayser, Paschen and Bergmann. somewhat generalized recently by Ritz and Paschen and the results used, following a suggestion of Rydberg, to predict the existence of "combined" series. The success which has attended this
may be judged
development
Annalen der
from papers by Paschen in the
PTiysik from 1908 to the present timej.
been described in the spectra of the elements have preceding chapter, p. 510.
The X-ray
Series in
Band
Spectra.
of relations between the emission frequencies observable in band spectra have been discovered by Deslandres. The frequencies of the lines which make up a single band are with the formula found to
A
number
agree very approximately j/
= A(m +
a
a)
are constants and
where A, a and
c
integral values.
As a
+c
m
(18),
takes various successive
rule this formula does not deviate seriously
more than from the observed values unless the band contains about 50
lines.
but occur in In general, emission bands are not found isolated of number a OU P S an(i in some cases there are related
V *
-|-
Report for 1888, p. 576. K. Svenska. Vet. Akad. HandL vol. xxm. No. 11,
Brit. Assoc.
p.
155 (1890).
+ See also Stark, Principien der Atomdynamik, vol. n. p. 50. 8
from 1902-1905. Pacers in Comptes Rendus from 1885-1891 and
Deslandres finds that
which exhibit relations with one another.
the following relation covers the whole system of groups of bands in certain cases
:
v
Here
2
=/(n p ,
2
)
x
m + BH* + 2
2
(p
(19).
)
frequency of a line in any of the bands, and / are characteristic functions, p, n and
v is the
constant,
jB
is
m
a
are
The number p is characteristic of the group and n is integers. characteristic of the band in which the line occurs, in the same
m is characteristic of the particular line in
the particular n of and becomes values identical with band. For specified (19) p = The values of p and n never exceed 10, whereas 0. (18) with a
way
m
that
may be comparable with
100.
No explanation of (19) has ever been suggested but it is important to observe that the numerical relation between the frequencies of lines in band spectra is of a character quite ;
different
from that which occurs in the series observable in line
With regard
spectra.
it is also
to the foregoing system of series of line important to remember that very few of the lines
spectra in the more complex spectra have been found to fall into these so that the possibility of the existence of other types of series ;
series of lines, as well as of lines
at
all, is
which do not belong to any series
one which we have to bear in mind.
The Zeeman
We
Effect.
have several times had occasion
to refer to the
change
in
the frequency of spectral lines caused by the application of a magnetic field, which was discovered by Zeeman. In Chap, xvi
we even
considered the theory of this effect in connection with We shall now discuss the theory the theory of diamagnetism. in what is probably the simplest possible existing case.
Consider an electron which proportional to its displacement
the atom.
is
subject to a restoring force
from the equilibrium position
in
The
restoring force per unit displacement is isotropic, that is to say it has the same value for different directions of displacement, in the absence of a magnetic field. Taking the
equilibrium position as origin the equations of motion of the particle, when there is no magnetic field, may be written
^z y m-# = -/^ m^fy, m^ = -/* 3
2
,
a
a
,
SPECTROSCOPIC PHENOMENA
519
The natural frequency n./Zir has the same value whatever the direction of vibration and is given by
w
2
=//m
................. .......... (20).
H
Now
suppose a magnetic field 9 parallel to the * axis is As the electron is in motion this will add to the forces applied. previously acting a new force proportional to the vector product of the velocity and H. The of this new force on the components electron are
respectively
The equations
of motion in the magnetic field are therefore
"*&&
eE^y m^x = -& m w fVJ+. ~ft'
&y
eHz dx
,
&z m w^-fa ,
......... (22).
These are solved by
x=0i cos (njk + x
=
a2 cos (n2 t z
where
+
fti),
y
=-
b2 ),
y
=a
= % cos (n
^
2
eH --
-n-^
2
Qt
c^ sin (nif 4- &i) ...... (23),
sin (n 2t
+6) 2
......... (24),
-1-63) ..................... (25),
=
2 '/?
..................... (26),
Equation (25) shows that the vibrations parallel to the magThe corresponding frequency ?z netic field are unaffected by it.
The six field is applied or not. which the first three of b 6 a. b constants a a 2) l} 3 1? 2) arbitrary determine the amplitude and the last three the phase of the has the same value whether the }
,
,
vibrations, depend on the initial conditions of motion and are of no particular interest in connection with the present discussion. harmonic Equations (23) represent two rectilinear simple and differ of are These at motions equal amplitude angles.
right in phase by a quarter period. They therefore constitute a circular vibration in a plane perpendicular to the magnetic field. The same is true of equations (24) except that the direction of the Since the difference between n^ or n and ?z rotation is reversed.
is
with either, very small in comparison
we can
write, instead of
(26) and (27),
Thus the
difference
between the frequency of each of the two circular
original vibration has the same magnitude The case and negative in the other. one in the positive field is quite independent of unit difference magnetic per frequency
vibrations
but
and that of the
is
the frequency of the original line and the universal electronic constant elm.
Now
consider the
electron
in the
moving
radiation
is
which
entirely determined
will
by
be emitted by an
manner we are contemplating.
Since the
circular vibrations are capable of resolution into simple harmonic motions, the nature of the radiation, although not necessarily its
amount, can be deduced from the simpler case of a rectilinear simple harmonic motion. By Poynting's theorem the rate of transmission of energy at any point is proportional to the vector product of the electric and magnetic intensities and the direction of transmission
is
the direction of that vector.
It follows from
the results of Chapter XII that in any motion in which the velocity and acceleration are collinear the magnetic intensity lies
and vanishes at points along therefore no radiation along the direction of the axis of a simple harmonic motion. Now turn to the radiation It in any of the directions which are perpendicular to this axis.
in circles about the axis of motion this axis.
There
follows from
wave in and the
is
Chapter XII that the
electric intensity in the radiation
this case lies in the plane containing the axis of motion radius. It is also perpendicular to the latter. The mag-
netic intensity
is
equal to the electric intensity and its direction is electric intensity and the radius. Since the
normal both to the
plane of polarization of the radiation is that which contains the direction of the magnetic intensity in the wave-front, we see that the radiation emitted in the direction under consideration
is
com-
pletely polarized in the equatorial plane. These principles together with equations (23) (25) are sufficient to determine completely
the character of the radiation which incident with, and normal
to,
is
emitted in directions co-
the lines of magnetic force.
Consider the light emitted along the lines of force first. simple harmonic motion given by (25) emits nothing in
The this
SWfiCTBOSCOPIC
PHENOMENA
521
direction; so that the original line will be entirely absent when observations in this direction are made. It is different with the circular vibrations (23) and The direction of observation is (24). no\v situated in the instantaneous equatorial plane for these motions. The observed radiation will therefore have its electric intensity parallel to the instantaneous direction of motion and its magnetic intensity parallel to the radius drawn from the centre to
the instantaneous position in the equivalent therefore exhibit
complete polarization.
orbit.
The
The
light will direction of the
rotation given
by (23) is in the clockwise direction as seen from the points along positive z axis, that given by (24) is in the direction. Thus the light emitted along the opposite positive axis of consists of two lines in
H
circularly polarized
opposite
the one which has the frequency n is right-handed, the other which has the frequency ?i 2 is left-handed.
directions
When
;
the light
is
observed in a direction perpendicular to
H
z
the vibration corresponding to equation (25) will -give rise to a line which is polarized in the plane to which the magnetic is perpendicular. The frequency of this vibration is n the same as that of the original line in the absence of a magnetic The circular vibrations are now observed in their own field.
intensity
plane and
,
may be
resolved into simple harmonic motions along
The former give rise to no emission in the direction contemplated, whilst the latter give rise to light polarized in the plane containing the magnetic intensity. and perpendicular
to the line of sight.
Thus when the emitted
light is viewed in directions normal to the the force The magnetic original line is converted into a triplet. middle component has the same frequency as the original line and
polarized perpendicularly to the lines of magnetic force, whilst the other components are at equal distances from it and are polarized in the perpendicular plane.
is
According to (28) n will exceed n Q if e is positive and be less than ?i if e is negative. Zeeman found that when the light was emitted along the positive z axis the slowest component of the observed doublet exhibited right-handed circular polarization. This shows that the centres of emission are negatively charged particles.
The problem of the effect of a magnetic field on electrons executing closed orbits in the atom is quite complex in detail;
but Larmor* has shown in a general manner that for small values of the magnetic force the motion consists in the combination of the original motion with a uniform rotation about the axis of This leads to equations (28) for the Zeeman shift. All of the foregoing conclusions f,
many
of
H.
which were predicted
by Lorentz, were found by Zeeman and other investigators to
be-
the case of a large number of spectral lines. accurately all the lines of the single line series of Paschen According to fulfilled in
spectra exhibit this relatively simple type of Zeeman effect. same is true also of many other lines. Thus Purvis finds
The
about 50 lines in the spectrum of palladium which give the normal I ought also to add that the value of e/m deduced from effect. the magnitude of the shift in these cases agrees with that given by the electrons furnished by the cathode rays and from other sources.
It seems fairly clear from these results that the hypothesis of vibrating electrons is an important step towards the explanation of emission spectra.
is
In the case of the majority of spectral lines the Zeeman effect more complicated than the "normal" type just outlined. But-
even the more complex cases exhibit certain relatively simple features which are of importance. For example, when the lines are observed in the direction of the magnetic field, the components of lower frequency exhibit right-handed, and those of higher frequency left-handed, polarization, showing that the vibrators Moreover EungoJ has shown that negatively charged.
are
although the frequency displacement ?nlH often differs from the theoretical amount e/%mc yet it is always a small integral multiple of e/2mnc where n is a small integer. This holds true even when,, }
}
as
we
shall see, the
field is
number
of
new
lines
produced by the magnetic
much
covered by
greater than two. Another important result,, disT. Preston is that all the lines of a given series and of ,
of different elements are decomposed homologous in field the same manner. magnetic series
*
by the
Aether and Matter, p. 341. The rather different treatment considered in Chap, xvi, when it is fully worked out, leads to results identical with those above. Of. Lorent/, Theory of t
Electrons, p. 124.
J Phys. Zeits. vol.
Phil Mag.
vm.
p.
vol. XLV. p.
232 (1907).
325 (1898),
vol. XLVII. p.
1G5
(18<)0).
.PHENOMENA
To illustrate this it will be best to consider one or two examples. The behaviour of the lines in the sharp series of doublets is principal
exemplified by the case of the sodium D lines. Their behaviour in a transverse magnetic field is exhibited in The letters p> Fig. 53. and n denote that the lines are polarized parallel and perpendicular to the lines of magnetic force, respectively.
n
n
p
n
p
Fig. 53.
The two components of sharp subsidiary series of doublets are decomposed in exactly the same way except that the less refrangible component of the one series replaces the more refrangible of the other and vice versa. This may be regarded as a confirmation of the correspondence in the structure of the lines of these two series which is indicated by the respective series formulae. The strong lines of the diffuse split up into triplets, whilst the
subsidiary series of doublets are satellites split into eight com-
ponents of which six are polarized perpendicularly and two parallel to the axis of the
The way
in
magnetic which the
field.
lines of the sharp subsidiary series of
break up in a transverse magnetic field is shown in The behaviour of the diffuse series of triplets is still more complex, as is also that of many lines which have not been
triplets
Fig. 54.
assigned to any series.
Paschen and Back* have recently observed that with certain double lines the character of the Zeeman
effect
depends very much
on the strength of the magnetic field. The arrangement of the lines in strong fields is in some respects simpler than that in weak fields.
Not much has yet been accomplished nation of the more complicated types of *
Ann.
tier
way of an explaZeeman effect, although
Pkusik, vol. xxxix. p. 897 (1912),
in the
vol. XL. p.
960 (1913).
the theory has been attacked from several different points of view by Lorentz*, Voigtf and other writers. One important point
brought out by Lorentz is that a magnetic field cannot of itself alone endow the vibrating atomic system with essentially new modes of vibration it can only cause the separation of existing :
This is an example of periods which previously were coalescent. the well-known dynamical principle that the number of possible modes of motion of a system is determined by the number of degrees of freedom, since the number of degrees of freedom is not by the magnetic field. Naturally, by imagining the
affected
motion of the electrons to be constrained in various ways or by assuming different forms for the expressions for the potential and kinetic energies of the electrons, it is not difficult to arrive at rather complex types of Zeeman effect. But the theoretical results thus far attained effects exhibited
do not seem to resemble very closely the
by the actual spectral
lines.
Fig. 54.
Most of the lines of band spectra which have, been examined do not exhibit any measurable Zeeman shift. In the few cases in which this effect has been observed in the lines of band spectra the results have been
more irregular than those given by line Thus Dufour J found in the banded flame spectra of the haloid salts of the alkaline earth metals some lines which, when
spectra.
*
Theory of Electrons, chap. in. t Magneto- und Elektro-Optik, chaps, n. and St. \rr\\ YTV rm lift 990
iv.
PHENOMENA observed in the direction of the magnetic ponents with circular polarization in the
field,
525 gave
rise to
com-
opposite direction to that
exhibited by the line spectra. Other lines showed the normal behaviour in this respect and in both cases the polarization was incomplete. The existence of circular polarization in the opposite direction to that given by the normal effect cannot be necessarily
interpreted as implying positively charged vibrators, since Lorentz* has shown that this effect can arise from the vibrations of negative electrons under suitable circumstances.
Theories of Spectral Emission.
The frequency and the
effect
of the occurrence of the normal type of Zeernan more complicated types are closely
fact that the
related to the simple type and to the series of line spectra, show that the oscillations of negative electrons play a very important part in spectral emission. The chief difficulty which now lies in
of
the
these
way
of the further development of the interpretation arises from our ignorance of the nature of
effects
the emitting systems. If we consider a particular series of lines, example, we can either regard these as the overtones which
for
accompany the fundamental vibration of a single system or we can look upon each line as the natural vibration of a separate In the latter case the other lines of the series are system. attributed to the occurrence of systems which are constitutionally
some regular way. Both of these views Another view, which attributed advocates. respective all the spectral lines to the natural modes of vibration of the normal atom, is certainly no longer tenable. As each electron can
related to one another in
have
fcheir
most only give rise to three natural frequencies the number of electrons per atom which would be required to furnish the spectra of elements like iron and titanium is quite prohibitive. Moreover
at
such a hypothesis would make the absorption spectra of metallic what they are. vapours quite different from
The experimenters who have adopted what one may
call
the
overtone view of the nature of spectral series, have come to quite different conclusions as to the nature of the emitting systems.
Thus Lenardf found
that the
upward stream of colour
*
Theory of Electrons, p. 123. + Ann. der Physik, vol. xi. p. 636 (1903), vol.
xvn
p.
195 (1905).
arising
from a bead of alkali salt in the outer regions of a flame is undeof the flame flected by an electric field, whereas in the interior case former the In deflected. only the lines the coloration is of the
principal
series
are emitted, whereas in the latter case
lines of the subsidiary series
may
He
also be found.
therefore
concluded that the emitters of the principal series were uncharged atoms of the metal and those of the subsidiary series atoms which
had
lost
one or more electrons.
Experiments with
in the arc confirmed this conclusion.
salt vapours the other hand Stark *,
On
made on the rapidly moving positive ions largely from experiments certain circumstances, (canal rays) found in vacuum tubes under has come to the conclusion that the emitters of
all
the
line,
series
are positive ions: and holds, on other grounds, that the band There is no doubt spectra are emitted by the neutral particles.
that the sources of the line spectra are in many cases in rapid motion, since Stark f has shown that they exhibit the Doppler
This fact alone does not settle the question, but there is no doubt that the streams of deflected positively charged particles do in general give rise to the emission of series lines. However, such effect.
.streams usually contain a fair proportion of neutralized particles which -might be the source of the emission. In fact it seems to
the writer that the bulk of the experimental evidence which has
been brought to bear on
this question
might bo interpreted
in
various ways.
The
truth of the overtone view of the nature of spectral lines In fact there are very R. LadenburgJ, who investigated the grave objections to it. dispersion of luminous hydrogen in a vacuum tube, found that it
cannot be said to have been established.
was negligible except in the neighbourhood of the red line II a this region satisfactory measurements showing the regular type of anomalous dispersion to be expected near a natural period .
In
were obtained, although no measurable
effect could
be detected in
the neighbourhood of the bright blue-green line Hp. An extensive series of measurements covering the dispersion of the vapours of the different alkali metals in the neighbourhood of the lines of the various principal series has recently been carried out by Bevari. * Jahr. der Eadioakt. u. Elektronik, vol. vm. p. 231 (1911). t Phys. Zeits. vol. vi. p. 892 (1905). Ver. der Deutsch. Physik. Ges. x. Jalirg. p. 858 (1908). Roi/. Soc. Proc. A, vol. LXXXIV. p. 209 (1910). vol. LXXXV. u.
f>4
Both these investigations lead
to
important conclusions of a similar
character.
Let us turn p. 148.
term aP
to the theory of dispersion given in Chap,
vm,
In the case of a gaseous substance we can neglect the which depends upon the polarization, on account of its
smallness, so
that the refractive index, neglecting the effects
specifically due to the absorption term in the equations of motion, will be given by (29), h ^
where
m
the charge, the mass, of the vibrators, s the natural p of the 5th frequency type of vibrator, p the frequency of the light, vs the number of vibrators of the 5th type present, and n the number of types. Bevan has shown that an equation like (29), with e is
one term for each line, represents the measurements for potassium as accurately as they can be made so that we shall not be led into No doubt .any serious error by neglecting the absorption term. ;
the same conclusions apply to the other substances investigated. in the case of the hydrogen lines e/m has the regular value
Now
Zeeman effect is of the normal Thus all the on the right-hand side of (29) type. quantities are known except v8 It follows that from measurements of the refractive index we can obtain the number of emitting for negative electrons, since the
.
particles.
Now Ladenburg
no
and
Loria's experiments show that there is appreciable dispersion in hydrogen until it becomes luminous,
whereas luminous
marked manner
hydrogen shows anomalous dispersion in a
in the
neighbourhood of the line
H
a
.
Thus
it
follows that the systems which emit this line do not exist in ordinary hydrogen but are only formed when the gas becomes
luminous.
Moreover luminous hydrogen does not show measur-
able anomalous dispersion near the line Hp, whence it follows that the number of systems which can emit light giving the line H$ is much smaller than the number which can emit light giving the line
H
a.
This conclusion must be valid even though
we have
from the shape neglected the absorption term because it follows, of the dispersion curves, that the inclusion of the absorption term affects the estimated value of v 8 seriously when we make use ;
only of frequencies very close to
.
Now
H
a
and H* are the
first
two
lines of
Balmer's
series, so
that this
argument
leads to the
conclusion, that the systems which can emit the different lines of It follows that the this series are not present in equal numbers.
different lines of
a given series are given out by different systems
and therefore presumably by atoms in different
states.
Bevan's results with the lines of the sharp principal series of the alkali metals confirm these conclusions.
The argument
not so strong because these lines do not exhibit the normal here we have not such good grounds for type of Zeeman effect, so that the validity of the simple theory of dispersion which leads to (29). The constants might be of the more general type given in Chap, is
However, even in this case, it is not 176, for example. of order the that magnitude of v# estimated from the simple likely vin,
p.
formula would prove to be seriously wrong.
As
to the actual numerical differences in v8
Be van
rinds for the
first four doublets of the principal series of potassium that the numbers of vibrating electrons are proportional to the respective
numbers
:
m = 0113, w =l-58xlO- m = rixl()^ 4
a
1
,
s
and
w. 4
=
:j
x 10-'.
some evidence that the value of w is equal, or approximately equal, for the two lines of a doublet. Results of a similar The proportion of character are given by the other alkali metals. atoms present is of emission to the number of total systems capable in the case of the centres which emit the sodium probably greatest There
is
i/
D lines.
Using data given by Wood for the temperature (>44 C. Bevan estimates that the proportion of centres to atoms is about
A
corresponding calculation has been carried out by and Loria in the case of hydrogen in a Geissler tube. Ladenburg of use the formulae (24 a) and (24//) of Chap, vin, By making in which the part played by the absorption term is taken into
1 in 12.
account, they conclude that about one centre capable of emitting a is found in every 50,000 molecules, under the conditions
the line
H
of their experiment.
As to the nature of the difference between the different atoms which makes them capable of emitting different spectral lines a number of plausible hypotheses may be considered. In the first, place the electromagnetic analysis (measurement of the deflexion produced by transverse electric and magnetic fields) has shown
SPECTKOSCOPIC PHENOMENA that the atoms present in the vapours of metals may lose varying numbers of negative electrons. Thus Sir J. J. Thomson* has recently found that the positive rays in mercury vapour contain mercury atoms which have lost 1, 2, 3, 4, 5, 6 and 7 electrons
In the second place there have also been shown to respectively. occur in vacuum tube discharges atoms which have gained one or more electrons in excess of the normal amount. Finally, systems
may be formed by
the
combination
of
two or
more atoms.
I ought also to add that Ritzt has shown that if the electrons are supposed to vibrate in the field of a small magnet, different
frequencies occur which obey the same kind of law as Eydberg's series, if the magnet is supposed to be built up of varying
smaller elementary magnets. The possibility of this last type of theory has received some support from Weiss's work on the magnetic properties of bodies considered in Chap, xvi, but it does not otherwise seem to agree very well with present
numbers
of
tendencies in the development of the theory of atomic structure H. A. Wilson J has pointed out that if we take (see Chap. xxi). the atom to be composed of a number of electrons in equilibrium inside a sphere of positive electrification of uniform volume density a hypothesis which, as we shall see in the next chapter, gives
a
fair
account of
many
of the properties of the
atoms
each atom possesses only one mode of vibration which
is
then
effective
producing any considerable amount of radiation, and the frequency of this mode of vibration depends only on the universal
in
m
and c, and the density p of the positive electrificae, tion in the sphere. According to this theory the frequencies of all spectral lines are determined by the density p of the positive constants
For a given atom the requisite changes in p are electrification. secured either by a deficiency in the normal complement of As electrons or by combination of the atoms with each other. these changes always take place in discrete amounts the frequencies can be made to depend on whole numbers in the same way as in
Balmor's
A
series.
of attempts have been made to construct vibrating have overtones resembling the spectral series. which systems
number
*
Phil.
May.
vol. xxiv. p. 0(58 (1912).
f Gexanmwltt! Wcrke, Paris (1911). J Phil. May. vol. xxin. p. 000 (1912)."
Ritz* has succeeded in constructing two-dimensional dynamical has expressed the form systems of this kind, whilst Whittakerf which the requisite energy functions must take in a very general
manner,
A
theory of spectral emission, due to Bohr, which depends on from any of the foregoing will be considered in
different principles
This theory offers a quantitative exthe next chapter, p. 585. of Balmer's series and is able to overcome most of the planation difficulties
we have
so far encountered.
Fluorescence.
Many substances when illuminated, let
us say by monochromatic are found to emit light of a different colour or frequency. light, This phenomenon is called fluorescence. Sir George Stokes, who researches on the subject, concluded that the was invariably of lower frequency than the light This generalization has been found "to be only exciting light. approximately true. The first exceptions to it were noticed by LommelJ. The more recent investigations of Nichols and Merrifct have shown that in the case of certain substances which they
made very important fluorescent
examined, the relative distribution of energy in the fluorescent spectrum does not depend very much on the frequency of the exciting light, and that a considerable proportion of the emitted energy
may belong
to higher frequencies
than that of the exciting
radiation.
Very often the fluorescent emission the exciting radiation has been cut off
some time after The phenomenon is then
lasts for
often termed phosphorescence. There does not, seem to be any very sharp line of demarcation between fluorescence and
phosphorescence, as the duration referred to may take almost any value from zero upwards with different substances and according to circumstances.
However,
in
the case of liquid and gaseous to be too small t.o
bodies this duration has always been found *
Loc.
cit.
t Roy. Soc.'Proc. A, vol. LXXXV. J Wied. Ann. vol. in. p. 113
p.
(1878).
Phys. Rev.
vol. xix. p.
18 (1904).
202 (1911).
SPECTROSCOPIC PHENOMENA
measure; so that they are
531
strictly fluorescent according to
the
definition.
Fluorescence is very susceptible to changes in the physical condition of the bodies exhibiting it. Thus some substances are fluorescent only when dissolved in certain non-fluorescent liquids
;
whilst others, barium platino-cyanide for example, are fluorescent in the solid state and not when dissolved. The luminous paints
are examples of an important class of substances whose behaviour has been investigated by Lenard and Klatt*. They find that
these bodies, which are of a saline character, only exhibit phosphorescence when they contain three different chemical substances.
Only small traces of two of the three need be present. These facts show that phosphorescence is not a unique property of the atom, or even of the chemical molecule. probable that phosphorescence is invariably accompanied This is a very important by the liberation of electrons f. of these phenomena. point and at once suggests the explanation It
It
is
now seems
fairly certain that the
exciting light
causes
first
the emission of electrons from some constituent of the material and that the fluorescent light arises from the recombination
The time factor will then be these emitted electrons. determined by the resistance to this recombination and will be than in liquids and gases, and at low than at high greater in solids These requirements are borne out by the solids. in temperatures In results. fact, all the observed phenomena seem experimental to receive a very plausible explanation on this view. of
fluorescence can be sharply disquestionable whether of kinds optical absorption except such as tinguished from other It is at least possible that in all arise from electrical conduction. It
cases
is
absorbed from the light by the electrons until The disruption might occur entirely within an occurs.
energy
disruption
is
atom and not give
rise to perceptible emission of electrons.
The
have any value including that frequency of the emitted light might There might even be a as incident the of particular case. light
no re-ernission of radiation except very
indirectly, the
whole of the
absorbed energy being stored temporarily in the atom. Ann. dcr Phyit. vol. xv. pp. 225, 425, 633 (1904). vol. ix. pp. 481, 661 (1908) f Stark and Steubing, Phys. Zeits. Ha.pln,nd. Ann. der Pluis. vol. xxvni. p. 476 (1909).
In fact
*
;
Lenard and
this hypothesis
referred to, facts
known
seems wide enough to cover
all
the phenomena
able to judge, then." are no at present which definitely contradict it.
and so
far as the writer
Wiedemann and Schmidt*
is
discovered
that
the vapours of
various organic compounds and of the metals sodium and potassium exhibited fluorescence. This list has been extended to include the additional elementary substances mercury, iodine and bromine by W. Wood. The fluorescence exhibited by the vapours of the
R
alkali metals
and found
and iodine has been investigated in detail by Woodf remarkable results. Although there is a
to exhibit
good deal of similarity in the effects exhibited by the different When substances, perhaps the case of iodine is most interesting.
by monochromatic light the fluorescent spectrum is not a continuous one but is in the form of a series of fine lines whose frequencies are approximately equidistant, with the stimulation
is
effected
'
the original line forming one of the lines of the .series. The of fluorescent of the the lines in a positions spectrum change remarkable manner as the frequency of the exciting light is
They are also intimately related to the fine structure of the very complex absorption spectrum of iodine. This spectrum has seven very fine lines, visible only with the highest resolving When the power, within the width of the green mercury line. altered.
green mercury line is used for excitation it. is found that a large of the fluorescent lines are made up of fine lines having a structure similar to the seven absorption lines covered by the
number
exciting spectrum, only the fluorescent lines are about thirty times on the frequency scale as the corresponding absorption
as far apart
Many other interesting peculiarities have been observed which promise important results when investigated further.
lines.
The line fluorescence shown by the elements alluded to does not obey Stokes's law. It possesses the remarkable property;" of being converted into a band spectrum by an admixture of inert, The intensity of the banded fluorescence amount and molecular weight of the strange No adequate theory of these phenomena has yet been
gases like helium. diminishes with the
gas added.
put forward, although *
Ann. der Phys.
it
vol. LVII. p.
is
clear that
447 (1896).
t Wood's Physical Optics, chap, xvm., "t"
WnnrJ
a.nrl
Fvarr>lr
T//J<>-
any satisfactory theory of
Slav 7\,, *../.;,
New York r>;.,...
/
(1005); Phil. M
L>
4
/imii
I'.IPJ.
spectroscopic phenomena will have to take account of them. The remarkable complexity of the absorption spectrum of iodine, which has been pointed out by Wood*, should also be remembered in this connection.
The Effect of Pressure on Spectral Lines.
In 1896 Humphreys and Mohlerf observed that the spectral were displaced towards the red end of the
lines of various metals
spectrum when they originated in the arc at a high pressure. The was approximately proportional to the pressure. J
shift
Humphreys
explained this displacement as arising from the Zeeman effect of the intermolecular magnetic fields. The writer pointed out that there should be a displacement, of the observed order of magnitude, arising from the forced vibrations in the atoms of the gas near the emitting centres. An explanation having a similar physical basis
A
was given about the same time by Larmor||. theory which is not very different in principle from the last two has recently been put forward by G. H. LivensIT, who, however, comes to the conclusion that the shift should be proportional to the concentration of the emitting particles and only to the pressure of the gas in so far as it influences the concentration. No doubt effects of the kind
contemplated by Livens may occur under favourable circumstances, but it does not seem probable that they have much to do with the in which it has been observed. pressure shift under the conditions
In the first place the experiments described above on the dispersion of light by luminous gases and by the vapours of the alkali metals indicate that in the case of many of the lines which exhibit the are very few centres in a pressure shift it is probable that there volume of the dimensions of the wave-length of light, so that It then Livens's analysis will not apply without modification. In the writer. as that same the much to reduces given by thing
the second place the shift should vary enormously for spectral lines of the same series to accord with Bevan's results. Finally, such a much view is hardly likely to lead even to so consistency as has *
Phil.
Mag. Dec. 1912.
t Astrophys. Journ. vol. in. p. 144 (1896). t Jalir. der Rad. u. Elektronik, vol. v. p. 324 (1908). Phil. (|
II"
Mag.
vol. xiv. p.
557 (1907).
120 (1907). Astroyliys. Journ. vol. xxvi. p. Phil. Mag. vol. xxiv. p. 285 (1912).
been recorded it
must be
in
measurements of the pressure
admitted, this
far
is
from being
shift,
although, that might be
all
desired.
The Stark
JB/ect.
Stark* has recently observed interesting changes
in
a number
of spectral lines when the emission takes place in a strong electric As in the case of the Zeeman effect the phenomena are field. different for different lines
and the observed
effects
depend upon
the geometrical relation between the direction of emission and the When the lines // and //^ are direction of the electric field. tt
tinder observation in a direction
perpendicular
to
the
electric
of each original field, it is found line. These five lines are symmetrical about the original line with which the central one is coincident. The two outer lines
that five lines apjx*ar instead
much
stronger than the other three and are polarized parallel The three inner lines are polarized perpenWhen the radiation is observed in field. to electric the dicularly the direction of the electric field three evenly spaced lines appear.
are
to the electric field.
These have the same wave-length as the throe lines which were polarized perpendicularly to the field in the former ease and are
now
unpolarized.
Of the helium
which have been examined those belonging sharp principal series and to the sharp secondarv series exhibit no noticeable decomposition. The lines belonging to the lines
to the
diffuse subsidiary series break
up into three or four
lines of
unequal
intensity, asymmetrically spaced with respect to 1he original line. None of these lines are coincident with the original line and they
are unpolarized. In the case of helium the lines observed in the direction of the electric field present the same characteristics as in
the transverse direction.
H
For the three lines (Hp, = 402(5 A for which the y and He X data are most accurate the displacement of the components is The helium line proportional to the strength of the electric field. *
Sitzungsber. d. k. Prems. Akad. d.
Physik, vol.
baum,
XLIII. p.
965 (1914).
ibid. p. 991, p. 1017.
)
Whs. Berlin,
Stark and Wendt, ibid.
1
91:?,
p. is:j;
p.
9.T2
;
Ann. dcr
Stark and Kirscli-
of wave-length 447 2 A, which is split up very asymmetrically, appears to deviate from this law. With a field of 13,000 volt cm.-1 the difference in the wave-lengths of the two outer components
was
3*6
x 10- 8 cm.
for
Hp and
5'2
x 10~ 8 cm.
for
H
y.
A number of
lithium lines exhibit effects of the same order as those given by hydrogen and helium. Here again the sensitive lines are in the
The lines of the two sharp series subsidiary series. exhibit only comparatively small displacements. The displacements shown by a large number of lines of calcium and mercury
diffuse
which have been examined are
all small.
An He
explanation of these effects has been put forward by Bohr*. points out that the external electric field will change the
amount
of energy in the atoms in the different stationary states (see p. 585), and thus involve a change in the frequency of the radiation which is given out in changing from one stationary state
Other explanations have been offered by Garbasso'f,
to another.
Gehrckef and SchwarzschildJ.
The Inverse Zeeman
Effect.
In our treatment of absorption and dispersion in Chap, vin to consider effects which may arise when an external This defect will now be remedied, but field is present. magnetic
we omitted
the treatment given will be very brief. Those who wish for fuller information about this and related questions may be referred to Lorentz's Theory of Electrons, chap, chap,
xvn and ;
i
v
;
Wood's Physical
especially to Voigt's Magneto-
Optics,
und Elektro-Optik,
passim. If
we wish
to take account of the effect
both of a magnetic
and of absorption, we cannot, in
to neglect general, afford electrons of the motion of the in equations any of the terms shall, however, simplify matters numbered (1) in Chap. vin.
field
We
kind of electron
per by assuming that there is only one whose motion is of importance. Let us first consider the when the external magnetic intensity R lies along the z *
Phil.
Mag.
vol.
xxvn. p. 512 (1914).
| Plnjs. Zeits. Feb. 1 (1914). :!:
Verh.
(L
Deutsch. Physik. Ges. Jahrg. 16, p. 20 (1914).
atom case axis,
which
that of the direction of propagation of the light. of motion, under these conditions, reduce to
is
equations
The
(38X
(35),
where
/c
=X
1
If there an*
of Chap. vin.
are given
by
.V
of the movable
components of the polarization
electrons in unit volume, tin*
Px = iV&r, Pv = Ney
and
P
= Xe- ............ (3(j).
I> :
Let the waves propagated through the medium depend on the time through the factor &** only; then, by making use of (3(5), equations (33) -(35) may be replaced by JBs
= (v + iS)Px -ieP
E
= (y+i&) P:
2
,
where
K -a7 = -v^ JVe-
7/0/"
f
.......
.................. (37),
lf
.
................... (311),
8= ^ -'' p, rs
,
IV e-
(,
,
t-
t
= H
iv
;/ f
...... (40).
Since the waves, supposed plane, are propagat<*
t
be independent of
and
a;
and
z
through the factor t'V'V-'i^ only, and will Thus the fundamental electromagnetic
y.
equations i
,
r/ r<)t//=
i-ot^^-
and
a/>
........
_
c
M
1
?llf
c
oi
give us
Substituting these values for
[tfT^ZTiand
[tf.gti
Kx>
(>+
E,, in (.S7)
8
l^
=
and <:{)
-iel'
:l
\vc
.l)t,ain
............ (4-2),
)]
~ (7 +
^, =
+
/'
............ (4).
<S)]
P,,=
iPx
........................ (44).
There are therefore two for a
given value of
Py
Px
solutions having opposite values of
To
interpret the imaginary sign consider the case, corresponding to the solution having the positive sign, where x is proportional, at a fixed point, to the real part of &&, .
P
or cos or
pt.
sin pt.
Then Py will be proportional to the real Thus the solution Py = + iPx corresponds
vibration of
P
}
and since
P is in
part of
i&&
9
to a circular
the same direction as the resultant
displacement of the electrons these also execute similar circular vibrations. The value of q for this vibration is given by substituting in either (42) or (43) from the equation thus given by
Py = + iPx
.
It is
In a similar manner we see that the equation Py = iPx corresponds to a circular vibration in the opposite sense, for which q is given by
L
= l+ cY * In general q
will
be complex.
,,
As
obtain the coefficient of absorption if
we put cq
where n and k are
in Chap, vnr, p. 165,
p and
we
shall
the index of refraction n
= n(l-ik) ........................ (47),
n and k
real,
k -
.................. (46). ^ /
will thus
be obtained by solving
the simultaneous equations
where the positive sign corresponds to the vibration (45) and the negative sign to (46). When there is no magnetic field these equations become n* (l
_ A?) =
1
+ ry-
~
-f
2wfc
,
=
" -v-js
...... (49).
Equations (18) and (19) of Chap, vin evidently reduce to (49)
when
the
notation
This
same assumptions are made and the is
allowed
result
establishes
behaviour of the two
magnetic
lield
slight difference in
for.
a
simple
correlation
between
the
circularly polarized rays in the longitudinal
and the equivalent unpolarized
light in the absence
The only difference between the equations the and (49) corresponding pair of equations (4K) is that 7 is, in the one case and by 7 in the other. Thus replaced by 7 +
of a magnetic field.
t>
t*
considering a given value of 7, the plarized ray (45), in the presence of the magnetic field, will have the same retractive index
and absorption the value 7 + 6
coefficient in the
as the unpolarized ray for which 7 had
absence of
a.
magnetic
clusions apply to the other ray if c from (40) that/) satisfies the relation
is
Similar con-
field.
replaced by
We
f.
see
so that the right-handed my, which corresponds to (45), will exhibit the same behaviour as the unpolarized ray of frequency p shows in the absence of a magnetic field, provided its frequency
/>j
in
given by
K N&fu__ a
pa-. ^ m \Ae-
The left-handed ray 1
the foregoing
if its
\
_,y7
= Ne-f -{. //i
-.
.
\Ae
J
exhibit the
(415) will
frequency
II
ic
.....
/
a
_,y_ cAt^
same behaviour
as both
II
Thus the curves that express the n 'tractive circularly
...<;>()). .
satisfies p.2 ic
coefficient of the
,,
rays
polarized
magnetic field as functions of the a frequency p 2 ~-/>i given by
AV
intlex in
and absorption
the
frequeiicv an*
l{
longitudinal
separated
by
.
V//.C
Each
of these curves
is
identical with the corresponding curve in
the absence of a magnetic field except for the displacement. original curve is almost midway between the displaced eurvrs. relationship is exhibited graphically in Fig. 55, p. 544.
The The
Comparing (52) with (28) we see that the displacement of the circularly polarized components of the absorption spectrum is just equal to the Zeeman shift in the emission spectrum for the same magnetic field. Also if e is negative the absorption band two
for left-handed circularly polarized light is shifted towards the low side. In the direct Zeeman effect we found that when
frequency e
was negative the component of lower frequency was left-handed,
we observed the emission
if
in the direction of the
magnetic
force.
the kind just specified were first observed by Maculoso and Corbino* in their experiments on the absorption of light by sodium vapour in the neighbourhood of the D lines, in Effects
of
the presence of a magnetic field. More complete experiments by and Hallo have shown that the phenomena are comZeemanf \ pletely in accordance with the theory, the development of
largely due
Drude and
which
Voigt. Becquerel has shown that are displayed by the very sharp absorption bands exhibited by the salts of certain rare earth metals at low temperatures, whilst Wood has found them in the fine lines of the
is
similar
to
J.
phenomena
||
channelled absorption spectrum of sodium vapour. In both these cases the direction of rotation for some of the lines corresponds that given
to
by the elementary theory
for positively
charged
particles (see p. 524).
We
shall now turn to the case in which the light is propagated at right angles to the lines of force of the external magnetic field which we shall still suppose to lie along the z axis. The equations
of motion of the electrons will therefore
still be given by equations the Let be (33) (35). light propagated along the x axis. The various electric and magnetic vectors will then depend upon the # (*-*). In the first place coordinates only through the factor l
we
notice that
when the
electric vibration is in
light is plane polarized so that the the same direction as the magnetic field,
For the the latter exerts no effect on the observed phenomena. between the polarization, the electric intensity and the
relations
motion of the electrons are given by equation (35) together with
D =P +E Z
Z
Z
and
P = Nez. z
Those equations, together with
dH = ^ t
dx
*
Comptes lieMlus,
1
,
G
dD2 cu
,
anc*
^
ox
at
vol. cxxvii. p.
dEz =
1 dff,, f
"
c
"'S*''
>
ct
o4H (1898).
Arch. Necrl. t Amtt>rd(wi /Voc. vol. v. p. 41 (1902) + Arch. Ntie.rl. (2) vol. x. p. 14H (1<)()5). ;
(2) vol.
vn.
p.
405
(1<)02).
!< determine completely the iH'haviour of the and none of them involve the external magnetic- field A*.
are sufficient
It
is
dhvt'ttiit,
light
otherwise with the light jxlarized in the perpendicular The e fleet of A* has u\v tu he considered, on account
Sinre
of equation (34).
div
rot //
h=
-
If r,
we have
= 0,
c'di v rot //
and since the waves are plane
always have the value which was present namely zero. Thus
so that /Jj will
had before the
it
light
,
From
this relation, together with Ct7), (&S)
and
-
/>
K + j> tl
Jlt
we
find
p =
'
'
7
"*'
8
'
+
(7
+ <7>)(I + 7 +
// "
(7
'"
i5.)(l+7-t-io)~i
Thus '
MII
(
7+
/
)
(
I
+7
'
-f /o
'
''
'
t:"
t
From
(56) the index of refraction /, and flu- corfiicieni of absorption kp/c may be obtained by equating real and
and solving the resulting simultaneous equations If
imaginary
for
/,
From (54) we observe that the ratio of /',. ),, /' we turn this ratio into the form
parts,
and'/'.
j
s
complex.
ratio of the
amplitudes and 6 the phase difference of the components of the Since 1>X and /' are polarization. and // proportional (,, .,.-
respectively, will
an
we
sec that in general the motion of
tin-
electrons
perpendicular to the direction of the magnetic be^in field. The eccentricity and orientation of the ellipse will evidently depend on the frequency of the light, since 7 8 and * involve this ellipse
,
nna.Yit.itxr
quantity,
To determine the
position of the absorption bands it is necesof k. value to know the Solving the equations we find that sary this quantity is given by
where
= (1 + 2 G= [y (1 + y) + S - e + (1 + 4e
7*
2
2 2 ]
The exact determination by
of the
2
)
8s
.
maximum
differentiation leads to formulae
values of k from (57) which are too complicated to
handle, but it happens that these maxima are given very approximately by the minimum values of C. These values are readily found in the case of very sharp bands or absorption lines, where the
region in which perceptible absorption occurs is so narrow that S e may be treated as constants compared with 7, which passes through the value zero in this neighbourhood and varies very
and
rapidly with p.
Subject to these approximations the minima of of k are given by or
7
= -4
The corresponding values
A = 48 e 2
2 ,
B=4
(1
+
Ve'2
~S
2
4-i
C and Px jPy V4e2 - 48 4-1),
of A, B+ 2
4e'
P0
G and maxima
s
(58).
are (7
=
S 2 (1
4- 4e-),
7;
r
Fhe simplest case arises when the absorption is relatively feeble and the magnetic field is strong enough to produce a separation of the absorption linos which is great compared, with
Under these circumstances S is largo compared with (see unity Chap, vm, p. 167) and e must be large; compared with 8 so that from (58) we have the approximate relation their width.
;
7=e=/ip
(60).
This shows that the original line is split into two absorption lines which are separated by the same frequency difference as the
inverse effect or the lines in the longitudinal lines in the direct effect. emission corresponding To the same order of accuracy the maximum coefficient of
corresponding
absorption is
is
evidently, from (57), the
same
for both
lines
and
given by
&- 4oc
(01). x
c
of the value of the maximum absorption just one-half coefficient in the absence of a magnetic field under otherwise
This
is
identical conditions (see Chap, vin, p.
107
).
Thus when the medium
traversed perpendicularly to the direction of the magnetic force, by light in which the electric intensity is also perpendicular to the is
the light will be absorbed to the extent given by (61) when the frequency has either of the values given by (GO). Corresponding to one of these vibrations the polarization of the
magnetic
force,
medium and by
the motion of the electrons will be given respectively and x = + iy, and corresponding to the other by
Px = + iPy
Px =
iPy and x
=
iy.
Thus the
electrons
move
in opposite
directions in circles perpendicular to the direction of the magnetic This motion agrees with that which corresponds to light force. of the
same frequency when the inverse longitudinal Zeeman
effect
In general these transverse effects will not be symmetrical about the frequency corresponding to 7 = because the expressions for n and k involve odd as well as even is
under consideration.
powers of
7.
Magnetic Rotation and Double Refraction.
The rotation of the plane light, when it passes through a
of polarization of plane
refracting
medium
in
polarixed the direction
an applied magnetic field, was discovered by Faraday* when he was engaged in trying to find the connection between electromagnetism and light. The explanation of this, the first magneto-optical effect to be discovered, follows at once from the theory of the longitudinal inverse Zeeman effect which of the lines of force of
has just been given.
.to
As is well known, a plane polarized ray of light is equivalent two rays circularly polarized in opposite senses and differing *
Exn. Res.
2152 (1845).
Thus the plane polarized my can bo regarded suitably in phase. two of as compounded opposite circularly polarized rays having But we have seen that these constituent the same frequency. in the direction of the rays are propagated with different speeds lines of magnetic force, so that when they emerge from the medium they will no longer have the same phase difference as
when they
entered. in
They will combine into a ray which is, in a plane different from that of the original
general, polarized This rotation of the plane of polarization is easily calculated ray. in the case in which the absorption coefficient kpjc is so small
This condition must certainly hold that we may neglect it. when we are dealing with transparent refracting media.
Turning
to equations (4H)
the refractive index
we
when k
see that
is
n
is
*(
1
=.! v
.
f
-=
and
+
.'
c
2r;
+
<)''
--1
Su
^"~
'
4-
2 c (7
c
ty
(7
(
fc')'-
4"
&w
If one of the circularly polarized rays gets
ahead of the other in phase by an angle the plane of polarisation of the resultant rav will clearly bo rotated through the angle c/>/2. The rotation r*>
of this plane per unit length of the medium, arising from difference in velocity of the two rays, will thus be
(lie
1
y (?4-) 4-# -f.
a
The
sense, of
force
Ii
by
f
is
this
rotation
(.'
(7-<0"4-c reverses
clearly reversed, because changing
and
vice
<:
(,he
when
sign of
the li
magnei
replaces
{
it-
,
versa.
The rotation near an absorption line or band, where k cannot be disregarded, is most Let AIU'IW readily obtained graphically. be the curve which Drives l/v as a, function of t.lie fn>om.M,M- ;,, i,..
SPECTROSCOPIC PHENOMENA
544
Then the corresponding quantity for polarized constituents when a longitudinal
absence of a magnetic the two
circularly
magnetic
field is
field.
present will be obtained, as
clear from the
is
ABODE
discussion on p. 538, if we simply displace or to the left by an amount corresponding to
+
to the right
Let the curves and A 2 B2 C2 D.,E (Fig. 55). In these e.
thus displaced be AJS^GJ)^ curves the ordinates represent l/v and the abscissae the values of p which is 27T times the frequency. The value of the coefficient of rotation
&>
be given as a function of the abscissae p by PQRST in which the ordinates are the differences
will
the lower curve
of the ordinates of the
two upper curves at the same value of
These conclusions hold good when the light
is
p.
propagated
along the lines of magnetic force. When the direction of propagation is transverse to the magnetic field a different phenomenon, verified by Voigt, mani tests itself. to We can then be plane polarized. the incident light Suppose resolve it into two components one polarized in the plane con-
which was predicted and
taining the magnetic field and the axis of propagation, and the other in the perpendicular direction. In the latter component, the electric vibrations are parallel to the magnetic intensity, so that,
the velocity of this constituent is the same as that of light of the same frequency in the absence of a magnetic field. This conclusion follows from the discussion on p. 539. -|~VCVW"\CkY"irll
/Ml O V I
r*f\t~ir\V\S\-r\f\irt4-
TO
Vl /-vtTT/-V1T^l..
rvZvmt,*
The 1-vvr
velocity of the
-1-1-^^v
-.nil,.,,
,
.
I'
obtained from equation
The plane polarized ray therefore (56). two splits up plane polarized rays, polarized in mutually perpendicular planes, which travel with different velocities. Thus into
for light
propagated transversely to the direction of the
magnetic
force a simply refracting
behaves, in a magnetic This effect does not reverse
a doubly refracting crystal. is reversed because (56) contains
field, like
when
E
through
lines of
medium
e
(and therefore R) only
its square.
Kerrs Magneto- optical In 1876 Kerr* showed that when
Effect.
which
light,
or perpendicular to the plane of incidence, poles of an electromagnet, the reflected
is
light
is
polarized in
reflected from the
becomes
elliptically
polarized when the magnet is excited. To explain this effect on the electron theory it is necessary to take into account the
modification of the laws of transmission and reflexion of light, arises from the change in the motion of the electrons in the
which
metal which
is produced by a magnetic field. Naturally, the more than that of the inverse Zeeman complicated theory is It discussed at length by Voigt in Magneto- und Elektroeffect. Optik, chaps, vi and vn. is
Natural Rotatory
Effects.
When a beam of plane polarized light passes through a plate of quartz so that the direction of propagation is parallel to the is found that the emergent light is optic axis of the crystal, it a different plane from that of the incident light. in polarized The rotation of the plane of polarization is proportional to the is somewhat different The general character of the
thickness of the plate and different frequencies.
for
light
effect is
of
thus
similar to the Faraday effect, but a difference in the nature of the two effects is observable if the light is reflected so as to traverse the
In the case of quartz direction. rotating system in the opposite whilst the emergent light is then polarized in the original plane, case the effect is equal to that produced by a in the
magnetic of the single path. path of double the length
Thus
in the quartz
relative to the axis of propagation is the same plate the rotation the plate; whereas this for both directions of propagation through *
*B. A. Report, 1876, p. 85
;
Phil Mag. May, 1877.
SPECTROSCOPIC PHENOMENA
546
not so in the magnetic case, on account of the reversal of the direction of the magnetic field relative to the direction in which
is
found that some quartz crystals rotate the the right and others to the left. The exhibit a similar geometrical dissymmetry, crystals themselves The relation between one being the mirror image of the other.
It the light travels. plane of polarization
them
is
that
like
is
to
between a left-handed and a right-handed
helix.
Similar effects are shown by other crystals and by a very large number of liquids and solutions. In all the liquid substances
which exhibit this constituent least
is
effect,
either
it
has been
compound
of
found
that
carbon
in
the
active
which
at
combined with four different chemical of some other element exhibiting an a or compound
one carbon atom
groups
a is
If this type of chemical structure equivalent chemical structure. is visualized in three dimensions it will be seen that there are of arranging the groups, so that the only two possible ways be displaced in space so as to become cannot molecules resulting These are so related that one of them is the mirror coincident.
image of the
other.
There
is
thus the same relation between the
molecules of these compounds as there is between the crystals of Corresponding to the behaviour of right- and left-handed quartz.
we should expect one of the two isomers, us they are. called, be left-handed and the other right-handed otherwise, since
quartz to
;
purely a question of spacial arrangeone would their other physical and chemical properties ment., expect to be practically the same. Large numbers of such pairs of optical
the structural difference
is
isomers have been isolated and their existence has led to great advances in our knowledge of structural chemistry since Pasteur*,
Van
't
Hofff, and
Le BelJ pointed out the importance of these
phenomena. been found that arrangements which rotate the plane of polarization of light may be constructed by taking a pile of thin It has
sheets of doubly refracting crystals like mica, arranging them so that their optic axes are all similarly situated with respect to the RecJterchen sur la dissyiiuitrie mohiculairc dex pnxliiita on/rr/n '///<'..
Paris (1860). t
La
Chimie
dam
VEapace, Rotterdam (1874).
mitiirt'l*,
axis of the pile
and then shearing the system about fche axis so that points which previously lay on a line parallel to the axis now lie on a It is possible that a sheared structure of this kind spiral might account but
crystals,
the rotatory properties exhibited by some obviously inadequate to account for the similar
for
it is
phenomena displayed by
A general explanation
liquid substances.
based on the electron theory has been put forward by Drude*. It is only necessary to suppose that the electrons, which move with equal freedom in different directions on the usual form of dispersion theory (Chap, vin), are constrained to move in helices in naturally active bodies. In that case Drude was able to show
that right-handed and left-handed circularly polarized beams of would travel with different velocities in the medium, the one
light
being quicker than the other according to whether the helicoidal paths of the electrons are right- or left-handed. The rotation of the plane of polarization thus arises in a very similar manner to that considered in the inverse Zeeman effect and is greatest when
the general dispersion
is
greatest.
Taken
as a whole the facts are
good general agreement with Drude's theory, but it is possible that other ways of introducing a helicoidal structure into the
in
behaviour of the electrons would lead to very similar
results.
Electro-optics.
When
various isotropic insulators are placed in a strong electric they are found to behave optically like doubly refracting This effect, which is exhibited both by solids and crystals. field
liquids,
was discovered by Kerrf.
The
difference in the refractive
indices of the ordinary and extraordinary rays is proportional to In dealing with the the square of the applied electric intensity.
theory of dispersion and related phenomena we made the assumption that when an electron is displaced from its equilibrium position in the
atom
it
is
displacement. optical
effect
assumption * -|
Lehrbuc.h Phil.
only true as a
is
tier
Hag.
acted on by a restoring force proportional to the VoigtJ has developed a theory of Kerr's electrowhich depends upon the supposition that this first
When
large
Optik, chap. vi.
[4] vol.
L.
pp. 387, 44(5 (1876)
vol. ix. p. 157 (1880); vol. xni. pp. 158,
+
approximation.
;
[5] vol.
248 (1882).
Magneto- und Elektro-Optik, chaps, vm, ix and
x.
vm.
pp. 85, 229 (1879)
;
548 are involved it is relatively large displacements necessary In to consider higher powers of the displacement, than the first. forces
and
this theory seems to be capable of embracing the general form
its
known
facts.
Peculiar electro-optical effects are exhibited by many crystalline Some of these appear to arise from the internal
substances. strains
produced by an external electric
field,
an effect which
is
the converse of the piezo-electric effects investigated by J. and P.
Curie*
an electric field during the omission of spectral considered (p. 534). been has already
The lines * vol.
effect of
Comptes Rendw, vol. xci. pp. 294, 383 (1880) pp. 204, 1187 (1881).
xcm.
;
vol. xrzi. pp. 1.%,
350 (1881)
;
CHAPTER XXI THE STRUCTURE OF THE ATOM THE way
in which the electrons form comparatively stable which exhibit the that characterize the groupings properties atoms of the various chemical elements is a problem which has
engaged the attention of a large number of physicists. It will be impossible to do more than briefly indicate some of the more interesting results which have been achieved in this field. Perhaps the most striking property of the chemical atom is its and permanency. Its properties are only temporarily affected by the very strenuous actions which accompany chemical combination and decomposition. Sir Joseph Larmor* appears to definiteness
have been the
first to
be expected to arise electrons
of
point out that this definiteness could not matter consisted of nothing more than
if
linear-
negligible
dimensions whose motions were
governed by the classical equations of the electrodynamic field. For consider any system in which the electric and magnetic vectors
by the equations (1) (4) of Chapter IX with p = 0. Let x, y, z and t be the space and time coordinates of any point of this system, E and being the electric and magnetic vectors. Now consider a second system in which the corresponding variables are denoted by the suffix 1 and are such that
are given
H
(a?,
y, s)
=
k
(,'/;,
,
y,
,
t
^),
=
lt {
E = mE
,
l
Then by substituting in the equations referred = n, seen that, provided k = I and
H = nH
l
,
to, it
is
.
at once
m
E = 0, rot E = - HJc div
div
l
1
where the operations now *
,
= 0, H, = EJc,
H,
rot
refer to the subscripted
Aether and Matter,
p. 189.
independent
THE STRrtTUKK OK THE ATOM
550
electrons
introducing the condition that, the charge of the same in both eases we obtain an additional
By
variables. is
to be the
and in, but there is still nothing to determine equation between A* these quantities. of value Thus, corresponding to any absolute the an infinite* number of other are there solution of the equations, solutions corresponding to linear transformations of the variables. If any solution corresponds to a state of steady motion or rest
there are an infinite
number
of such possible states.
Thus the
and general definiteness of the atom must be due
finite size
something outside the equations referred This conclusion
is
valid only if the
to
to.
dimensions of the elements
of electric charge are negligible compared with the dimensions The radius of an electron, as deduced from its of the problem. inertia, is
about
10~" 13
cm. and
therefore about 10
is
"'
times that of
We
should therefore expect that, as an approximation an atom. towards the problem of atomic structure, we may treat the nega-
endowed with inertia. must be something which determines the
tive electron as a point charge
case there
and definiteness of the atom, which the negative electron It
field.
may
lies
itself as well as
In that finiteriesw
outside the properties of
outside the equations of the
therefore be expected to depend
of the positive
electricity.
One way
in
upon tin* properties which this additional
requirement has been met is by supposing the positive electrification to be limited to the region inside a sphere of atomic dimensions. This hypothesis was first introduced by Lord Kelvin*.
have been worked out verv fully by Sir J. J. Thomson and the question has also been considered by Lord Other possibilities will be referred to later. (See Rayleigh}.
Its consequences ]
pp.
580587.)
A
number
of writers, in addition to those referred
to
above
and in the preceding chapter, have considered the question of the emission of series of spectral lines by atoms made up of electrons. Among these are Jeans, Nagaoka and Seliott'L One of the !
*
I!
Phil Mag. Phil. Mag. Phil. Mag. Phil.
1PM.
vol. in. p.
257 (1902).
vol. xi. p.
117 (1906). 421 (1901);
vol. n. p.
f Phil.
vol. xi. p.
Mag. vol. vn. p. 445 (1904). Mag. vol. xn. p. 21 (1906);
.Vi
vn. p. 237 (1904).
604 (1900).
vol. xin. p.
189 (1907): vol. xv.
p.
438
JL Jil
XV 4. V/1U
objects of these investigations has been to find systems of electrons state of equilibrium will rise
which when disturbed from a
give
to a series of vibrations in
which the overtones are related to the fundamental in the same way as the different members of
Rydberg's series are. In this respect the researches referred to can hardly be considered to have been particularly successful, and it
now seems
as though this view of the relationship between the
must be a mistaken one (cf. Chap, xx, p. 525). the other hand On they have greatly added to our knowledge of the structural conditions which must be satisfied for systems of lines of spectral series
this
kind to exist permanently.
The question
of the definiteness of the atom, already alluded
carries with it that of the definiteness of frequency, or the This fineness is of a very high degree. fineness, of spectral lines. to,
Recent researches* have shown that in the case of many lines, practically the whole of the observable width is due to the Doppler effect arising from the thermal motion of the molecules.
The
actual emitters
must
therefore be instruments of very great
Moreover, as Jeans f has pointed out, they must be precision. atoms, or at any rate systems whose symmetry is such that they are incapable of acquiring through molecular impacts any appreciable motion of rotation about any axis except that of the For instance if the system contains an emitting emitter. is rotating with angular velocity 27no about a fixed axis, the observed light will consist of three lines having o> instead of the single line of frequencies p + &>, p and p
doublet whose axis
frequency
p
emitted by the system when at
rest.
If
&>
has the
range of values given by the Boltzmann-Maxwell law of equiin the production of a partition of energy, this effect would result rather broad band fading away at the edges, instead of a line of the sharpness which is usually observed. In the case of monatomic molecules &> is inappreciable, as is shown by the specific heats of
monatomic
gases.
Disregarding the perturbation due to rotation it is necessary how the frequency p of the atom at rest can be
to consider
sufficiently definite.
positions of statical * [
If the electrons are
Buisson and Fabry, Journ. de Physique, Phil.
Man.
somehow arranged
equilibrium, and the spectral
vol. n. p.
422 (1901).
vol. n. p.
lines are 442 (1912).
in
caused
JL
o
fin.
JL Jtfc
uul u
their defmiteness in accounted by vibrations about these positions, On the other hand the existence of paramagnetism leads us for.
the presence, at least
infer
to
closed
in
in
certain
atoms, of electrons
orbits.
Consequently numerous revolving the origin of spectral lines in attempts have been made to seek Unless rather the kinematics of revolving rings of electrons. and somewhat strange assumptions are made, however, it rapidly
special
definiteness of frequency in this impossible to secure sufficient
is
way.
The
difficulty here,
which has also been emphasized by Jeans, following discussion which is due to
the
brought out in Consider a system containing a ring whose radius is r consisting of n electrons, evenly spaced about the circumr ference when undisturbed, and revolving with velocity J =/3c.
is
Schott*.
Approximate
stability
assured by the presence of a central
is
of positive volume electrifipositive charge or a containing sphere rise to an attraction towards device some other or cation giving
the centre of the ring.
The equations
of motion an*
+ &)K+W* = 2^-(1 rr-
and where
U
is
a function of
??
and
/3
which
shown
to be
'
............... (-2),
r is
proportional to the
W
K
and in (2) rate of radiation from the ringf. The terms in represent the force which arises from the electrons in the ring itself,
other than the one whose motion
W
function of n only and of and from the rest of the atom and arising
P
centrifugal force.
whose form
it is
l
may
//.
is
l\
/3.
considered. is
term on
tin*
t.hc
ri^ht
be considered as a function of
not necessary to inquire,
m
of the form ~<\^(/3), where a
is
its
is
a
is /*
the into
the mass of an electron
g2
is
K
the central force
radius.
Now
nU, being
C" Ct
proportional to the radiation, is always positive if the system radiates at all. Thus if a is constant aquation (1) shows that (3
cannot be constant for a When /3 is given r radiating system. may be obtained from (2), so that the radius of the ring also will *
Phil.
Mag.
vol.
t Of. the value of
n
IQd.
HQA7\
xn. p. 21 (190G).
U
in Phil.
May.
vol.
xn.
p.
22
(11)00)
and
of
R
in vol.
xm.
ATOM
:
55.3
vary with the time, when the system radiates. Subject to the constancy of a, there can therefore be no state of strictly steady motion which is compatible with the emission of radiation. If a is allowed to vary with the time then can (1) and (2)
be
satisfied
by constant values of
definitely steady motion
energy radiated
is
V
possible.
or
$ and
obviously
r and a state of
This, however,
makes the
come from the
internal energy of the electron and, so far as one can see at present, there is no other warrant for
such a hypothesis. In any event any part of the radiation which does not arise from the expansion of the electron would, by its emission, cause changes in /3 and r. Thus the expanding electron might account for the sharpness of some lines, but it is difficult to see
why
practically all of
them should be sharp on such a
view.
In constructing a theory of the atom it is not necessary either that spectral lines should originate with revolving electrons or that the lines of spectral series should correspond to harmonics.
The
existence of para- and ferro-magnetic substances does, however, appear to necessitate the existence of revolving electrons in
some atoms. radiation
is
Such
electrons
practically zero.
under these circumstances
/3
may be
so
arranged that their
Equations (1) and (2) show that and r may be constant. The devia-
determined by the amount of radiation, so that an atom may have revolving rings of electrons in it and This constill be quite definite provided these do not radiate.
tion from steady motion
is
by placing a
dition can be secured
sufficient
number
of electrons
with great rapidity any one ring, as the radiation decreases can easily be which This increased. is number result, as the the obvious is also established calculation, physically, since
in
by
amount
of radiation
static field.
we
On
is
determined by the lack of the corresponding
the other hand, for the emission of spectral lines be unstable, as it is known not to which
may require a system have a he the normal atom, but which must radiate and must
definite frequency.
Power
quency are precisely the
of radiating
qualities
and definiteness of
fre-
which do not concur in systems
of revolving rings of electrons. of matter If revolving systems occur in the ultimate parts of the matter and their axes must be determined by the structure is turned. must turn in space when the matter as a whole would bodies and magnetized Otherwise the properties of crystals
IJtljb
OJLJaUVJl
UIVi
When
in space*. depend on their orientation
contain systems
of
electrons
in
orbital
bodies whose atoms motion are turned, a
The fact that such to be experienced. gyrostatic couple ought an effect has not been observed cannot, I think, be regarded as of such systems. The moment of momendisproving the existence is so small that one could electrons tum of the revolving only
out special experiments expect to detect such an effect by carrying of a delicate character, and this has not yet been done.
In the papers of Schott to which we have referred, the quesof, and the amount of radiation from, rings
tions of the stability
the calculations the only assumption
made
as to the nature of the
force attracting the electrons to the centre of the is
In most of
both considered in detail.
of revolving electrons are
Some
of the usual electromagnetic type.
atom
is
that
it
of the results are
more general than the corresponding ones given by the investigation which we shall now describe. therefore
is properties of the type of atom which a inside of electrons sphere large coplanar rings
The
made up of
indeformable positive electrification have been worked Sir J. J.
Thomson f and
lead to very interesting results.
of
uniform out by If 6
is
the radius of the large sphere and p the density of the electrification in it, then the force, acting on an electron whose charge is e at a distance a from the centre of the sphere and due to the positive charge of the latter,
is fyrrpett.
the neutral atom, the charge
E
If there are v electrons in
in the positive
E = ve = 4?rp b
sphere
is
n
j*\.
Hence, in terms of the number and charge of the electrons, a from the centre due to
force they experience at a distance
positive sphere
is
ve 2 a/&.
the. the.
In addition to this the electrons are
acted upon by forces clue to their mutual repulsions. If is the centre of the sphere an electron at will be acted upon by an electron at B with a force tfjABr and the radial component of
A
this =~T-77 cos
GAB,
If
OA =
OB,
i.e.
if
the electrons are in the
same
Hence, if we ring, this repulsive force is e~/4
4
1
Cf. Jeans, Phil. Mag. vol. xi. p. 606 (1906). T />/ flit ff alc/-\ 7'?>/> /Tn/k.MKUAi ?/.- '/'I, ,..,. ,
*\r..t* ___
^\ _____
...
x*
_____
\?,...i.
-\t\i\n
the circumference of a
due to the others ,
4aa
(
cosec
IT/In
circle,
them
the radial repulsion on one of
is
+ cosec %7r/n /
+ cosec STT/W 4- ...... /
-f
cosec -
\
Calling the sum within the bracket are at rest under their mutual forces,
v&a
Sn we
have,
if
.......
-
w
the electrons
~
e*
..............................
W-
This will determine a state of possible equilibrium if the electrons If they are rotating round the ring we shall have to
are at rest.
take account of the centrifugal force and the equilibrium condition
becomes e
,,
2
/A
ci
\
These equations only determine a state of possible equilibrium. tell us whether the equilibrium is stable or unstable. For such a ring to be a possible part of a normal atom it is
They do not
necessary that this arrangement should be stable, otherwise, the To ring would break up under the action of any external force. find out
whether the equilibrium
is
stable or not
it is
necessary to
calculate the forces called into play by an infinitesimal displacement and to see whether its direction is such as to cause the
displaced electron to
must be done both
move back
for radial
and
to
its
original
for tangential
This
position.
displacements
in
the
plane of the ring and for displacements perpendicular to the- plane of the hitter. Let us illustrate this by considering one or two
simple eases. In the case of a single electron it is evident that the centre of The equilibrium the sphere is the only position of equilibrium. in tins ease is evidently stable. In the case of two electrons it is
evident by symmetry that the only equilibrium position which they lie along the same diameter of the. sphere.
is
that in
If they are at rest at a distance a from the centre the repulsive force is 2e~ (i
g2
v- and the attraction
4*7 ~
is
~^., '.
rr
Hence the
total force acting on
THE STRUCTURE OF THE ATOM
556
each electron, reckoned positive the sphere,
directed towards the centre of
if
is
a.
This vanishes
if
a
= ^6.
If
a becomes a ,
4
As the quantity the equilibrium
The
.
o
c>
+ du, R
increases
by
\
s (I J
\b*
in brackets is positive when 6 = ~2u, it follows that is stable as regards radial displacements.
by a displacement
reaction called into play
tion perpendicular to the radius
may
in
be calculated as
any
direc-
Let
follows.
the angular displacement from the equilibrium position be dd] The attraction of the then the linear displacement is add. so that the tangential positive sphere is still towards the centre, the will be to other electron. The force due entirely restoring //"
force
between the two electrons
the line joining them, gential
component of This
if
is
repulsive and equal to
we neglect small
this
is
quantities.
--..sinJfW or
, t
,
dti
along
The
tan-
the
first
to
directed towards the position of equilibrium of the displaced electron; so that the equilibrium is stable for lateral As this is true whatever the direction of the disdisplacements.
order.
is
placement, the arrangement of two electrons along a diameter, each at a point halfway between the centre and the circumference, is one which satisfies all the conditions for stability.
We shall now consider the general case. Take the centre of the atom as origin and let the position of an electron be given by the cylindrical coordinates r, 6 and z. r is the projection of the radius on the plane of the undisturbed orbit, $is the angle r makes with a fixed line in that plane and z is the displacement perpendicular to the plane of r and 0. The coordinates are supposed to
undergo small
z=
oscillations
/v
=M
about the steady values r ^/,
If the suffix s refers to a particular electron fore put rs = a + p s and 0.
*
we can
and
there-
ZTT {5),
where n
is
the
number of electrons in the ring, p and n nnrl A .^/l j. ....... n ..... .
it
r/ymrfl:rWI wit.li
_______
.
i
c.
are small
-,1
_-
/
.
The
radial repulsion exerted
= -
2
T|T
a
- pp
I
sn
where
+ r - 2rr \
(i
= (p
5)
-
*
by the sth electron on the "* 2 ~
+
cos **
2 sin 2
~"
^)th is
-
-
1
.
?!/
The
tangential force
($
p8
tending
to increase
P is
=_ e:A
e."s
- ^* Pj> - 4 -"-(*T *P) r/ The
perpendicular to the plane; of the undisturbed orbit
forc'e
111
(7).
a
a
(8) higher powers than the and z have been dropped.
equations (6)
quantities
The
p,
Rp
radial force
ring on the
jpth
exerted by
all
and perpendicular
the other electrons in the
and
.,
1
/
/
fftS B
.S-TT-X
f
.
2 (l/sin^), lsin / V /
^ MC .
Jw
/J
sin
!
;i
,S'7T/
.-1,=
Sa[ \
,VTT
.
sin
......... (10),
.................. (11),
^'^/'.. 4rr
/'
i
on the same particle
forces
% = XjftH# ^^ + ^i = VJ-2^ M x/^
(
" -
of the small
is
The tangential are respectively
,S'=
first
is
//
. ., 1
.
,,.W\
/am- n
/
/
sin-
n
\
,S'7T
(12).
,
sin-
1
and
D=
^
./>,. *
of motion of the pth electron are
The equations
'
m
r
(15).
From
w
(13) since
is
the value of
when the steady motion
-~~
is
undisturbed
agreement with
in
becomes,
By
use of this relation, (18)
By making
(4).
to the first order in the small displacements,
similar treatment (14)
may
made
also be
same
linear, to the
of p and are the degree of approximation. Thus the values solutions of a set of 2n simultaneous linear differential equations To find the frequencies assume that p lt and of the second order.
p
vary as
e iqt
Equation
.
(A - mf) pp
4-
-Ai/VH
(1(5)
then becomes
+ Ap v +
-h
-
-
.
+ aBi c^, +1 + a &,$ where
A=
-r-^S
+ A'.
+
,
lt
Treated in a similar way *
(
. . -
'
~
0.
.
.(
1
7
),
14)
a
a
+ (C-mf)
p
-
C\<j>p+l
-
Cnhn,,
-
...
=
()...( IS).
There are n each of the equations (17) and (IS) obtained by giving
p
the successive integral values
1, 2, 3,
...
//.
These equations can be solved by the following artifice. If = = a and a< for all values of if from to a.nd , identically; the nth roots of unity and is equal to ,
l
t
I
//
1
,
where k
may
(A
pp
an integer between
is
we
have,
and n
Whatever value
1.
see that equations (17) and (18) reduce to
- mq*+ OLA.+ a?A + ... + ^"M^) + p a (- Zim&q + a#i + ^B, + + of 2
1
.
.
.
1
j}^) = O
and p p (2im wg
-
a^ - a 5 -
-
2
2
. . .
a^- 1 J3 n _i)
+ ^a (C-nif-ad - a
2
Thus we
-
...
- a^C^) =
see that all the 2n equations (17) and (18) are satis= a# n for all the values of p between 1 p
by pp = oL^p n and
fied
<73
<
and 71, provided (19) and (20) are also satisfied. By eliminating p and pp from (19) and (20) we see that the frequencies q are <j>
given by the biquadratic equation 2 (A - mq + o4x +
a2
A+
.
.
.
+ oP^Aw) n
This
may
_^
. .
.(21).
be written more concisely
Lk - A -
a3
-
Lk =
.
where
j\rfc
6 cT
==J
s
cos
In these equations k
and may
is
the integer in a
= cos
w-1.
If
have any value from
to
2&7T
.
-M
n-k
is
.
sin
^ written
differ only given by the equations the values all Thus same. the are in sign, so that the frequencies = ... (n-l)/2 fc 2, 1, 0, of the frequency can be got by putting If n is odd there if n is even. ... and k
for
if
,
A;
n
lre
in (22) the values of q
is
odd,
=0,
1, 2,
?i/2
w _i)/2 equations of the type (22) leading (
to
independent
M
and equation (22) rek= values of the frequency. When k = 0, are 2 = 2?i there that 4(?i + l)/2 duces to a quadratic, so altogether
When n
values of the frequency. tions,
but
M
k
=
is
even there are w/2
+
1
equa-
when k= or n/2, so that two of these reduce Thus we see that whether n be odd or even
to quadratics. there are 2n possible frequencies for vibrations in the plane of to the 2?i degrees of freedom of the the orbit,
corresponding
electrons, in this plane.
In a similar way the n frequencies q of the vibrations at of the orbit may be shown to be right angles to the plane given by
+ Pi -P -u/=0
Pk = 71
1
where
H-l
V
^ r~T
f
z
8ttV-i
OS'TT /I
*dOA//J
cos
............... ,
n
.................. (24),
sin"
n
/
"*
t
............... (2o).
I
The values of L, I/, ^V and P have been worked out- in a number of the simpler cases and the corresponding frequencies calculated *.
The importance of this investigation lies, for reasons which have already been given, not so much in the fact that it enables us to evaluate the frequencies of the vibrations of the electrons, as in the fact that the frequency equations (22) and (24) enable. us to determine whether the equilibrium is stable or not. the values of q given by (22) and (24) are real then all
all
If
the
p, and z are periodic functions of the time, and the system will only execute small oscillations about the steady In this case the steady motion is stable. If, configuration.
disturbances
on the other hand, any of the roots contain an imaginary part, the corresponding values of p, c/> or z will contain factors of the. ht where h is a real positive constant... Thus any such type e disturbance will increase indefinitely with the lapse of time and ,
the presence of complex or imaginary values of (/ shows that the equilibrium is unstable. The condition that the equilibrium should be stable is therefore that all the values off/ given by the equations (22) and (24) should be real for values of k. *
Cf.
Thomson, he.
cit.
all
the admissible
THE ATOM
J;-
The
5(31
stability of the
motions can always be provided for o> so far, at our is, disposal. Transposing equation (9) we have
since the value of the rotation
and, substituting this value of
T * * k =
where
(24) becomes
*
1* g2
-y
a,
--8a P p, and y ^ ^~ dI1U
-*
*<)
is
always negative, so that the right-hand side of (24) can obviously be made positive by choosing o> large enough. Thus q can always be made real and the z motions stable. 3P*
o
Now
turn to the motions in the plane of the write equation (22) in the form
We may
orbit.
whore niA
A,
B,
=
S + L k - L,
-
|
and
(J
1) are real
mB = N, - Nk mC= Mk and
is positive.
We
D = 2o>.
and
,
can make
C/D
we
and at the same Let A have any
positive, value,
then C/D can always be chosen so as to be less
as
as
snia.ll
please by suitably choosing &>, time preserve, the stability of the z motions.
tli
an
when
A
-f
(/
-
or
H- /!-.
lies
Now f(q)
is
positive
between +-4- and
when q =
positive axis four times
C/J).
+ B^
when
or
q
=
oo
A- and
The graph of/(g) thus
,
negative B-, and
crosses the real
If A is negative there are only 00. between two such intersections. For all the roots to be real it is therefore This is the necessary and sufficient that A should be positive.
x'
condition for stability.
When Mo^it.ive
number of electrons in a ring becomes considerable, values of A begin to appear; so that no amount of can make the ring stable. The greatest number of the
rotation electrons which are stable in a circle inside a sphere, contain-
ing
an
equal
total
quantity of
positive
electricity
uniformly
THE STRUCTURE OE THE ATOM
562
Six or more electrons in a single ring are .never in stable equilibrium however great the angular velocity Such a ring may, however, be made stable if a negative is. distributed,
five*.
is
If this negative charge is effect of the additional the carried by p electrons, equal to that introduced is to increase A in the frequency repulsion thus
introduced at the centre.
is
charge
equation (27) to
by taking p sufficiently From what has been ring
to
This can always be made positive
.4+3^-*.
be stable
is
large.
said
it
is
A
-f-
that
clear that the condition for a
~=
/}/s
(Y\f>~
3
|
should be positive. The greatest value of when n is even and for k=(n l)/2 when n of electrons
which would have
ring to ensure stability value of p for which
is
'Df**
-
(L n
L
t)
is
-Lk
Lk)
is for
+
3
i
=
^
.,
?//2
The number
odd.
be placed at the centre of the
to
by the
therefore given
least integral
~
P0" ^ *~
3
r
>Lr
US
when i
Ln
4
;/
even
is
'
(.1
7T
...
%*>
and
C2,H).
"
_
_
r
r ~~ -^ n **>
CL'
~~ 1
t>
wm n
>\
i
i
-
"
ls
(V
of these equations the least value of j> for a given value of n can be calculated, p increases very rapidly for largo values of n as is shown by the following table of corresponding
By means
values
n
:
5
p
When p
G
7
8
9
10
15
io
i
i
i
a
3
15
;ji>
exceeds unity the additional electrons
:jo
101
cannot.- all
lie
at the centre of the ring but must separate under* their mutual Thus when n = 9 and p = 2 there will bean external repulsion.
ring of seven and two electrons inside situated alon^ a diameter of the sphere at equal distances from the centre. Wh^n /> is largo is possible for the electrons to arrange thomsolvos in stable equilibrium in rings rotating about a common axis and lying in On the assumption that the eiVect of external parallel planes.
it
rings of electrons can be neglected and that the effect, of infernal rings is the same as if the total charge on flic electrons were
THE STRUCTURE OF THE ATOM
563
collected at the centre of the sphere, the number of electrons in the successive rings can be calculated by the application of the It is probable that this treatment will give representation of the properties of atoms containing a
foregoing principles.
a
fair
number
of electrons, even if the latter are not arranged in rings The conditions for equibut, for example, in concentric shells. librium are of the same general character in both cases. If a shell
contains a large number of electrons it will be unstable some of them are inside, just as the corresponding ring shall now consider the structure and properties of the
unless was.
We
atoms which have
different
numbers of
electrons,
when
their
determined by the equilibrium conditions to which constitution we have been led. The numerical computations will be omitted is
;
they
are-
given in the paper by Thomson to which reference has
frequently been made.
We
have seen that when there
only one electron it will lie whilst if there are two, they at the centre of the positive sphere, lie along a diameter at equal distances from the centre. is
always
Three electrons
will
arrange themselves at the corners of an
situated on a circle whose radius is given by -equilateral triangle The arrangement of four electrons at the corners (3).
equation of a square
is
unstable
if
the electrons are at rest.
It
is
found
jump out of the plane of the others and that the stable arrangement for four electrons is at the A similar occurrence takes corners of a regular tetrahedron. number of electrons is greater than three. place whenever the Thus three is the greatest number of electrons that can exist that one of the electrons will
are at rest. By equilibrium in a ring, provided they of electrons the velocities, to sufficiently high rings <-issi"-nini( can be increased to as it appears that. (,he number in the ringwithout five ring of six or instability setting in. many as if even Six is unstable rotating. number, however, n-nv in
stable,
A
Beater
electrons can be in stable equilibrium and the other five rotate tin*
sphere
single
electron
at
the centre
is
if
one
is
at the centre of
A
a ring around it. sufficient to make a ring of in
seven or eight stable, thus accounting for nine electrons altoo-ether.
With
ton electrons
it
is
necessary to have two in the With eleven three go to
.centre and a ring of eight outside. U,c eentre and this holds till we
sret
to
fourteen,
wen
THE STRUCTURE OF THE ATOM
564
With seventeen
in the centre increases to four.
number
elec-
found that the stable arrangement is an outer ring trons We have seen, however, that a six inside it. with of eleven one of them goes to the centre that so of six is unstable, is
it
ring
leaving a ring of
We
five.
have thus three concentric systems
in this case, containing respectively one, five and eleven electrons. When the number of electrons in the outer ring becomes con-
number which
it is necessary to place inside the ensure For stability increases very rapidly. ring n in the ring the number p inside, number the of values large
siderable, the
order
in
varies as
to
if.
actual arrangement of the electrons with some of the The number numbers is given in the following table. of number the the columns numbers horizontal rings gives
The
smaller of
;
number of electrons in each ring. The total number the atom is thus obtained by adding up the vertical columns.
are the
4
3
2
outer ring
1
outer ring inner ring
5
outer ring 2nd ring 1st ring
11 5
11
11
6
7
I
1
1
outer ring 3rd rmg"
15 15
3
3
4
14
II
IT,
lf>
10
IO
1<)
11
f>
f>
f>
I
17
17
17
17
17
13
11
14
i:>
15
1111112233
9 3
K)
10
lo
10
10
11
:>
f>
17 15 11
15 11
19 16 13
20 If?
13
20 16 13
2
3
3
I
20
1
20
20
20
20
2O
20
10
17 13
17
17
17
17
13
13 lo
II
14
10
10
10
lo
3
4
f>
f>
13
8889!)
2
<>
3
3
21 17 15
22 20 16
23 20 17
17
11 5
13
13
15
8
10
1
1
2
3
Thus 93 odYMCiC!
13 in
17 13
11
56
1st ring
lo
17 13
1
3
Ki 16 H> 'Hi 16 11 12 12 12 13 H 8 8 7 7 7
13
<)
13
1st ring
of 5, 11
I
13
H 3 'lit
5
2nd ring
1
5
13
12 8 3
12 8 2
11
13 8
2nd ring
outer ring 5th ring 4th ring 3rd ring
12 8
12 7
10 5
10 4
10 3
1(5
11
1st ring
outer ring 4th ring 3rd ring
f>
678889 1111233
2nd ring
in
(
21
21
r,
:
r>
24 21
1
r>
electrons arrange themselves in six concentric rings and 24 respectively. <)4 eleetrons begin the ui.nm*t~\ vi n rvo Voxivwv -f-Vi/i *,. /<.,.....%. M II r ?
15, 17, 21
Txrii-Ti
.1
i
4.
t
I
I
1
J
UF THE ATOM
ODD
Numbers like those in the preceding table showing the arrangement of the electron in the atom can be obtained in the It is first following manner. necessary to determine a sufficient number of corresponding values of p and n. This can be done by means of equations (28). Having done curve which gives p as a function of n. It
this is
we can draw the
of the general form
exhibited by the accompanying figure.
Suppose it is required to find how a large electrons will arrange themselves in stable rings. along the axis of p and draw PiQi inclined at 135
N
of number Take OPi = N to OPi inter-
Then if QiP is perpendicular to OP l} secting the curve in Q lf QjPassPjjPj. By the property of the curve QiP2 electrons in u ring require OPo inside to make them stable, so that JV = OPi electrons
will
arrange themselves with an outer ring of P^2 P electrons. In general OP 2 will not be an
l
.surrounding OPo integer
;
in
this
instead of
Thus we
higher integral will arrange themselves so P./ PI
electrons.
OPo we take OP/ the next
case,
value.
We
can
see
that
N= OPi
electrons
outermost ring contains the distribution of the OP/
that the find
electrons by repeating the process. From P/ draw P/Q-j inclined If OP/ ut 135 to OP, and draw QJ\ perpendicular to OP^ the next integer higher than OP the OP/ electrons will is 1
;{
arrange themselves with an outer ring of P/P/ electrons surrounding the remaining OP/. In this way we can proceed until the whole N electrons nre accounted for. on the preceding page we see that the ttiiocessive atoms formed in this way possess features analogous
Referring to the
to
table,
those properties of the chemical
elements which are sum-
marized by the periodic law. Thus it will be observed that the first element with seven rings is the same as that with six except
THE STRUCTURE OF THE ATOM
566
the outer ring of 24 electrons; this in turn is the same as We that with five except for the outer ring of 21, and so on. same the with internal elements structure might expect that If so, we should expect them would have similar properties, for
be separated by groups of elements having widely different lines of investigation If, as a number of different properties. to
lead us to believe,
the atomic weight
is
proportional
to
the
exactly what happens.
number electrons, As we proceed through a list of elements this
of contained
weight
we
is
of increasing atomic character find elements of similar periodically ap-
different elements. This kind of pearing separated by entirely exhibited be to such by properties especially relationship ought as the frequency of vibration of the electrical constituents of to be determined largely by contained electrons. of the It is the geometrical arrangement series of elements of well known that the lines of the spectral
the atom, which
we should expect
the same chemical family are closely related.
Another kind of resemblance elements
is
brought out in a
to the properties of the chemical
still
more marked way
if
we can
consider the equilibrium of the electrons in the successive artificial atoms and confine ourselves to the case when* there is a
coustant number, for example, twenty, of electrons in the outer Starting with the first member, that with o9 electrons rings. altogether, this will only just have enough electrons inside to keep the outer ring stable. It will therefore very readily i(ive
When it has given this oil', however, thenbe 58 electrons left, a number which is not only ^rcafto have an outer These will therefore enough ring of twenty.
off
one electron.
will
Now o,S arrange themselves with an outer ring of nineteen. is the number which can have an outer greatest ring of nineteen, so that the stability of this atom as regards emission of electrons will be very high, more particularly as it. lias an excess of positive charge. The atom with 59 electrons will thus be
capable of emitting one negative electron and so form a mon
the strongly electropositive elements hydrogen, lithium, sodium, potassium, etc. The next atom with 60 electrons will be some-
two electrons before the number
falls
to
5H and
it
becomes
Thus it will exceedingly stable as regards further emission. resemble the divalent electropositive elements of the alkaline In a similar way the next element will not so an emit electron but it will be able to part with three easily before reaching the very stable condition with 58 electrons. It will therefore be less electropositive but will be capable of giving rise to a trivalent positive ion. It will therefore resemble the. trivaearth group.
lent earth metals
the
number
In this way we see that, as which have an
such as aluminium.
of electrons increases, the elements
outer ring of twenty become continuously less electropositive but have a continuously increasing electropositive valency. When we come to the last atom with twenty electrons in the outer ring and (57 altogether this would theoretically be able to emit
nine electrons, but practically it will be so stable that, it- will be incapable of emitting any except under very great forces. It may thus be considered to resemble the inert gases helium, argon, etc. which ant incapable of entering into chemical combination. Again, this element will he unable to combine with
a free electron; for
if it did so it would have IJrS electrons altoand these would gether arrange themselves with an outer ring of 21. As the system with (>8 electrons and an outer ring of 21 is very unstable and liable to emit an electron when will be still more unstable when carries an excess neutral, it.
it-
of negative charge, so that this atom with (>7 electrons will be Itwill incapable of combining with one additional electron.
thus have zero electronegative valency. the inert, gaseous elements.
In
electrons will
tend to combine with one electron;
the atoms are neutral, the element with
that with
()<>.
this respect-
The element
resembles
It.
will
not be able
to
it, again with (>(>
since,
when
more stable than combine with more than <J7
is
would possess (iS electrons altogether, an as we have seen, is exceedingly imstable which, arrangement even when neutral. This element, will thus behave like the strongly electronegative monovalont elements fluorine, chlorine, one,
for
if
it.
did
it,
The atom with bT> electrons will have a less strnnHv o marked tendency to combine with an additional one but- will be able to combine with two altogether before reaching the
etc.
7
limiting condition.
It
will
thus
resemble the elements oxvgen,
than ihe that
til*
have an
those which
Pnwerdinor
-lenients.
pmvding
element with
4
numU-r
lens
sneivssi\**ly
has
negjitive element
electrons and
is
way we
this
see
of electrons, out of greatest <wtT ring of twenty* has zero electro-
As th*' negative viilfwy. diminished tin* valency steadily become
in
number
tht-
*l**ctrns
*f
than
less
successively the elements
Th*
rl*'ctron*'gative.
nil**
is
whilst
inciv;tM*s
inns!
eleetro-
maximum number
tin*
of
mnval'nt.
furnish u striking analogy tu the variation of valency and rWtrK*h'mie.al |ir|i-ertit's as \\' pass through a
These
series
pro|x*rt.i**s
of eleiaenis
tli*
ill
Starting with
tabl*.
jHTiIi<*
the
element {KKHHessing 5H rlrctnns, tin* ina\innun number with an the electn>outer ring of nimt*rn, the folhtwing table >h*\vs
hw
\al'ncies
positive mifl electronegative inereasi*d
rini!
Il> f
iminlHT
...
"...
-
H
- IJ
- 11
-'
:,u
f*ii
*;i
*;^
tor
Atomic weight Electropositive valency Elect ront^ati v
NV
Su
ilO'iJ
xJ.'l'O"
M
is
...
u
;ji
;:*
-'
-'*
- 11
IM
;:
r.i;
;:
-
1
(is
.sho\sn in th- following table:
A! r
,"i
Si
JV1
H '3
I
-2
3
4
u
7
r
r>
I
the
sum
H
S
1*
:tr
-
H-'l
.l.vri
^
7
;!
1
Ar H!)'j
'
k
valency
each
number
tin-
the series of fli-ments between neon
and argon, omitting potassium, are Element
-
^'
ThecorreHjH>nding valencies
In
chang*
:
NumW in nutir Tttiil
as
cast
1
of
the
:i
iliavimmn
{
i-leetropo.siti\i'
and
constant, and th- rli-m-nt^ become olcctroncgat.ive. as the |-ctrop>^iti\r Huccessively more electrtnegati\e and \t^\ valencies
mass of
The
tht*
atoms
numbers
is
increases.
rl-eiropositive n-pivsrnt th and the electronegative valencies of the different j-hi-mical elements are not in every case verified by taets. They are rather to be
purporting
to
j
taken as representing a law which >ummari/.e^ a general Thus so far as the writer is aware no compound ot
tendency.
sodium
due
is
known
in
which
to the fact that it
is
it
is
difficult,
heptavalrnt. to get
elements to combine with one another. elements are more adaptable in this res
HUN
is
probably
stronglv eleetropositive
Tbe poet,
electronegative
and they furnish i'
instance the following compounds electropositive and
which
exhibit the
maximum
electronegati ve valencies:
Electropositive valency Electronegative
SiCl 4
PCI,
SCL
SiH 4
PHs
SEU
HCl'
PNa3
Na*S
KaCl
In each case the sum of the valencies
is
Cl>0
constant and equal to
eight.
The numerical
relations obtained above are to be taken as
than as indicating the actual number of electrons which the atoms of the respective chemical elements If the electrons were contain. arranged in shells instead of in of conditions the stability would lead to results of the rings, rather
illustrative
same general character but different.
properties
the numerical values might be quite In order that the periodicity in the electrochemical and the observed changes in the electropositive and the
electronegative valency should be found, all that is necessary is that there should be a sudden, increase in the stability of the etc. electrons in atoms which have certain numbers 2 ly s
N N N ,
,
of electrons and a gradual diminution as we pass from any one For example, an atom of these numbers to the next higher. which contains N'., + n electrons would lose n electrons rather easily,
but
nitude
to
it
would require
forces of a different order of
+ 1 electrons. Its On the other hand it
dislodge w
would thus be
?/.
mag-
electropositive valency would have a tendency
combine with any number of electrons until the total number Its electronegative valency would therefore became (Mjtia.1 to N r same be -Y -(A a + '/0. Tlle sum of tlie two valencies has the
to
.
lt
:l
the elements with numbers of electrons between In discussing the question is equal to 9 -N*. A', and N.A and the effect of va-leney it, is not, necessary to consider specifically
value
for
all
N
are of positive or negative charge when electrons This effect will change ,vmov<*d Irom, or added to, a given atom. It is also electrons. of or excess t.he deficiency rro-ularly \vil.h of valency is one of observe*, that the question n.rrsssiry to which is uncharged as a <M,uilibri'imi in a system, the molecule, atoms The Affect, of the. total charge on the individual whole* to a very limited iu come , different, will therefore only )( near together. It would be rather <.xt,rnt sincv they an. so very the possibility of the lomzation if w> wer, discussing
of
t.he
,
l
exWss
m
of a
iv<'ii
atom.
elements which lie monovalent elements between the very ronegative like fluorine and the next higher strongly electropositive monovalent element, e.g. sodium, are the gases He, Xe, Ar, Kr and Xe which have no chemical affinity; whereas those which lie between elements like manganese, which show some analogy It
is
worth
the
while remarking that strongly elect
with the chlorine group, and the corresponding element like to sodium, exhibit the highest electrocopper, which is analogous This is shown by the of the elements. valency of any positive
oxy-compounds and various complex amirio-compounds of the metals of the iron, palladium and platinum groups. It
is
convenient to have a
name
tendency of an electron to leave an cussed above. Following Sir J. J.
for
the
measure of the
atom which has been disThomson we shall refer to
this as the electronic pressure of the
atom.
Chemical Combination. It has been pointed out that
Lord Kelvin,
"Aepinus atomized" (Phil. Mag.
vol.
in his
in. p. 257,
paper entitled was the
190:2),
to suggest that the chemical atom consists of a sphere of uniform positive electrification containing negative electrons of much smaller dimensions embedded in it. In that paper lie dis-
first
cusses the forces which will
come
into
play
when some of
the
Considering the simpler types of atoms are brought together. of which contains that all, only one electron, it, simplest type is
evident that two such atoms will exert no mutual force
spheres spheres
lie
A
sphere of sphere of
entirely outside one another.
penetrates another
.#,
If,
if
the
however, one of the
then, since part
of the positive
B lies inside that of A, the repulsion of the positive B by that of A will be less than the attraction of the
negative electron at the centre of A. Thus the two spheres will It is clear, however, thai the negative attract one another. electron at the centre of each sphere will still be in equilibrium there until the centre of one sphere lies within the circumference If both the spheres are equal, the mutual repulsions of the negative electrons then exceed their attractions bv the .the line positive spheres; the electrons therefore move,
of the other.
along
joining the centres of the two atoms so as to centres, but remain always within the atom.
lie
We
outside
thus
the
<>vt
n,
HTKUUTUKE OF THE ATOM
571
neutral (uncharged) combination of the two atoms which may be regarded as the simplest of type elementary molecule. If the sizes of the spheres are unequal, the story is somewhat different A calculation of the forces shows* that, whereas the electron in the smaller sphere moves outwards along the line of centres, that in the larger sphere moves towards the smaller. This movement towards the smaller sphere is gradual at first, but at a certain position the equilibrium becomes unstable and the electron proper to the large sphere makes a sudden jump into the smaller sphere.
In the case in which the
ratio of the radii of spheres is three to
when the distance between the centres and 27 times the radius of the smaller sphere.
one, this instability occurs lies
between
2'6
After this has occurred both electrons remain inside the smaller sphere even if the. larger sphere is taken away. We thus get a case which
is
analogous to the formation of a neutral compound when this molecule
molecule, and, as in the case of electrolytes, is
subsequently broken up one atom
is
positively
and the other
negatively charged.
We
have seen that the important differences between the all probability determined by the differ-
chemical elements are in ence in the
number
of electrons in the
atom rather than by the
We
have every reason difference in size of the positive spheres. to believe that the atoms of most of the elements contain a considerable
number
of electrons, so that there is
no evident reason
why they should behave in the same way as the extremely simply constituted atoms just now under discussion. It seems clear, however, on general grounds that if the atoms contain a large number of electrons they will attract one another whether they are like or unlike, and so will tend to coalesce into groups of more than one
atom.
That the
more or
between uncharged atoms will in general from the fact that the electrons are Under the influence of the electric field due
forces
be attractive appears less mobile.
to follow
n neighbouring atom, these will arrange themselves so that their potential energy is diminished. It is to be remembered that are electrically neutral there will be intense although the atoms of force in their immediate neighbourhoods owing to the to
fields
and different geometrical distribution of the positive atoms The attraction between uncharged electricity.
*
Cf. Kelvin, loe. tit.
negative is
similar
between an uncharged conductor and a charged sphere Thin effect will take place whether the alums are .similar or disto that
similar.
The combination
of two atoms in this wav will, in general he a transference of electrons from one to the accompanied l>y other The way in which this takes place is must conveniently described in terms of the idea of electronic pressure. Consider the atom which possesses N% -f I electrons. Let us denote this hy A
and suppose it to be brought into immediate juxtaposition with the We have seen that the atom By which contains -V electrons. electronic pressure of the alum A is very hi^'h while that of //is 1
;,
In other words less work will be required to drag an A than out of IL Fnder these circumstances we
very low.
electron out of
should expect an electron to pass from A t< ft: so that in the compound thus formed A will carry one electronic unit of positive and B one electronic unit, of iterative electricity. After this transference has taken place A will contain .V, electrons so that its electronic pressure will be very low. Thus no m*re electrons will
pass over to B, especially
would then have
r
A
;i
-f
1
another were to be transferred H electrons, and this numU-r cirn's]Hin
an atom with a very high electronic pressure. Tbej-e is reason why no further transferenee should tak' plaee, that the transference of the
between the atoms which tends If another of the
B
to stop
another
nl that
produce an electric
am
is
field
furtluT exchange.
were brought into th- neighbourhood would not he able transfer another
.1
it
It,
despite
T
its
if"
it
r
I
}
is
Tin, ad ton would
particularly likely to shoot itV one elect run. certainly result in this cast; since it would be field
:t
own lii^h electronic pre^ure. For would then have A + elctrons. an arrangement \\hich
electron to
did so
atom of
molecule Alt
electron
first
si ill
h.-lp..d
B
from
to A.
element and molecule
AB
tendency
to
B
Thus
is
A
by the electric
murrains? ,-ieetropoMtive a monovalent electron. --a m,. element. The is
clearly a
a fully saturated nmL-ciile and uill >!iou no enter into further cotnbinatmn with am ,,tli,. r is
elements.
Now
consider the interaction between an
^ + 2 electrons N 2 electrons. 2
that oi D,
and an atom
The we should
D
atom r
of another element
electronic pressure of
<<
lM-in-
,-ontainin'.-
cont-iinin-
hi&.v than '
exn<^.t,
nn
*,i/^.f
..,,..
*
.
.
,
,,
.
r,
STKUU'JL'UKJbJ UJj
G
1
AJLUM
TJtlJiJ
+ 1 electrons and one unit of positive charge, have JV3 -1 electrons and one unit of negative charge. Now if C and D were uncharged their electronic pressures would be respectively higher and lower than before the then have N,2
will
D
whilst
will
transference. to pass from
Thus there will be a tendency for another electron Cto D, despite the opposition of the electric field esta-
blished by the transference of the first electron. place there will be no further transference, since
N* and
N
electrons respectively carry an additional electron would be opposed 3
;
If this takes
C and
D
now
so that the transference of
by the discontinuities in the
electronic pressure as well as by the electric field produced by the two electrons already displaced from C to D. The compound CD is thus completely saturated. If C and B were placed in contact,
an electron would evidently
at once be displaced from
C
to
B\ but
despite the increase in the electronic pressure of C thus caused no further transference could take place. For this would involve increasing the
number
of
jB's
electrons to
with a very high electronic pressure. be assisted by an intense interatomic
^H-
1,
an arrangement
Since this pressure would electric field, the two com-
bined would effectually stop any tendency for a second electron to If, however, an uncharged atom of B were placed in the
go to B.
neighbourhood of CB, this restriction would not occur. This atom would huve a low electronic pressure and would readily abstract an additional electron from C.
Thus the compound molecule C
3
which has preBy would have no tendency and saturated be would this coded, fully In a similar way the element t,o combine with any other atoms. similar reasoning to that
would be formed.
I)
would form the
combine with A. electropositive and It is to
molecule fully saturated
C and D
D
DA
2
when allowed
are typical divalent elements,
C
to
being
electronegative.
be borne in mind that there is a difference between liberated when G combines with B to form CB and
the energies
when CB combines with
B
to form CB,.
The formation
of GB,
from CB is opposed to some extent by a preexisting electric field, this whereas that of CB from C and B is nofc. In some cases formathe be sufficient to prevent ready restraining influence may action may stop at the CB the and CB 2 the of tion compound valencies furnish the elements with all ,
stage.
Nearly an effect of
f.v'imt)les of
high
this character.
The intermediate stages
are
known
compounds. A number of examples are elements of different chemical families, in the
as unsiLt united
given, selected fro in table.
accompanying
Maximum
Element
Chlorine
CompnumiH
valency
n,O,, no,,, -
4-7
t
-i
nri
+!
s.,n.j,
-j
n,,s
Ph<sih(ruN
+">
i^'lj, PC'!,
(..'arhon
+4
sulphur 1
'!,<
>,
(?l.,O-
'
sn,"s
SIM,,
si,,
'
1
Iron
t
% 5
.
Copper
-f-
PotaHsituu
+1
to be observed, as
we have already
se**n
indicated by a general way, that the maximum valency of aii element depends on whether it occurs in the compound as an Thus consider the electropositive or an electronegative element. It
is
i*
this theory in
r element with A
;!
2 electrons.
We
have se-n that this can only
take up two additional electrons be fop reaching the unstable Two is therefore the maximum valency *f the element, .stage.
when
occurs us the electronegative part of the compound. On '1 X electrons before it the other hand it can part with *V it
:
,
reaches the other unstable configuration of its will
;
so that this
maximum
electropositive valency. this property is very well
show that
A glance
is
at
illustrated
the value the table
by the com-
i* obvious from It pounds of chlorine, sulphur and phosphorus. these considerations that the terms elect i\e and electroroposit
negative are merely relative.
An
element
maybe
electropositive
and electronegative to another-; in thU IMM- it lose an electron when combined with the first element and one when combined with the second.
to one element
The
will i^ain
function of the electrons in the
elementary molecule* such CL, Br,, L, etc. is a matter of great interest. have seen that similar atoms will show an attraction for one another, and they may be in equilibrium in pairs without the
as
H
2}
(X,
N
2
,
W
transference of an electron from one of the atoms to another.
It
not however certain that such a state of equilibrium would be stable without the transference of one or more electrons. Suppose is
that the uncharged atoms contained
N
electrons,
where
N
is
a
number lying between N2 and N3 We have seen that such an atom has a tendency both to emit, and to combine with, one or .
more
electrons, the precise
number depending on the value of N. (N 1) (JV-f 1) would be
conceivable that the arrangement more stable than the arrangement NN.
It
is
In the former case one
atom would
carry a positive and the other a negative charge, whilst in the latter case both atoms would be neutral. On the
whole the evidence seems to be in favour of the elementary diatomic molecules containing oppositely charged as opposed to neutral atoms. The facts which bear on this question may be
summarized as
briefly
follows
:
Walden has found that in certain solvents the elementary Br2 and L can be electrolysed and equal quantities of
(1)
solutes
bromine or iodine are liberated
at each electrode.
In structural chemistry the bonds which combine like atoms are treated as being in every way similar to those which (2)
hold together unlike atoms. inert gases, helium, neon, etc. which do not enter into combination with other elements have rnonatomic molecules. the same time the metals have monatomic vapours for the most (3)
The
At
to combination with one another, although part and are averse with electronegative elements. combine
readily
they
and dispersion of a substance to whether the atoms occurred might differ considerably according There is no evidence of any in oppositely charged pairs or not. of elementhe between optical properties well-marked difference of those and a as compounds. class, substances, considered (4)
The molecular
refraction
tary
which unite we accept the view that the chemical bonds involve the transference of different atoms of the same element of possible isomeric forms an electron, it follows that the number number indicated by the than is substances If
of
greater
many
ordinary structural chemistry.
Take
for instance ethyl chloride
of chlorine obtained by substituting one atom structure The ethane in C,H 6 of hydrogen fen-one of the atoms
C H
Cl
This
is
.
of ethane thesis
would be represented according to the present hypo-
by
+ 11 +
H_ _
'
c
+11
+
_
i
-a-
!
-
1 1
+
!
+H
+ ii
while that of ethyl chloride would be either
11
H+
H-f
H+
H+
H+
-
+
-r
4-
H
('
+H
-ci
the chlorine
entered
+
;
1
-C-
C+
11
+
-
-- ('-
7
+
+ii
-n
-
II
+
combination
with the according carbon atom containing a positive unit of electric charge or not. It is obvious that ordinary structural chemistry dors not admit of as
into
this kind, and its position in this respect is facts. the Although attempts have been mudr to supported by as such ethyl chloridr in different wavs, so as to compounds prepare
any dissymmetry of
isolate different isomeric forms if they existed,
been unsuccessful.
thry have invariably
The ordinary
structural formulae appr.-ir to }>( of all tin* ditfrivnt is>mrric fnrms of account, taking quite capable of carbon compound which can be prepared. This, however, is no
very conclusive evidence against the view abovr as to thr nature of the bonds, for in most cases it is clear that onr of thr i\vo possible forms would b(. much more stable than the othrr; so thateven if the less stable form were produced at first, it would imme1
1
We should r\prrt thr number diately be changed into the other. of structurally possible compounds always to be greater than the number of those which can be actually isolated. Il is clear that, the number of these possible electrical isomers will increase rapidly with the number of carbon atoms in the compound. This method of definite physical
They represent the atoms. The bonds tive electron
and
looking at
meaning
to the,
chemical
combination gives a bonds of structural chemistry.
directions of the electric fields
are
to be regarded
as starting ending on a positive or negative
bond extending between two points may positive
or negative.
This quality
is
beUeen on
a
the
nega-
A charge. therefore he cither
not taken account, of
in
OAIVU \jjLUJtui
j-.ii.iy
ur THE ATOM
577
The position of the end of the ordinary structural chemistry. bond where the negative charge is may conveniently be indicated by an arrow pointing towards it. This enables us to omit the cumbersome positive and negative These would then become
H
H
H
H
1
J
I
H_^0-*C* H
may
H J
H-^C ^C^-H
ftH H The
signs in the formulae on p. 576.
J Cl
H *C
f
t
H
i
123
H 1
^C^
H
J
Ji
method of drawing structural formulae be illustrated by the successive chlorine substituted
application of this
also
methanes, as follows
:
H
Cl
i
H
H
^G^
H
t
*C-
Cl
H
H
t
t
H
t
H 4
5
Cl
Cl
t Cl -
H
C t
H
^Cl
CH
t
C
^Cl
J 01
that the total positive charge on an atom in electronic units is obtained by subtracting the number of bonds It,
is
clear
which point towards the atom from the number which point away from it. Thus the carbon atom in each of the successive com- - 2, 0, + 2, + 4 electronic pounds 1 to 5 will be charged with 4, The formulae have been drawn, units of respectively. electricity
of course, on the supposition that each hydrogen atom carries one chlorine atom one unit of negaunit of charge and each positive
tive charge.
the valency electrons existing in a compound molecule are saturated, there will still be a considerable external an external as a neutral electric doublet gives rise to
Even when
field, just
all
field.
These
forces
would not be so intense as those which come
into play during chemical combination proper, but they are probably capable of accounting for such phenomena as hydration, cohesion,
surface
tension,
latent
heat
and
the
properties
arising
from
molecular forces generally.
the
In considering the nature of the bonds connecting atoms of same element it is interesting to observe that the phenomenon
is only shown to any considerable degree by two and carbon silicon, and that both these elements lie elements, in the centre of their respective series in the periodic exactly
of self-combination
table.
Their electrochemical
properties
are
neither
therefore
markedly electropositive nor electronegative, and they ought to show an almost equal tendency to enter into combination in either This is just the kind of condition of the atom that we should expect would give rise to self-combination, on the view that we have adopted as to the mechanism of this action. For sense.
such an atom would be almost equally stable whether the positive or the negative end of a chemical bond.
it
formed
would show a greater degree of adaptability to the effect of the remaining groups with which the atoms were combined than would atoms which were not so constituted. If this view of chemical combination
is
the
It
correct one,
it,
becomes a matter of great importance to determine the sign and magnitude of the charge carried by each atom in different compounds. The most generally applicable method is, of course, the In the case where one atom is deposited at the electrolytic one. cathode since, so
presumably carries a positive charge in the compound .far as the writer is aware, no cases are known where the
it
;
is capable of reversing the polarity of the ions formed by a given electrolyte.
solvent
The phenomenon shedding some
of magnetism also seems to be capable of on this phenomenon. Townsend has shown light
that the magnetic permeability of solutions of salts of iron containing the same amount of iron has the same value for all ferric salts; it also has the
same
value, but one
which
is
different from
the preceding, for all ferrous salts, whilst for the ferricyanides it has uniformly the value zero, i.e. the ferricyanidcs arc non-
magnetic.
pounds
is
These results show that the magnet-ism of iron coman atomic property but indicate that, it depends on
the electric charge carried by the iron atom. reasons we believe that the iron atom in ferric
From
chemical
salts carries three
units of positive electricity, whereas in the ferrous salts two and in the only ferricyanides it occurs in the
it
contains
electronegative
the molecule.
part of
Thus by making experiments on the
magnetic properties of any iron salt we could tell what the electric charge in the iron atom in the molecule was.
Thomson
suggests that considerations of this kind may account the magnetic properties of elementary It is well oxygen. known that both oxygen and ozone are strongly paramagnetic. On the view of atomic combination above, the oxygen molecule
for
may be
expected to contain one atom which
is positively charged. of oxygen it probably functions as the electronegative constituent. Even when combined with
Now
in all the
known compounds
the strongly electronegative element chlorine, oxygen appears to be electronegative. This is indicated by the high valency of chlorine in the higher oxides C10 2 and C1 2 7; and by the oxide CL2 is being to some extent acid forming. Thus elementary oxygen
the only form in which an electropositive oxygen atom appears to are the only exist, and this may explain why oxygen and ozone
substances in which oxygen
is
paramagnetic.
Attention has recently been called by K. G. Falk and J. M. Nelson* to a number of facts in structural chemistry which seem the to support Thomson's view of directed valencies. Among these following may be mentioned :
(1)
A
hydrocarbon which
is
thought to have the constitution
the electric current when in solution. As JCJ((JH s ) 3 ]o conducts the electric current does not appear to produce permanent Chemical change the ions have been thought to be -C(C 6 5 )3
H
and
4-
(2)
(CH
5 ) :f
.
The symmetrical
<;o,,H(OH. ) / ,-CH a (
saturated dicarboxylic acids of the type show definite differences in CO
-(CH
2 )p
a
H
are compared as a group with their physical properties when they - (CH^CO.JL In the rather similar acids COJ3(CH 2 )P the.
former case the directed valencies
may be
arranged symmetrically
whereas in the latter they cannot. (3)
The
quantitative yields
of the various isomers formed
when the asymmetrical hydrocarbons *
Proc. Amer. Chem.
of
8oc. vol. xxxii. p.
the
ethylene series
1637 (1910).
combine with the haloid acids an* readily interpreted on
this view,
as also are the chemical properties of tin* dia/.o-eoiiipminds and number of other .substances.
a,
It is necessary to add that most of these facts have been accounted for by chemists in other ways which seem fairly satisfactory.
Reviewing the whole question broadly it seems quite likely that in a great number of eases of ehemieal combination transference of electrons between the atoms will not occur. There seems to be
little
doubt that the forces between uncharged atoms
can be sufficiently great to account for the energy of chemical iMhiitencss of valency combination in a great number of cases.
can also be accounted for in this way. For instance, if the neutral carbon atom possesses four electrons arranged at the corners of a regular tetrahedron, the directions of maximum electric* intensity
in the field of the neutral
atom
will
be along lines possessing
a.
The safest course to adopt at symmetry. to be that of restricting the would appear present interpretation of valency bonds as representing electronic transference to those
similar tetrahedral
cases only in which the possibility of electrolytic dissociation has
been demonstrated.
The Structure of
The
the Positive
/sArfnV////.
foregoing, necessarily brief, review of ehemieal phenomena.
shows that there
a very close correspondence between the elements and those required lv tin- atms properties considered by Thomson. It, is not, likely that the hypothesis of
is
the
of a sphere of positive electrification of a uniform volume density is essential in order to arrive at conclusions of (he same general
character as those which have been indicated. It is probable that somewhat similar conclusions about questions of stability would hold if the volume density of the positive* electrification," instead of being uniform,
the other hand, a definite
were greatest
we have
atom out of
seen
at,
the.
that, it
centre of the sphere. is
impossible
to
<
)n
construct,
indefinitely small elements of positive
negative electrification acting on each other according
to
and the
UJj
classical
ThLJE
ATOM
581
laws of electrodynamics. It is necessary to introduce else in order to account for the actual size of the
something atoms.
The sphere of continuously distributed positive electrification has the merit of lending itself readily to calculation and, as we have seen,
it gives a satisfactory account of many of the properties of the chemical elements. On the other hand there are some
rather striking properties which it leaves unexplained, at any rate in its simplest form. Consider for example the phenomenon transformation, The chemical and spectroscopic of the radioactive elements radium, thorium typical properties and uranium are not sharply different from those of the elements
of radioactive
which do not exhibit
radioactivity; so that
it
does not seem likely
that they have a constitution radically different from that of the But we know that the atoms of the radioactive elements others. are continually emitting atoms of helium and turning into other elements of lower atomic weight. Thus an emission of part of the itself positive sphere To account for these
a possibility which has to be contemplated. phenomena Thomson has suggested that the is
atoms of the elements of higher atomic weight are made up combinations of sub-spheres like that which may be supposed
of to
constitute the helium atom; but on such a view it is not easy to see why the atoms should possess a different order of stability else from that of their compounds, without importing something consideration some deserves which Another into the theory. point
an upper limit to the weights of atoms. bismuth (208) are All the elements of higher atomic weight than of this The striking feature radioactive and therefore unstable.
is
the fact that there
is
it appears to affect the positively charged part instability is that the negatively charged well as as, if not more than, the of atom,
constituents.
As
it
no account of the mass stands, the positive sphere gives of of estimating the number Unless all the methods
of the atom. electrons
which l
is
a contingency the atoms are entirely misleading, in view of the excellent agreement given very unlikely the whole of the mass methods, in
v entirely different
of matter
Now
must belong
the electromagnetic
and Thomson
is
practically o it. to the positively electrified parts Kelvin of inertia of the positive spheres
negligible compared
with that of a single electron,
so that the greater part of the
by
mass
is
entirely unaccounted for
this theory.
The following hypothesis appears to otter a possibility of explaining these facts and at the same time of retaining the important features of the positive sphere. Suppose that the positive electricity, instead of being uniformly distributed, and mass Af, where the form of electrons whose charge is
E
is
E
in is
very small compared with the numerical value of the charge e of Since the electromagnetic mass of such the negative electrons.
R
is comparable with proportional to E'-jR, where were small enough the positive their volume, it is clear that if electricity could be made to carry most of the inertia of the atom.
particles
is
l
R
If e
= vE,
then r/R
ve/ui
the negative electrons.
+ E/M, where
If the
the small letters refer to
hydrogen atom contains only one
negative electron, thj3n E/M is the value of the corresponding would be quantity for the hydrogen atom in electrolysis. /
R
in the proportion of something comparable with greater than 1000 v, so that the positive electron would have to be confined to a much smaller space even than, the negative. This would lead to
the difficulty about the definiteness of the. atom already alluded to, if the law of force between these positive electrons were that of the inverse square. Let us suppose that at very small distances this law does not hold but is replaced by something more complicated, let us say
+ where
>p > l. c
p.1
l
a
1)
;--,,-
At very
+
c
,*;
small distances the third term would
give a repulsion and keep the positive electrons from /joining together, and the first term would give the usual law of force
The middle term would cause the positive at large distances. electrons to attract one another at certain distances. This would
make them aggregate
into clusters which,
if
the constants were
of suitable magnitude, would be of dimensions comparable with that of the atoms. The positive electrons would be regularly distributed inside so that such clusters would behave very much like
a continuous distribution of positive electrification, provided
were
K
sufficiently small.
On this hypothesis there would be a definite positive atom capable of existing without the presence of a negative electron.
Such atoms would be able
2
to coalesce if the a/r term were neutralized by the presence of a negative electron, but not otherwise. In this way more complicated atoms could build up, and the conditions for the equilibrium of the electrons in them would be similar to those of Thomson's theory. There would be a limit
atoms which could be built up in this way, because it would be difficult to arrange the negative as to prevent a whole unit of positive electricity (vE)
to the size of
with the larger atoms electrons so
from ever becoming unstable. It will be urged that there
is no experimental evidence in favour of the existence of particles carrying a charge less than e. That is true, but if E/e is very small it is questionable whether
any experiments which have been made would be capable of detecting their existence. Their mutual attractions would prevent an electron from getting away from an atom with any considerable
number
Another objection is that all known chemical either neutral or carry a charge which is an integral e. It may be that the elementary positive atom multiple of has a also charge which is equal to an integral multiple of e,
atoms
of them.
arc
1
somehow makes the stable systems neutral. be admitted that these suggestions are not more than the hypothesis that the atoms are provided with
or that the law of force It
must at
artificial
a
least
sphere of positive electrification just sufficient to neutralize the
eleetrolis present. ( Collisions between systems of this kind would be rather different from those between atoms made up of positive electricity of uniform
density, and
might be expected of sharply deflected
percentage Rutherford and (Jeiger.
We
shall not.
to give rise to a relatively high a rays, such as was found by
pursue this subject, further.
The
deflexions of
X
the a rays through large angles and the scattering of rays by in Chapter XIX, agree much light: atoms, which were considered the positive electricity in the eoneentrated in a minute region of it than with the
better with Rutherford's view
atom
is
This position is made considerations brought forward in the two be noticed that if the. linear dimensions o'J
uniform sphere of positive stronger
still
next, seet ions.
by It,
the,
will
that,
electrification.
the nucleus are small enough the whole mass of the atom of eleetroinainietie ori trill.
may
Vj
The Radioactive Elements and the Periodic Law.
The study
of
the
chemical
the
of
properties
radioactive
elements* has brought to light a number of facts of the highest importance which bear on the relation between the chemical of the elements
properties
and their atomic weights.
Soddy
pointed out that when a radioactive element A was converted into a second element B with accompanying emission of an a ray,
then the chemical properties of B would be those of an element in the next column but one before A in the periodic table. Russell -fshowed that if a /? ray were emitted instead of an a ray the new
element would be found in the next column beyond that of the element. Thus, to illustrate the case by considering one chemical property, the emission of an a ray diminishes particular
first
the electropositive valency by two, whereas the emission of a /3 ray increases it by one. In the case of the thorium series for example,
we neglect branch
products, the successive changes are exhibited in the following table if
:
a
Th
-*.
IV
llayless Msth 1 -* Math II III II
The numerals underneath
a
a
/3
-^ Ra Th
-*
Th X
IV
-*-
a
a
Em
-*.
-+
VI
II
represent the
A
number
ft
H -. IV V
of the column in
the periodic table (or the value of the electropositive valency). The character of the change is indicated above the arrows.
But the matter goes further than this. In some eases an a ray change is followed by two successive changes in which an a ray is not expelled. The last element then occupies the same column in the periodic table as the original element. In such cases these two elements have, so far as can be ascertained, identical chemical properties,
and are incapable
of being
separated
methods.
There
identical.
These results are surprising at
is
by chemical
evidence! also that their ('mission spectra an* first,
sight since
{In-
atomic weights of the elements in question must differ by the atomic weight of helium approximately, or about: four units. In *
Soddy, Chem. News, vol. cvn. p. 97 49 (1913); K. Fajans, PJnjs. Kelts, Deutsch. Phys. Ges. vol. xv. p. 240 (1918). F.
vol. xiv. p. d.
t Chem. News, vol. cvn. p. 49 (1913). Russell and Rossi, Roy. fioc. Proc. A,
(11)11$)
;
G.
vol. xiv. pp.
vol. LXXXVII. p.
v.
Huvi-sy, r/n/s. Xrit.i.
l.'il,
130
(111
478 (1912).
1
3);
fV/7/.
their chemical properties, the radioactive properties of two such elements are quite different. These phenomena receive
contrast
t.<>
u plausible explanation on the view, advocated by Eutherford, that the atoms an/ built up of electrons revolving round a massive central nucleus of small dimensions, if we adopt the hypothesis that both the a and /? particles are ejected from the nucleus itself.
]<W the chemical and spectroscopic properties of such an atom will be determined almost entirely by the charge of the nucleus and hardly at all by its mass. The ejection of an a particle, which positively charged helium atom with twice the electronic charge, will diminish the total -positive valency of the nucleus by two, and the ejection of an electron will increase it by one. The is
a
resulting nuclei will be those appropriate to atoms whose positions in the periodic table are those actually found,
This position
Thus
evidence.
is
strengthened by a number of other lines of Thomson* has discovered that neon consists
J. J.
of two gaseous elements having the same spectrum and chemical di tiering in atomic weight. We have seen also that properties but.
the X-ray spectra of the elements are not accurately functions of the atomic weight but are accurately determined by successive
whole numbers.
atoms
The
sequence- of these
numbers
is
that of the
order of ascending atomic weight except for the elements anomalous. It seems positions in the periodic- table are
in
whse
u atomic numbers" as measures of the interpret the These facts do the carried by positively charged nucleus. charge Kelvin-Thomson on the obvious receive not any theory explanation
natural
to
of the atom.
Itulirtt
in Atoms. Thenrif of the Itehaviour of Electrons
This theorvh which frankly discards dynamical principles, has ahvadv been referred to in Chap. xvi. As its chief success so tar
has been
in
the explanation of the frequency of spectral lines which intimately on atomic structure,
and other phenomena
depend
be permissible to recapitulate here the chief assumptions of negative electrons It, is based on the idea on which it rests. nucleus of negligibly small revolving round a, massive positive it
may
*
Kmi. ,SW. /'w. A, vol. LXXXIX.
t
N
'Bohr, Vhil. Mnn. vol. xxvr.
p. 1
p. 1,
.
xxvn. p. 506 (1914). 470, 857 (1913), vol.
dimensions, the mutual force following the electrostatic law of the inverse square. Stability and dertniteness of frequency, which, as we have seen, are dynamically impossible with such a structure, are secured by the introduction of the two following hypotheses: (1)
That the stable motions an* those
momentum (2)
of the electrons
That radiation
is
is
emitted when there
possible configuration of the
thus omitted
is
assumed
to
which the angular A/^TT, and
for
an integral multiple of is
a,
change from one
elect-roils to another.
be monochromatic and
W
The its
radiation
frequency v
determined by the equation W^~hv, where h is Planck's W and W^ are the energies of the system in the two configurations under consideration.
to be
constant and
l
l
These assumptions lead immediately to the following formula which may be emitted by the simplest
for the possible frequencies
atom (that which contains a single electron when neutral):
where r2 and r are whole numbers. a
When
ru
=
2 and r
l
takes
The strong successive integral values this gives Balmer's series. point in favour of this theory is that it determines the absolute constant in the spectral series in terms of the fundamental constants
//?,
e
and
h,
and the agreement
within which these constants are known. of the constant
best values of factor
/a
jK"
e,
in Balmer's series
m
exact to the accuracy spectroscopie value
The
3'2!.M)
When
x HP.
the
and h are used the calculated value of the
in (29) is 3*26
known hydrogen
is
is
series
is
x 10 in
.
.By [Hitting r,
=
:>
another
obtained.
When
the central charge has twice the value just considered a single electron revolving round it., we should expert, to get the spectrum of a helium atom which had lost, both of its electrons (an a. particle) and subsequently combined with a single
and there
is
As a matter
electron.
of fact the series calculated for this system
include Pickering's* series arid certain series recently observed by Fowler f. None of these series have been obtained in the laboratory
except in the presence of helium. * f
Astroplnjs. Joiirn. vol. iv. p. 369 (1K%), vol. v. p. 92 UH97). Month. Not. Rni/. .J./r Knr vnl T.VYMT MOVJ\
UD
THE ATOM
587
Another
consequence of these assumptions is that the constant in Rydberg's formula for spectral series should have the same value for every element. The theory also gives the correct order of magnitude for the dimensions of atoms and for the work necessary to liberate ions from atoms collisions. It is also in
N
(}
by agreement with Whiddington's law connecting the atomic weights of the elements with the velocity of cathode rays which will excite the emission of characteristic X-rays from them.
In the last paper (Phil. Mag. modified this position somewhat.
vol.
xxvu.
p.
506) Bohr has
The
energies in the different are as empirically given treated configurations (stationary states) the laws of series and of the constant the value by spectral
K
is
determined from the condition that
for
very slow vibrations,
corresponding to large values of r, the frequencies must agree with those calculated from the classical electrodynamics. The condition (1) above then appears as a particular case restricted to circular orbits; in general it is replaced by Tn ^\nkw n where Tn is the mean kinetic energy of the electrons, and & n the frequency, ,
in the ?ith stationary state.
Whatever view may be taken
as to the character of the under-
no doubt that this theory has been lying assumptions there is in much more successful accounting quantitatively for the numerical of spectral lines than any relationships between the frequencies tried. been has Although the other method of attack which yet with dynamical ideas they are of a very The fact that they conflict character. simple and elementary to them, with dynamics does not appear to be a valid objection as there are a number of other phenomena, the temperature
assumptions
conflict
is inadequate as radiation for example, which show that dynamics behaviour. a basis for a complete explanation of atomic
CHAPTER
XXII
GRAVITATION Gen end
THK almost
Chtt
position of physical theories of gravitation to-day is as speculative as it lias been ever since the days of
Nevertheless we cannot atlord entirely to overlook tin* of the.* electron theory mi gmvitut ional aetitm, particularly bearing as we have found the theory to be capable of giving a fair account
Newton.
of the other
may
seem,
known
it is
Paradoxical though it. physical phenomena. chi*f olitlinihifs in the of th* quite likely that onr
of a physical theory of gravitation lie* in the <\irem' .simThe prospect of plicity of the known laws of gravitational action. addition to our knowledge of gravitation In e\p'rinn*nt is not
way
very hopeful on account of the .sniallness of tin* forces rowrrned. These are large enough, of course, when we ar* dfaling with th<*
enormous aggregations of niatler
fiuniliar to
are exceedingly small with masses which <*an
a^tnno!nv, hut they IM*
ci.utrollrd
in
th
i
laboratory.
The fundamental law
of gravitation,
\\hich
'nunciatcd
\\as
by Newton, may be expressed by means of th* iMjuation ,
T
.
dntdi/t' /''
where
F
the attractive forte l)et\veen
two material particles dm' at a distance apart, and /r is the constant which is to b'1 x 10 in r.;.s. units. gravitational equal is
whose masses are
dm and
/
*
This inverse square law of attraction
is
astronomical calculations of the orbits of
As
these are very exact
great precision.
it
the foundation of the
the heavenlv bodies.
follows that tin* index of
/
is
"2
with
The value nature of the
of k
is also very accurately independent of the attracting materials. This is proved by experiments
made with pendulums
constructed
These show that the weight surface, divided by its mass
from
of a body, at
different
any point
substances.
of the earth's
a constant independent of the
is
nature of the body. This law of proportionality between weight and mass is one of the most accurately known of the experimental
laws of physics.
The
inverse
electrostatic
square law at once suggests a relation with If neutral matter consists of oppositely
forces.
charged elements it attract one another
that two uncharged particles will the attraction between unlike elements
is clear
if
slightly exceeds the repulsion between like elements.
however,
a
well-known
electrical effects
which
difference
There
is,
between gravitational and
calls for consideration.
The
space inside
found to be completely shielded Iron the effects of an external field of electrical force, whereas an electrical conductor exerts no such shielding effect from the action
a closed
electrical
conductor
is
i
Nevertheless this is not a valid objection of gravitational force. It is to the view that the two forces are of the same nature. from arises that the electrostatic shielding qualitatively obvious the the opposite displacement of the two kinds of electricity along A inside. as to tend to annul the field conductor in such a
way
shows that the compensation is exact if the precise calculation the case of law of electrostatic force is the inverse square. In be to the effect of a conducting screen will gravitational action and bodies material two between increase the attraction
A
B
Fig. 57.
screen
,!,,,. ,
t
,
h(
.
C surround*
as
let the than to diminish it. For, a conductor he. in The fundamental property of figm ,.
Lot some of the mobility of some of its ultimate electrified parts. and others matter (Q) repelled. these (P) be attracted by neutral The effect of A on the screen will be to make the parts P move and the parts Q towards E. Thus the force urging B increased by the presence of the screen owing to a D and at attractive of an the formation repulsive layer at layer
D D
towards towards
is
(./,
It is evident that this additional force will be
E.
exceedingly In a similar way gravitating matter might give rise to a field of electric force inside a hollow conductor, but with any of gravity on electrons the probable hypothesis about the action minute.
resulting electric forces are too small to detect.
Comparison with Electrostatic Force*. Before proceeding further
it
desirable
is
to
emphasize the
smallness of gravitational attraction compared with the forces between the electrons composing the attracting matter. We may illustrate this by comparing the gravitational attraction between
two material particles of masses iiii and in with the electrical attraction between two particles of like masses, of which the first
(m^
solely of positive electrons and the second The magnitude of solely of negative electrons.
consists
consists
H
gravitational attraction is (H> x IO""" the number of positive electrons in
=
Let us
m
for the
{
and
//.,
and
c aiv
negative electrons
= 5'4
x 1(F
m...
charges on the electrons,
a
in
K
negative electrons in
m
m^nJr- dynes.
Thru if we have
take, as a lower limit for
KjM
the
Let A
('///,,)
the
r ,
number
be of
the respective
K.s.
units.
for the positive electron, the
value of this quantity for the hydrogen atom in elect rolysis.
We
then have J
._J..^
w,
M
3 x 10" eleetrostatie units.
So that the gravitational attraction
= Looked
4*1
x 10~ 40 times the electrostatic
attra,ctj"on.
at from this point of view it is rather surprising that the electrostatic forces do come so near to balancing one another and
it is
quite clear that
change in the law
it will
only be necessary to make a minute between electrons in order to modify it
of force
sufficiently to take account of gravitational attraction. The change which we shall have to make will be quite beyond anything which could be detected by direct on the
experiments
between
electric
forces
exerted
charges.
Law
of Force between Electrons.
Perhaps the simplest assumption that we can make in order to account for gravitational attraction
between two
electrons,
W V
1
not accurately equal to
may be
It
that the repulsive force
is
whose charges are
that the force
^
2
as
,
E E lt
respectively,
we supposed
in Chapter
is
i.
not determined entirely by the charges
is
of the electrons but depends to a slight extent on the geometrical configuration and the state of motion of the charges as well. In order to take this into account we may write the force between
P --W^F(E E^) 1
E
1 }
and
in
2?a
form -
the
1
}
where
F
denotes a
E
function of the configuration and state of motion of l and E.2 can only differ very slightly from unity. of .
F
The magnitude
P
and Q at a distance r us consider two material particles of consists from each other and such that J^i positive electrons and n negative electrons of charge e l and the correof charge Let.
P
E
{
l
sponding magnitudes
for
Q
are indicated
evident that the total repulsion between *) 4-
by the
P and
Q
suffix 2.
It is
is
N
l
N&
N
= - n^ and 2E.2 = -.&,. the two particles are uncharged, like sign are identical in of Lrt us also suppose that the electrons If
both bodies and that charges.
all
the electrons of whatever sign have equal
Then
E = E = E = -e = -e^~e l
2
l
and the repulsion becomes
f x {F(&\EJ + F(e A)-F(E^ - F(E.^\
...... (2).
477T"
we can make is that the electrons assumption that in whatever substance they of like sign arc in the same condition The
.simplest
occur.
In this case the functions
teristic
of the
F (E$i) = to unity,
It
electrons.
Since
jP (#!&/)
we may
F
be constants charac-
will all
follows
also
by symmetry that the functions are very nearlv equal
all
write
where a, b and c are very small constant quantities. The repulsion between two neutral particles is therefore equal, on this supposition, *V J E* This will be negative if 2o is greater than to 2c). (a + b
N
'
\
4>7rr
a
-f 6,
so that in order to account for gravitational attraction
it is
between two unlike charges necessary to suppose that the attraction is slightly greater than the repulsion between two like charges.
On
the further hypothesis that
electromagnetic
it will
all
the mass of the substances
be proportional to the numbers
.Y,
and
is
N.,,
which the particles contain, except in so far as the electromagnetic mass of an electron may vary somewhat in If l/i and If, are the masses of the particles different atoms. of the electrons
M
at
P
m
that of a negative electron, then
and
Q
respectively
HO that the repulsion
and
if
is
the mass of a positive and
^Y,
=
'
M,/ + in
and
*Y.
=
:
M./ 4
in,
becomes
J/,Jl/a 47T?*-
[
\(
E*
H
(f
fc_*,J ~
4- //A)"
j
Since the (juantity in brackets is a universal constant, subject to the hypotheses which we have made, the attraction accords with the Newtonian law.
The preceding
We
result is based on tin* hypothesis that the mass not appreciably different in different, substances. shall sec in the next paragraph that we cannot, be sure that-
this
is
of the electrons
is
the case.
The Atomic Wei
in
is
now
is
AY^/mv//.^
number of negative electrons about half the atomic weight, or, at all equal with the atomic As the value of comparable weight..
any atom
events,
tin*
fairly certain that the is
to
the negative electrons is very large it. follows that only a very small of the mass of an atom can arise from the proportion If the mass of electromagnetic mass of the negative electrons.
elm
for
the atoms
is
of
electromagnetic origin
entirely with the positive
it
must be associated almost
As we have
seen, there is a considerable body of evidence in favour of the view that the positive electricity is confined to a very minute region of the atom, so that practically the whole of the atomic mass will reside in this nucleus. The estimated linear dimensions of the nucleus are very small (about the same as the linear dimensions of the negative electron) so that considerable overlapping of the fields of the constituent positive electrons should theremay be expected. fore expect the mass of such a nucleus not to be electromagnetic electricity.
We
equal to the sum of the masses of the constituent positive electrons on account of the mutual interference of their electric fields. It is impossible to say precisely what difference would be expected to but it is likely to be only a small fraction of the
arise in this way, total mass.
The expulsion of a rays from the atoms of the radioactive elements shows that the nucleus is made up of separable parts. However, the a particles are atoms of helium and have very nearly times the mass of the lightest known positively charged Thus unless we accept the ion. particle, the positive hydrogen
four
than one kind of positive electron hypothesis that there is more we cannot regard the a particle as such an ultimate unit. If we take the hydrogen ion to be the positive electron, the value of its linear dimensions are about e/m which it possesses shows that 4 10~ of that of the nuclei. If such positive electrons were distributed fairly uniformly throughout the. volume of the nucleus influence on the the, overlapping of the fields would have little
The electromagnetic mass of an. atom electromagnetic mass. to the sum of the masses of the positive close be then wo uld very and the truth of the statement at the end of the pre....ctrons,
We
of cannot, however, be would follow. ee sure^ ceding paragraph constitution the of are this deduction as we completely ignorant of the hypothetical nucleus. It
that further information might be obtainmight be thought
atomic weights of the elements. able from a consideration of the with sufficient accuracy -as treated be If the mass of an atom can
and negative of the masses of the constituent positive as can be. of atom regarded simply electrons and if each kind to units then, such of number "Mitral combination of a given a niv the
sum
the name degree of aecuntey, the atomic weights of ?he different
be proj>ortional to whole numbers. This is on the of one is kind and on*- kind of only positive // an* different If there kinds electron. of electrons of negative masses w lt w a etc., then the masses of the atoms will In* of the form
elements
will
supposition that there
,
MI from MI
-f
M:i-f
a
zero.
...
It
is
where well
e/j,
known
etc.
that
an* integral numbers starting the atomic weights of the
elements exhibit an approximation to whole* numbers which cannot be fortuitous*; but the deviations from integrals, ur from the linear law just referred to, cannoL now be regarded as likely to
flrtroma#nt*tie mass of the indiSoddy and others (S*M p. ">iS.*{) on the chemical properties of the radioaefhv eliMnrnts has shown that elementary bodies having identical chemical properties and
arise
from modifications of
The work
vidual electrons.
different
tin*
of
atomic weights frequently coexist. It is therefore likely so-called chemical elements an* mixtures of many
that a great
atoms having the same chemical properties but differing in mass. The experimentally determined atomic weight is, on such a view, the average of the weights of the constituent atoms and depends on the relative proportion in which they are present. There
is
connection.
another
point,
The two
which deserves consideration
in
this
charged particles with which we are familiar are the hydrogen ion and the ^ particle. According to the views of Rutherford and Bohr both of these lightest
positively
The a particle has twice the actual charge* and half the specific charge approximately of the ion. It is (e/m) hydrogen quite possible that tin* a particle of is made four ions cemrnted into a nucleus up hydrogen
particles consist only of positive* electricity.
means
of two negative electrons, but
has not, so
it
by been shown
far.
that the a particle can be broken up into anything smaller; so that we shall examine tentatively the consequences of supposing that it is a fundamental element of positive clectricil \
independent
of the hydrogen ion. In that case we ha\e to deal with two posi tive electrons, one of which has associated \\ith a iMVfn
charge
approximately twice
sis
much mass
as the
i,th,
T
On
.
p.
Ml
\\e
considered gravitational attraction from the point of \ iew that it arose from a minute lack of compensation between the a! tractions and repulsions of the ultimate electric charges. By supposing the *
Strutt, Phil.
J/tfi/. [(>]
vol.
r.
p. :u
1
t
liini
i.
additional forces to be determined
by the electric charges we arrived at equation in (3) agreement with the Newtonian law of attraction. This was on the hypothesis that the value of e/m for all the electrons of given sign was the same. If some of the positive electrons, let us say the a particles, have a different value of ejm from the others, hydrogen ions, the same
gravitational
will lead
to the conclusion that the argument weights of different elements will not be proportional to their masses if their respective atoms contain hydrogen ions and a particles in different proportions. Thus if we are to retain the law of proportionality between mass and weight, which has always been found to hold with high
accuracy, it is necessary to suppose that the modification of the law of force does not depend simply on the charge of the electrons. It must depend quite directly on the mass which is associated
with a given ultimate element of electric charge. The necessary change in the law of force will be considered in the next section.
be remembered that the necessity for considering this different question depends entirely on the hypothesis that there are There is no electrons having specific values of ejm. It is to
positive
any such hypothesis since, as we have seen, the nucleus of the helium atom may prove that quite possible to consist of four hydrogen ions held together by two negative it
logical necessity for
is
electrons.
Forces between Charges Modified by Mass.
We
shall
now
consider the consequences of the hypothesis that ultimate elements of electric charge is not
the force between two (ic'termined solely
by the quantity
cements but depends to a slight The results of tluTc.with as well. contradiction so far as the
concerned.
Such
We
of electricity present in the extent on the mass associated this hypothesis are free from
which have been reviewed are seems a natural one from another
facts
a hypothesis
the
form of the electro-
ordinary might regard force between neutral systems the makes which of force law static ultimate elements of is attained when the v anish as an ideal which an infinitesimal have and devoid of relative motion
standpoint.
clvmri'
arc.
electron theory these conAccording to the Another in any actual material system. ditions are not realized
volume
density.
382
of describing very much the same position is to say that the electrostatic law of force represents what would oeeur if the
way
infinitesimal elements of charge
were at rest
the undisturbed
in
The classic aether, that is to say in aether devoid of energy. electrostatic law of force would then represent exactly what, would But in the immediate occur if the aether were homogeneous. element of electric charge then* neighbourhood of an actual disturbance immense an is going on and, as we have scon, the of the energy of (his disa measure is element mass of the
We
might therefore expect that tin* field of force emanating from such a region would be modified to an extent depending on the local disturbance; and also that the reaction, to turbance.
an external
of an element of charge thus situated would be Under these circumstances the force between
field,
similarly affected.
two positive electrons at a given distance apart will not necessarily be equal to the force between two negative electrons of equal charge separated by the same distance, on account, of the difference of the masses; neither will the attraction between unlike electrons necessarily be equal to the repulsion
between
like electrons carrying
equal charges.
Unfortunately there is an apparently fatal objection to any view which makes the modified law of force depend on what is practically nothing more than an alteration in the effective charge
produced by the local energy or mass. For the law of force will then involve thy product of two factors each of which depends on the state of only one of the two separate elements. Iff//// denotes the mass which
take 06
is
associated with the element of charge
we can
as a measure of the intensity of the local disturbance.
Using Taylor's Theorem we
see
that
the
/ T
'
*'
30'
two
/
(
dE and
between
force "
elements
t/<\
( '
' ( ( '
instead of being equal to
will
be
4-Trr"
dEde' 47JT-
J\
1
+a
dm'
.
4.
de
A
dM x --i c)E
.
4.
aA
Inn! ,
,
fU/ :
j .
, T
}
.
.
.........
(
* K
.J.
)
( iJ<J
where A and a are fundamental constants which must be equal, by symmetry, when the charges are of like sign; otherwise ibis is not necessarily the case. Keeping the capita,! letters for positively elements and the small letters for charged negative charges \i.
GRAVITATION
597
follows that, to the accuracy covered force between mixed elements
by the terms above, the (dE+de) and (3E +de ) is r
f
'
Assuming
that the condition for neutrality
+ a3m'}
.
.
.(5).
is
the force between neutral systems becomes
adm^AdW + admT) dM, dm,
AW
and dm' are fi
/j
except that
positive
............ (6).
but otherwise
arbitrary
'
-m = -TTT>
,
so that (6)
is
This
essentially positive.
corresponds to a repulsion and is incompatible with an attraction unless either A or a has an imaginary part.
we are to get an attraction it is necessary that the change to the masses should not enter as a separate factor for each of due the elements of charge concerned,- provided we are limited to If
The law
functions whose expansions contain only real constants. of force may, for instance, be of the form
3w
3w'\
dede'
L
+a
dm + "^
,
b
3m'
w*
/3m \
As before, a = 6 and c=/i by symmetry, using Stirling's Theorem. no longer any necessary for charges of the same sign, but there is have the same value constants the If between a and relation
g.
of the sign of the charges concerned, independently
i.e.
if
value for the then the expression above leads to the following A and + (3Fde) forces between two mixed systems (dE
^
1
^ 7rr "
o
(
+ (f)/y"
j-
M\(?\
M 4- ^mM4-
f fi\
/
fdM ^tf, om (3E+ de] (xrvdM + -^7
mA
I
UKAVITATION*
598
If the condition for neutrality
between neutral systems
is
y (?3f
-f
As g may be
?w
}
is
?K + fa-fih" + ce - 0,
(c *)/' 4-
cm')
negative, this agrees with the
the force
4*77T~
{
Newtonian
9
).
law.
According to formula (8) the terms involving a and r might make an important difference to the force exerted on an electron
by a gravitational field, but would have no effect on the attraction two neutral particles, which is determined entirely by the term
of
containing
(j.
The
introduction of this term
lent to attributing the Newtonian law of pair of elements of charge however small.
is
practically equiva-
mass attraction
to
each
Conditions for Neutrality.
The introduction
of gravitational forces complicates electrostatic The condition for electrical problems very considerably. equili4
brium in
any
any conductor
in
direction at
is
that the stream of positive electricity
any point should he equal
same
electricity in the
negative this condition
to the
direction at that point.
stream of In general
may be very complicated. If we may suppose thatonly the negative electrons are free to move, as the positive current, is zero the If we are also free negative current must be zero also. to neglect effects arising from differences of pressure of the electrons at different points, the condition will he satisfied if at every point,
the electric force on an electron tational force, the
exactly balanced bv the gravi-
is
average force on an electron thm thus be a potential gradient in a con-
resultant,
being zero. There will ductor in equilibrium
a
in
gravitational
field;
this
potential
however, too small to be detected experimentally. The way to define a neutral material is not an obvious
gradient
is,
particle
matter under th use circumstances.
We
might, for example, define a neutral material particle as one which exerted no fore,, on a negative electron; but equally valid reasons could he urged for making the vanishing of the force on a positive electron the condition fC If both these conditions were neutrality. satisfied there would bo
no
force
between two neutral
particles, so that, they an* not compatible with the existence of gravitational attraction. In accord-
ance with
the. p.rmKi/Wofi,
k
L,
,.*
4l,..
i....i
,-
,..,.
GRAVITATION
599
equal accelerations on both positive and negative electrons. Taking as an illustration the formulation of the forces on p. 592 the force on a of negative electron at B, due to a particle consisting
JVi positive
and
^ negative
electrons at A,
may be
written
and the acceleration
will be obtained on dividing this expression In a similar manner the acceleration of a positive electron
by m. at
J3,
due
to the material particle at A, is
n e(l+<0} 1
If these two accelerations are equal and
NI n
Under
these
the particle at
A
were equal.
_
a
bM
E
e
cm
,- -
must have
N^^
this ratio
still
for
N
and l Previously we supposed case we see that the be to this suppose
be neutral.
we
v
cMam .........................
suppositions to
If
if
......... (10).
= (bM + am) I(M + m). Since with Zc>a + b if a exceeds b.
c
M
exceeds m, this
is
compatible
attraction to a slight excess of the attributing gravitational the repulsion of attraction of the oppositely charged elements over the to led are we paradoxical result the elements with like charges of electrification amounts small that the addition of sufficiently increase the will of like sign to each of two neutral particles it. Consider, as attraction between them instead of diminishing
By
an example, the specification of the
bmi
The
discussing.
force
repulsive
is
particles
E
which we have just between two neutral
forces
*
\N,N
(I
+ a) + n,n 2 (! + &)- (#1*1 + N&) (I + c)].
47T?'"
If a small negative charge
1)mmu Wl4 .S Wl and n 2 >
nna ,lt,red. trality,
or
t(>
Taking
will
^- ^
is
become
and
the additional repulsion
the
first
both the particles given to n,
^ = n, 2
is
order of small quantities
+
^
^
n, will
and A, being
as the condition for neu-
GRAVITATION
600
were added, instead of negative would be
If equal
positive charges charges, the additional repulsion (HI Stta
Since 2c exceeds a
Thus
+w
+ b,
a
&/,) (a
either a
- c) jE c
or
3
a
/47rr
6-
c
.
must be negative.
at least one of the cases considered gives rise to an attrac-
This attraction will only be experienced when very small since unity is very largo comquantities of electricity are added, pared with a, I or c. Thus the neglected second order term, which tion.
will very soon corresponds to the usual electrostatic repulsion,
overcome the attractive forces under discussion. If the negative
electrons are alone capable of motion,
the
interior of a large solid conductor in equilibrium under its own gravitational field will contain a slight excess of negative electricity,
since the gravitational force on an electron has to be supported by If we neglect tho pressure that arising from the electric intensity. of the electrons, this condition will hold right
whatever the
total charge
on the conductor
up
may
be.
to the surface
Any
electric
or magnetic effects arising from this excess of negative charge would be very minute, even with conductors of tho magnitude of
the earth. General Considerations.
A
review of the preceding discussion shows that the electron theory is not in a position to make very definite assertions about the nature of gravitational attraction. It seems likely that tho Newtonian law of attraction between elements of matter is one
between elements of mass or confined energy and that is of a very fundamental character. It is doubtful if it can bo replaced by a modified law of electrostatic force between electrons or it.
elements of electric charge, unless the modified law includes the associated mass explicitly. Even so, the case does not
appear
very simple. A number of alternative possibilities could be eliminated if the acceleration of a negative electron in tho earth's field of gravitational force
could be determined experimentally.
If gravitational attraction
residue of the electrical forces,
is,
as
it
were, an uncompensatod
we should expect
it,
like all electrical
actions, to be propagated with the velocity of light.
Lorentz*,
UKAVITATION
who tion
t)01
has considered this question very fully, finds that if gravitain the same manner as electrical actions it will propagated
is
give rise to effects practically identical with those which follow from the usual Newtonian law. The differences which arise are too small to be detected from astronomical data and they are also incapable of accounting for the recognized irregularities in the
motions of the heavenly bodies.
The view that
gravitational attraction is an electrical effect at The definite form of it, that particles of quite old. uncharged matter contained equal and opposite charges and that the attraction between the unlike charges slightly exceeded the
bottom
is
repulsion between the like charges, seems to have been first put forward by Mossotti*. The application to electrons in atoms has
been considered by
J. J.
Thomson f.
The Relative Theory of Gravitational
The
Effects.
about the relation following speculations, due to Einstein J, are of considerable
between gravitation and some other phenomena,
discussion sets out from the empirical law that in a uniform field of gravitation all material bodies move with equal have seen that this law is likely to be of a very accelerations.
interest.
The
We
Bather similar
fundamental character.
by Abraham by a somewhat
results
have been obtained
different argument.
Let us consider two separate regions of space. In the first is a uniform fluid of gravitational force. This space is provided with at rest, and the lines of gravitational force run in a set of axes ,
of the the negative direction parallel to the z axis. The magnitude is free of second The is 7. space region gravitational acceleration of axes *' from gravitational attraction and is provided with a set direction the in positive which move with uniform acceleration 7
/
The
of motion of
in
any particle equations material particles on it other of action the either system (provided the same, if the equations for each may be disregarded) are then
along the
*
axis.
0. F. Mossotti, Sur
les
la constitution intime des corps. forces qui regissent
Turin, 1836. vol. xv. p. 65 (1908). f Camb. Phil Proc. Ann. dc.r Physik, vol. xxxv. p. 898 (1911). ;{:
8
1 (1912). Physik. Zdts. 13 Jabrg. p.
to the set of axes which we have associated system are referred with that system. They are clearly
Now let us assume
that our two systems are physically identical,
between them by any impossible to distinguish to denying absolute is tantamount whatever. This physical means acceleration in the same way that the principle of relativity denies The equivalence of the two systems is obvious absolute velocity. that
i.e.
if
we
it
is
Newtonian mechanics. This is not when we turn to some of the results which we
confine ourselves to the
the case however
have deduced from the principle of relativity.
shown on p. 318 that the mass of any system involves a E is the radiant electromagnetic energy of the part Ej& where system and c is the velocity of light. Eft* is mass in tin* sense of It is
9
a coefficient of inertia
but we have not proved that
it is
subject to
in the way that the ordinary mass of the gravitational attraction, is not subject to gravitational attraction this mass is. If system
evident that the equivalence of the systems tc and K will not hold exactly; systems in a uniform gravitational field will only fall uniformly provided the inertia of their electromagnetic energy may be neglected. A formal proof that if K and K are exactly it is
it is necessary that /;<;- should represent- gravitating mass, as well as inertia, is easily constructed
equivalent
Let the diagram represent the axes * in the gravitational force. Let S 2 S, represent two
field of
,
sufficiently
material systems at a distance h apart along the z axis. S l are small enough, h will remain invariable. The
unifo iorm
minute
and
If
systems" will
simply slide along the z axis with uniform acceleration, they were held apart by a rigid connection of h.
as though
length
Suppose allowed to send a certain amount of radiant energy E which is received by S,. The energy is measured at S.2 and $ by instruments which can be brought together and compared. As we do not yet know anything about the effect of on
that S,
is
gravitation
electromagnetic energy we are unable to say anything about what takes place during the transference. We can, however, find out a groat deal about it if we admit that the system K is equivalent to the system K. For in the system K' let us measure the energy of the radiation with reference to a set of axes /c that are at rest with reference to K at the instant at which the radiation leaves
S
the energy thus measured is E when theory of relativity*, be equal to E (1
it will,
by the
by the time
it is at
For
*SV
S
l
it
leaves 2
-f 7/i/c )
then moving with reference
is
the relative velocity v = yt = 7/i/c. This 2 first order of the small quantity 7/i/c
S2
,
2.
If
to the axes KQ with
result is true only to the
.
(
t
)n
the hypothesis that the two systems are equivalent, exactly will have to hold with reference to the system of
he same result
In this case 7/1 is equal to $, in i-h< gravitating system. between the points the in difference he gravitational potential a given quantity of the are and if so that l energies of K* and
axes K
4
>
t
E
#,:
radiation at
tt,
and
f
S
,
respectively,
then
of the conservation of energy in If \VP are to retain the principle 2 is done on the to 2 c/)/c eases if, follows that work
#
equal
these
nu ii;mt energy by the gravitational
forces.
This
is
the same as
if
mass equal to ^/c We th' radiant energy possessed, gravitational of relativity shows the which principle h rl vi;>re nmelude that A /e?, mass its 2
.
T
,
to
}
M
.
n lr m j
,
rtl
mass of
radiation, represents
gravitating
als...
Kinstcin, Ann.
tier
PlnjM
vol.
xvn. p. 913 (1905).
Field. The Velocity of Light in the Gravitational
Eeturning to the system
*',
let
j/a
be the frequency of the
a change in the gravitational field will therefore produce radiation which leaves S., to the value frequency 2 of the
the
z>
at a place 8l where the gravitational potential differs by $ from These results are true when the clocks used to its value at $>.
measure time at $> and
But the
together. that, if
^
agree with one another is constant in the //
distance
when brought /c
the radiation were being continually emitted
system
by
$,
;
so
and
received by $>, there would be a continuous accumulation or depletion of waves in the space between them, if v^ and r., were unequal.
This contradiction indicates that the times at different parts of a field are not correctly given by clocks which agree gravitational when brought together. To give times which do not lead to a -f $/c~ times slower than 2 must go are the clock at Si when they brought together and compared. Measured with such clocks the frequencies at X, and *S, become* the
contradiction the clock
at.
?
1.
is same and the number of waves in the stretch independent of the lapse of time, when the emission and reception are steady and //
continuous.
Now
if
we measure the
velocity of light in different places of
the system K with clocks which agree when com pare* together we If K and /c' are equivalent,, get always the constant quantity c. the same is true of the gravitating system K. Hut. these clocks do I
not go at the right rate in different, parts of * whence that the velocity of light is not the. same at different ;
but
varies
with the gravitational
potential
it
follows
parts of tc according to the
relation Cl
= c,fl4\
Thus the Einstein
principle of the constancy of the velocity of light which basis of the principle of relativity does not hold
made the
for gravitational fields
according to this theory.
GBAVITATION It
gQ5
easy to calculate the curvature of a ray of light produced by gravitation on this view. Take two points on the wave front nt distance dl apart. At time dt later the wave front will be tanis
gential to spheres of radii cdt and (c
The
the two points. thus
If 8w
is
an element
+
|
dl} dt described about
rotation S0 of the wave front in time %t
of the normal to the
wave
front
is
cdt^&n,
jw^iac^icty
dn~ cdl^tftt
to a sufficient approximation.
Let us apply
$
this to the case of light passing within a distance
of the centre of a star of mass m.
Let k be the gravitational
constant, r the instantaneous distance of the ray of light from the centre of the star and ^ the angle between r and the per-
pendicular from the star on the straight path of the undeflected
Then
light.
<
= fan/r and 1 = -J
f3<jf>
-~f C*J dl
the small total angular deflexion , dn = -
is
knit** 13r -57 2 C J
.^rSZ
km Th<
k
xion
is
directed towards the
star, so
that
if
the light
<:
the sun's disc passing just outside
it
amounts to 083 second
of arc. so In a later paper Einstein* has extended the investigation and electromagnetic as to c>vcr Uio relation between gravitation the and mass velocity of radiation, using other than
phenomena,
to smallness of the expected effects appears in the verification the possibility of their experimental
similar methods.
The
preclude near future.
*
Ann. dur Phyaik,
vol.
xxxvm.
p.
355 (1912).
INDEX OF NAMES Abraham, M.
122, 228, 230, 231, 2311, 242, 202, 282, 483, 601 Adams, E. P. 439
Airy, Sir G. B. 134, 271, 275, 285 Ampere, A. M. 86, 302 Arago, F. 271 ff., 275, 283
Becquerel, J. 539 Berpj, 0. 452 Ber^mann, A. 517 J.
Berzeliufi,
Beascl, F.
W.
33
551
Brag,'
G. 270 270
11.
J.
509 1
BroK'Hf, M. 509 P. 360, 507
Dby,
Ih'Hlandn's, H.
517
Dirs*'Ihorst, H. 411 Dopplfr, C. .">26, "M!, 604
Divcq, M.
:$r>r>
P.
Drudc,
134, 136, 137, 407, 432, -138,
539, f>l7
Duam, W. 122 Du Hois, II. 392 Dufour, A. r>2i Dulunx, P. L. .T;7
Boltamann, L. 334, 352, 379, 399, 508,
Bradley,
43
375, 378, 390, 392, 394, 548
Darwin, C. G. Daw, Sir H.
De
12, 42,
493, 495, 496
A.
1
Bevan, P. V. 527, 528, 533 Blcmdlot, E. 122 Bohr, N. 395, 413, 434, 530, 535, 585, 587, 594
Boscovicli,
J.
54H
P.
487, 488, 490, 497, 499, 501, 502, 505, 509 Bartoli, A. 334 Beatty, B. T. 504 Beck, P. N. 397
123
Coulomb, C. A. Curie, J.
Barkla, C. G.
M. 539
().
Cornu, M. A. Crowther,
Back, E. 523 Baedeker, K. 439, 440, 453, 457, 403, 466 Baeyer, 0. v. 468 Balmer, J. J. 513, 514, 528 ff., 585, 586
Corhino,
W. H.
477
IT.,
Kinstcin, A. 295 ff., 31 t, 322, 3">4, 357, 359, 360, 42J, 47,'J, 171, tidl, 6();UT. Kttin^shausni, A. von 4.'i4
Fabry, (!. P22, ;M! FujauH, K. 583 Falk, K. ti. 579 Faraday, M. 1, :$, 39, 43, 542,
481,
490,
509
498,
100, 121,
H(',
r>i:>
Fcnnat, P. d<; 271 I ery, C. :$">." Fit/herald, (}. F. 1>
W.
L. 50H, 509
Broglie, P.
De 509
Brown, F. C. Buclierer, A.
Buisson, H. Byerly,
W.
6,
442
H. 239, 242, 316
551 E. 33
Campbell, N. H. 273 Cauchy, A. L. 155 Cavendish, Hon. H. 39, 43 Coblcnte, W. W. 159 Cole, A. D. 124 Cornpton, K. T. 474 r.onVA R T, AA-\
2S9, 291, 323 H. L.
Fizfiiu,
232, 273,
2S(),
271,
2K2, 2K3, 276,
3()l>
Fleming,
.1.
A.
124
Fourifr, J. Baron Fowler, A. f>H6
Franck,
J.
3,'i,
327,
4.1I>
532
411, 423 Fredcnha^cn, K., 442 Frasncl, A. 271, 273 IT., 2H5 r Friodricb, W. 507, >OH Fran/,, U.
,
2K5,
INDEX OF NAMES K. F. 20
,H,
Larmor, Sir
J. 2, 109, 170, 229, 258> 286, 291, 522, 533, 549
28, 30, 41, 42, 43,
ft.,
98, 204 (lehrcke, E. 535 GehrtR, A. 409
Geigor, H.
Lane, M. 325, 507 Lebedew, P. 215 Le Bel, J. A. 546 Lecher, B. 452 Leduc, A. 434 Lees, C, H. 412
494, 583
5,
W.
Gerlacli,
355 485
Gray, J. A. Given, G. 23, 32, 117, 190 Gallon, C. 122
Lenard, P. 482,525,531 Leroux, F. P. 154 Liebreich, E. 441 Lidnard, A. 244
139, 432
HaK*n, K.
Hull, K. H. 409,43411,463 Hallo, J. J. 539 HallwacliH, "W. 469 Havelock, T. H. 161 HwiHide, 0. 13, 15, 218 Ht'lmholt/M H. v. 2 IhTtss,
H.
HTWI%
Lodge, Sir 0. J. 276 Lommel, E. 530 Lorentz, H. A. 2, 3, 149, 156, 161, 162, 167, 177, 183, 228, 233, 239, 241, 276, 282, 286, 289 ff., 295, 315, 316, 345,
122, 4(50
509 H(v<>Ky, G. v. 583 Hick*' W. M. 515 HimHtt'iU, F. 122
373, 374, 413, 433, 522, 524, 525,535,
600
Lorenz,L. 149,411,423
Holhuigal, H. 159, 359 K. 392 Morton, K. 4(7 HU^lu'K, A. LI. 475 Hull, (1. F. 215 J. 533 HuTiiphrcyB, \V. Hurmuwwu, D. 122
Honda,
Loria, S.
527, 528
Lummer,
0.
356
S. B. 434 Maclaurin, E. C. 162 MacLean, a. V. 122 Maculoso, B, 539 Marsden, E. 494 Maurain, C. 393
McLaren,
411
%
ff J. C. 2, 6, 14, 35, 81, 149, 101, 102, 107, 115, 121, 122, 142, 224, 230, 155, 207 ff., 212, 215, 228, 551 '232 271, 307, 413 ff., 417, 508,
Maxwell,
M
lojin"' ?'ll'
'
3lf>',
358, 360
Lindemann, F. A. Livens, G. H. 5S3
J.
.lacgrr, (i.
607
t>i
411,
:$'.)'.,
103 105 122 B43
422,
'
423,
'
430, 434,
rfolf.,
441
F.
,
.r.
429, 433
10(>,
378 424 Preface
Kanu-rlin^hOnn,H,lL Kaufnuinii, W. 7,235,239,316 Ku\v,(i. W. ('-. 4Hfl KaVs-r, M. 513,517
282 Mfflikan, E. A.
5, 6,
J"
W
Mowley, H.
282 274, 277, 278, 280, J. 509, 510, 511
&
Mossotti, F. 0.
570, r7t.
356
Minkowski, H. 325
601
Nagaoka, H. 550
Kirsrhbuum, H. 5.M Khiti, V.
531
KWnwn,
H.
47H
!>
_
.
.
Nicholson, J. Nicol, J. 501
W5
KiirlbiHiiii. K.
H.
W.
395
86 Oersted, H. C.
(
4(i .( ,
,,,,,.,,,, H.
;*so, T
392
.,,,1,,,.,,
-215
!'!
Kiiiull, A.
Kiistni'V,
179,180,
E.L. 580
l>
S
BMl, 527, S28
10 ManiuiH de IK,
E. Oxley, A.
392
523 Paschen, F. 517, 522, Pasteur, L 546 J. 436, 440 Patterson,
424, Preface
INDEX OF NAMES
608
412, 426, 429, 430, 436, 459, 462, 463
Peltier, J. C.
455
ff.,
Perot, A. 122 Perrier, A. 378 Perrin, J. 6 A. T. 357
Petit,
Pickering, E. C. 586 Planck, M. 267, 276, 322, 344, 347 352, 354 ff., 360,395, 407, 424, 473
ff., ff.,
585 Poisson, S. D. 26, 28, 30, 31, 42, 43, 47, 52, 82, 119,
Poynting,
J.
Steubing, W. 5.11 Stewart, B. 331, 4i, 512 Stirling, J. 353, 402, 597 StokeH, Sir (}. (r. 4, 91, 95, 101, 271, 275 1!., 280, 28,1, 4H,1, 502, 5.10, 5,12 Strutt, Hon. It. J. 594 Stuhlmann, 0. 478, 479, 481
218
H. 203
if.,
208, 213, 215,
221, 484, 520 Preston, T. 522
12 Pringsheim, B. 356 Purvis, J. E. 522 Priestley, J.
Eankine, A. 0. 282 Bayleigh, Lord 149, 155, 175, 343, 345, 347, 350, 355, 412, 434, 550 Biecke, E. 407 Eighi, A. 434 Eitz, W. 517, 529, 530 Boentgen, W. C. 477, 478, 482 Bosa, E. B. 122 Bossi, B. 584 South, B.J. 464,492 Bowland, H. A. 99 Eubens, H. 139, 158, 159, 359, 432 Sunge, C. D. T. 517, 522 Eussell, A. S. 583, 584 Eutherford, Sir E. 5, 6, 3f>6, 482, 490, 494 if., 583, 584, 594 Bydberg, J. E. 311, 517, 529, f>51, 5H6 Sadler, C. A.
Stark, J. H, 513, 517, 526, 531, 534 Starke, H. 7 Stefan, J. 333, 3,14, ,141, 355 Steinberg, K. 439
497, 499, 504
Saeland, S. 531 Saunders, C. A. 122 Schmidt, G. C. 532 Sehott, G. A. 550, 552, 554 Schoute, C. 452 Schuster, A. 432, 483 Schwarzschild, K. 535 Sellmeier, W. 149, 15f., 156, 161, 177, 179, 181 Soddy, F. 583, 594 Sorninerfeld, A. 325, 4H,1, 484
Sir J. J. 3, 4, 7, 122, 209, 218, 229, 232, 2.14, 242, 249, 422, 4*2.1, 426, 429, 430, 4,13, 438, 456, 458, 482, 483, 4H5, 490, 494 ff., 505, 529, 550, 554, 560, 579 ff., 5H4, 5H5, 001 Townsend, T. S. 4 Trouton, F. T. 282
Thomson,
Trowbridg*', J. Vuigt,
W.
440, 491, 570,
122
524, 535,
von Bacyer, O.
20H, 407,
5,19,
544, 545, 547
468
von KttiugHhaus*>n, A. A. 434
Walden, P. 575 \\Vlwjr, W, ,162, 407 Webster, A. G. 44, 22.1 Wehncit, A. 441 Weiss, !. ,INO, ,isi, :jsi,
:wr> t :w7, :WH,
390, 3111, ,195, 397, 4&J, 462
\\VrnU,
534
(\.
504, 505, 509, 580 Whittab-r, K. T. 12, 276, i>8.1, 530 \VU!h*;rt, E. .1, 2.14, 244, 4H3
AVhiddiiiK'ton, H.
Witnicmann, K. 53'2 (i. 411, 4*2: J -
(i.
4f>9
,127,
346,
Wiwlmaiiit, \Vii-n,
W.
3.14,
Preface
\VillianiHou, H. 317 Wilson, C. T. U. 3, 4
~~~
4.14,
Wood,
H.
A. 5, 277, 28:i, 442, 529 R. w. r>28, 532, r.i:t,
Zccuian, 5*2
i IT.,
547
P.
2, .1, 5'27, 5 '28,
:S7.1,
5M
ff.
.175,
431,
430, :\\\"\
t
r.iu
51.1,
5,11), 54*2,
518, 545,
INDEX OP SUBJECTS a rayfl, scattering of 490 Aberration of light 269, 306
Charged system,
;
Conducting electrons 463 Conducting media, propagation of electromagnetic waves in 135; reflexion at boundary of 137 Conduction in metals, electron theory of 408, 413, 463 ; simple theory of 409 Conductivity, electric, of alloys 465; electric, of bad conductors 467 ; elec-
;
491, 494, 549, 554, 564; theory, apweights of eleplication to the 6 ments, and gravitational .attraction 592 weights of radioactive elements
trical
;
410, 420, 422;
periodic forces heat 411, 421
;
f>83
Constant,
number
theory of
;
504 590
Avogadro'n number 6 Axes, moving 286, 297
490
X
rays *503 Hand npoctra 517 Black body radiation 330, 433 Ilohr'H theory of atoms ctra 5H5 r'H
497
X
absorption
of
field
electric, with variable velocity determination of 3,
electron,
Diamagnetism 361, 368 Dielectric constant,
due to 247, radiation from 257; '24'); accelerated, mitral, of atom 494, 510, 583, 585;
on
in magnetic field 114; magnetic 113 ;
polarization 98 Currents, induction of 100
ff.
Charge, accelerated,
magnetic 389
force due to element of electric
395,585; theory
rays,
23
;
of,
experiment 239
Characteristic
Law
Crystals and X rays 507 Curie's Law 378 Current, convection 98 ; displacement 97 electric 96 ; force on an element
;
Horondiiry
for
metals for
JBoltzmann's 406; Planck's
Critical temperature,
by electric field 236; by magnetic field 237, 240
rays, deflexion Hcatlc.ring of
ft
electrical,
of
382
Coulomb's dc'tlexion
;
Contraction, Fitzgerald's 280, 291 Correlation, principle of 286, 290 Corresponding states, of magnetization
Attraction, gravitational, compared with
ft
430
356, 469, 506, 509, 585 Contact difference of potential 454 Contractile electron 232
395, 585 of electrons in 488, 494, 495,
electrostatic
motion
'
;
Bohr's
uniform
Charges, fictitious 57 Chemical combination 570; and transference of electrons 571, 574, 580 Collisions between particles under eentral forces 417 Combination, chemical 570 Conditions at boundary between two dielectric media 45
Absorption, feeble 166; of characteristic X rayH 497 ff. Absorption of light 144; coefficient of l()f> near critical frequencies 178 Acceleration and force 260 Aether 2(>8 and the principle of relativity 823 Acthor pulse theory of the Koentgen rays 483 Airy's experiment 271 Alloys, conductivity of 465 Arago's experiment 271 Atom, definiteness of 549 ; of Kelviu and Thomson 554 structure Atomic, numbers 510, 583
Atomn,
in
217
AbHorbing media 163
243 356
;
electron theory of
74 Dielectric
of 47
;
media 39; Poisson's theory electron theory of 58
vecDifferential equations satisfied by tors when charges are present 186
152 ; anoDispersion, theory of 142, malous 154; of rock-salt 161; geneof luminous ralized theory of 169 ;
Fl u oreseenco 53 ) Force, exerted on electric charges 205; <
thermooloctromotive 459 activity of the '201 ; between 595, 599 ; conparticles tinuously operative, on principle of relativity 320
Forces,
gases 526
due to 67 Displaced electrons, potential 343 Displacement Law, Wien's Distribution of energy, molecular 403 Distribution of potential energy 404 Doppler effect '284, 304 Double refraction, magnetic 531), 544 Doublet, potential due to 58, 69 9 different elements of heat of 429, 448, 451, 452 force between 10 Electrified Electricity,
;
specific
particles,
Electrodynamics, law of 101
first
law of 95; second
Electromagnetic, equations 307 7, 228 ff., 234
;
inertia
Electromagnetisrn 86 ; dynamical theory of 102 Electron, accelerated 199 ; at rest 197 ; determination of contractile '232 ; charge on 3, 356 ; expanding 552 ; in uniform field due to rigid 222; motion 197, 218; positive 8, 482, 580, 593; relativity theory of mechanics of 310 revolving, magnetic force due to 362, 367; theory, origin of 1 Electronic, conduction, kinetic theory of 398; conductors, equilibrium theory of 441, 444 .Electrons, conducting 463 ; emission in equiof, from hot bodies 441 ; librium with conductors, temperature ;
relation 444, 447, 450, 454 ; law of number of, in force between 591 reflexion atoms 488, 494, 495, 504 ;
;
by conductors 468 Electro -optics 534, 547
material
Galvanomagnutic effects 434 Gas, paramagnetic 379 Gases, luminous, dispersion of 526 Gauss's theorem 2011., 41. effect of, on Gravitation 5H8, 600 light 001, 604; relative theory of ;
601 Green's theorem
23
Hall effect 434, 437 Induction, electric 40, 51 ; magnetic 82 Inertia of energy #10 4 Intensity, electric 14, 20; of reflected light, minimum value of 181 Inverse Zeeman effect 535 Iron, magnetic properties of 391 Isotropic radiation, pressure exerted by 211
Jeans and Bayleigh's radiation formula 343 Kaufruann'8 experiment
*235
Kerr's electro-optical effect 547 ; magneto-optical effect 545 Kinetic theory and thermodynamics
399 KirchhoiT's
Law 330;
Holution
1H1)
of,
;
;
;
of
;
particle 31*2; of oscillator rate of loss of, by aether 203
moving
349
;
Laplace's equation 1H, 4*2 operator, transformation of 20 Law, Curie's 378 ; of Stewart and KirchhoiT 330; of YYiedemann-PYanKLoren/, 411 Least time, principle of 271 Ligbt, aberration of 261), 306; velocity of, in the gravitational field 001 ;
Energy, electric, of charge in uniform motion 220, 252 in electric field 34, 44; inertia of 316; in field of accelerated point charge 251, 255 magnetic, of charge in uniform motion of a system of charges 33 221, 252
4
4
Line of
force,
equation of
16
Entropy 399; and probability 400; of resonators
253
Magnetic, double refraction
Equations, electromagnetic, for moving systems 286, 290, 307 Laplace's 18, Poisson's 26, 42 Lorentx's 183 42 of line of force 10 of propagation 115; universal 182
field,
effect
of,
on
53'J, 544 ; electric, currents
specific heats
4:54, 437, 431); force due to moving electron 362 properties, abrupt, changes in MO; rotation of plane of polari/ation 535, 512 Magnetism 77; application of thermodynamics to 377; on electron theory
Ferromagnetism 361, 380, 393, 397 Field due to uniformly moving charges 218 Fitzgerald's contraction 280, 291 .Fizeau's experiment 273, 306
reaction mechanical Magnetization, caused by 3',)5; permanent 381, 3H7 Magneto-optics 373, 51 H, 523, 535, 539, 545 Magneton 394
;
;
;
;
Ferromagnetic substances,
;
of 397
;
361
Mans, electromagnetic, of rigid electron 228 electromagnetic, variation with velocity 234 ; longitudinal and transverse 229 ff. variation with velocity on the principle of relativity 312 ;
;
Material particles, conditions for neuforces between 595, 599 trality 51)8 Maxwell's stresses 35, 81 Mechanics of electron BIO Michelson and Morley's experiment 277 Moment, magnetic, of revolving electron 367 ;
Momentum,
and
radiation
;
particle
314, 319
286, 297
Natural rotation of plane of polarization 545 Negative electron 7 Neutral material particle 598 Newton's law of gravitation 588 Number of electrons in atoms 488, 494, 495, 564 Numbers, atomic 510, 583 Optical, convection, Stokes's theory of
275
;
effects
due to motion
of
re-
fracting medium 273 OHcillator, energy of 349
Paramagnetic gas 379 Paramagnetism 361, 376, 379 421), 455, 457 Periodic law, and radioactive elements 583 and structure of atoms 565
Peltier effect
;
Permeability, diamagnetic
371
Phosphorescence 530
603 Radiation, and temperature 326, 433; complete 328, 433 energy density of complete, in refracting medium 332 ; formula of Planck 347, 355; formula of Rayleigh and Jeans 343; formula ;
Wien
of
354; from accelerated electric
charge 257; isotropic 211; pressure and momentum 212 pressure of 209 ; reaction of, on accelerated charge 263 ; reflexion of, at a moving mirror 277, 335; secondary, asymmetrical emission of 477 ff types of 476 Radioactive elements and periodic law 583 Ratio of electromagnetic to electrostatic unit 122 Rayleigh and Jeans' s radiation formula 343 Rays, residual 157 Reaction of radiation on moving charge 263 Reflecting power, of metals 140; of quartz 159 Reflexion, change of phase on 138, 168 ;
.
;
;
electromagnetic waves 129; of of light electrons by conductors 468 by moving mirror 277, 335; of light near critical frequencies 178 ;
Point charge in uniform motion, due to 248
field
PoisHon's equation 26, 42; solution of equations of propagation 118; theory
media 47
Polsiri/.ution, electric 51; electron theory of 04,70; magnetic rotation of plane
natural rotation of plane of 51 J5, 54*2 of 5-15; of X rays 488; potential due to lit); variable 54 Positive) electricity, structure of 580 ;
'
Positive electron 8, -182, 580, 593 Potential, at internal points 19; conelectric 15; tact, difference of 454;
magnetic, due. to electric current 86; propagated 193; scalar 194; vector I'M, 1117
Poyn ting's theorem
Quasi-stationary motion 262
Radiant energy, gravitational mass of
of
Photoelectric action 469 Planck's radiation formula 347, 354, 355
of dielectric
Pyrrhotite, magnetic properties of 384
pressure
212 effect of inertia of energy on 320; electromagnetic 208, of charge in uniform motion 221, of isolated system 215, of rigid electron 227; of moving Moving axes
Principle of correlation 286, 290 Principle of least time 271 Principle of relativity 296, 322 Probability and entropy 400 Probability, of distribution of energy among oscillators 352,357; of statistical distribution of a gas 401 Propagation, equations of 115 Pulse theory of Roentgen rays 488
20ii
Pressure of radiation 209 Pressure shift of spectral lines 533
Refracting medium, optical effects due to motion of 273 theory of propagation of light in moving 284 Refraction, of compounds and mixtures 150; of electromagnetic waves 129 Refractive hidex 149, 165 Relative theory of gravitation 601 Relativity, principle of 296; dependence of physical quantities on velocity 322 ; experimental test of 242, 315, 318 Residual rays from absorbing media 167 Resistance, change in a magnetic field ;
439 Reststrahlen
159
Roentgen rays 482
ff.,
496
ff.,
507
ff.
of a Scattering, of a and /3 rays 490; and |3 rays, Rutherford's theory 494 ; of a and (3 rays, Thomson's theory
491; of
X rays
485
Thermomagnetie
Secondary rays 477, 496 Series in band spectra 517 Series, spectral 513, 585 Shell, force on a magnetic 88
Thomson
452
low temperatures 858
;
Debye's theory 360; Einstein's theory 357; of ferromagnetic substances 397 Spectra, baud 517 Spectra, Bohr's theory of 585 ; X ray of elements 510 Spectral emission, theories of 525, 585 Spectral lines, deiiniteness of frequency of 551; displacement in electric field
534; effect of pressure on 533 Spectral series 513, 585 Spectroscopic phenomena 512, T>17, <>25, 551, 585 Stability, of electrons in atom 552, 554, 558, 562 Stark
effect
534
Stefan's law 333 Stirling's
approximation 402
Stokes's theorem
91; theory of optical convection 275 Stresses, Maxwell's 35, 81, 207 if. Structural formulae, in chemistry 575 Structure of positive electricity 580
Temperature and radiation 326, 433 Thermionics 441 Thermodynamics and kinetic theory 399 Thermoelectricity 425, 448, 455, 401 Thermoelectrornotive force 459
CAMBRIDGE
:
PHTXlTTr.n
effects
434
429, 448, 452
Transparency of quartz 159 Tubes of force H, 22
Shells, polarized 62 Specific heat of electricity 420, 448, 451,
Specific heats, at
effect
Unit of current H9 Unit of electric charge Units, electrical
13
109
Valency, chemical, and atomic strueturo 506, 573 Values, aetuul and mean 65 Velocity, addition of, on principle of relativity 302; of light in the gravitational field 004; of propagation of electro-magnetic field 121; of secondary ft rays emitted by characteristic X rays 504 wave 252 ;
Wave
of reorganization
258
Wave, plane-polarized electromagnetic; 124
Wave-length of X rays 510 Waves, electromagnetic 115 Wien's law 337; displacement law 343; radiation formula 354
X X
ray spectra 510 rays, and crystals 507; ft secondary 503; characteristic, absorption of 497 IT.; characteristic, velocity of secondary ft rays emitted by 504 ; polarization of 4HH; scattering of 4H5; wave-length f 510
459,
nv .Tmm nr.AV
Zeeinan effect 373, 51H, f23; 535; theory of 373, 5 IK
vr
inverse
CAMBRIDGE PHYSICAL SERIES written
mcde?n modem
hv^rUT?h ^ ? well needs
Modern becond
e
ambr d g e Ph ysical
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