&-i LIBRARY 1VERSITY OF CALIFORNIA. Received Accessions No.
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THE MATHEMATICAL THEORY OF
ELECTRICITY AND MAGNETISM WATSON AND BURBURY
pontoon
HENRY FROWDE
OXFORD UNIVERSITY PRESS WAREHOUSE
AMEN CORNER,
E.G.
ms THE
MATHEMATICAL THEORY OF
ELECTRICITY AND MAGNETISM BY
H. W.
WATSON,
D.Sc.,
F.RS.
FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE
AND S.
H.
BUBBUBY,
M.A.
FORMERLY FELLOW OF ST. JOHN'S COLLEGE CAMBRIDGE
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ELECTROSTATICS
AT THE CLARENDON PRESS M DCCC LXXXV [
All rights reserved
]
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I.
PREFACE. THE
exhaustive
character
the
of
late
Professor
Maxwell's work on
Electricity and Magnetism has necessarily reduced all subsequent treatises on these
subjects to
the rank of commentaries.
Hardly any
advances have been made in the theory of these branches of physics during the last thirteen years of which the first
But
suggestions may not be found in Maxwell's book. the very excellence of the work, regarded from
the highest physical point of view, is in some respects a hindrance to its efficiency as a student's text-book.
Written as
under the conviction of the paramount importance of the physical as contrasted with it
is
the purely mathematical aspects of the subject, and therefore with the determination not to be diverted
immediate contemplation of experimental to the development of any theory however fas-
from the facts
cinating, the style is suggestive rather than didactic,
and the mathematical treatment
what unfinished and
obscure.
is
occasionally some-
It is possible, therefore,
that the present work, of which the first volume is now offered to students of the mathematical theory
of electricity,
may be
of service
as an introduction
Its aim or commentary upon, Maxwell's book. to, is to state the provisionally accepted two-fluid theory,
and to develop
it
into its mathematical consequences,
PREFACE.
VI
regarding that theory simply as an hypothesis, valuable so far as it gives formal expression and unity to experimental facts, but not as embodying an accepted physical truth. The greater part of this
volume
is
accordingly
occupied with the treatment of this two-fluid theory as developed by Poisson, Green, and others, and as The success of Maxwell himself has dealt with it. this theory in formally
explaining and co-ordinating only equalled by the artificial
experimental results is and unreal character of the postulates upon which is
based.
bilities,
The
electrical fluids are physical
it
impossi-
only as the basis of mathematical
tolerable
calculations, and as supplying a language in which the facts of experience have been expressed and
results
calculated
and
anticipated.
These
results
being afterwards stated in more general terms may to suggest a sounder hypothesis, such for
serve
instance
as
we have
offered
to
us in the displace-
ment theory of Maxwell. In the arrangement of the treatise the first three chapters are devoted to propositions of a purely mathematical character, but of special and constantly recurring application to
an arrangement
it
is
electrical
hoped that the
able to proceed with the in
due course with as
By such reader may be
theory.
development of the theory
little
interruption as possible
from the intervention of purely mathematical processes. Few, if any, of the results arrived at in these three chapters contain anything new or original in them, and the methods of proof have been selected with a
PEEFACE.
Vll
view to brevity and clearness, and with no attempt at
any unnecessary modifications of demonstrations
already generally accepted. All investigations appear to point irresistibly to a state of polarisation of some kind or other, as the
accompaniment of
electrical action,
and accordingly the
properties of a field of polarised molecules have been considered at considerable length, especially in Chapter XI, in connection with the subject of
physical
specific induction
posite dielectric,
and Faraday's hypothesis of a comand in Chapter XIV, with reference
to Maxwell's displacement theory.
last-mentioned hypothesis
is
The value of the
now
than any
universally recogas of more promise generally regarded other which has hitherto been suggested in
the
of placing electrical theory
nised,
and
way
it is
physical basis.
upon a sound
CONTENTS. CHAPTER
I.
GREEN S THEOREM. ABT. 1-2.
PAGE Green's Theorem
3.
Generalisation of Green's
5.
Correction for Cyclosis Deductions from Green's
6-17.
Theorem
...
1-2 .
.
.
3-5
5-6
Theorem
.
CHAPTER
.
6-19
.
II.
SPHEEICAL HAEMONICS. Harmonics
18-19.
Definition of Spherical
20-24.
General Propositions relating to Spherical Harmonics Zonal Spherical Harmonics
25-34. 35-37.
38-39.
.
.
Expansion of Zonal Spherical Harmonics Expansion of Spherical Harmonics in general
CHAPTER
.
.
.
.
.
.
.... ....
20-21 21-26 27<-38
39-40
41-44
III.
POTENTIAL. 40-46.
Definition of Potential
45-51
47-50.
Equations of Poisson and Laplace
51-54
51-63.
Theorems concerning Potential Application of Spherical Harmonics
64-68.
CHAPTER
54-67 to the Potential
.
.
67-73
IV.
DESCRIPTION OF PHENOMENA. 69.
Electrification
70.
Electrification
71.
Electrification
74
by Friction by Induction by Conduction
72-79.
Further Experiments with Conductors and Insulators
80-81.
Electrical
Theory
75
76 .
.
77-83 83-86
CONTENTS.
CHAPTER
Y.
ELECTRICAL THEORY. PAGE
ART. 82-87.
88. 89.
90-91. 92-94. 95. 96.
97-98.
99-103.
Properties of Conductors and Dielectric to the Two-fluid Theory Principle of Superposition Case of Single Conductor
and
Media according
Electrified Point
.
System inside of Conducting Shell Explanation of Experiments II, V, VI, and VII Case of given Potentials General Problem of Electric Equilibrium Electrified
.
87-90 90 90
.
.
.
91-92
.
.
93-95
....
Experimental Proof of the Law of Inverse Square and Fluxes of Force
.
96
96-97
97-100
.
100-107
Lines, Tubes,
CHAPTER
VI.
APPLICATION TO PARTICULAE CASES. 105.
Case of Infinite Conducting Plane
106.
Two
107-117.
Infinite Parallel Planes;
108
two
Infinite Coaxal Cylin-
ders; two Concentric Spheres Sphere in Uniform Field; Conducting Sphere and Uni-
formly Charged Sphere;
Infinite
CHAPTER
109-110 110-123
Conducting Cylinder
VII.
THE THEORY OF INVERSION AS APPLIED TO ELECTRICAL PROBLEMS. 118-122. 123-124.
125. 126. 127.
Theory of Inversion; Geometrical Consequences; Transformation of an Electric Field Problem of Conducting Sphere and Electrified Point solved
by Inversion The converse Problem Case of two Infinite Intersecting Planes Case of n Intersecting Planes
124-129 129-130
....
130
131-132
132-133 .
.
.
133-134
130.
Case of Hemisphere and Diametral Plane Two Spheres external to each other
181.
Distribution on Circular Disc at Unit Potential
.
.
135-136
128-129.
134
132-133.
Deductions therefrom
136-138
134-135.
138-140
136-140.
Case of Infinite Conducting Plane and Circular Aperture Case of Spherical Bowl
141-142.
Effect of small hole in Spherical or Plane Conductor
140-143 .
143-146
CONTENTS.
CHAPTER
XI
VIII.
ELECTBICAL SYSTEMS IN TWO DIMENSIONS. ART.
PAGE
143-144.
Definition of Field
145-146. 147-150.
Definition and Properties of Conjugate Functions Transformation of Electric Field
151-152.
Example
153-154.
Inversion in two Dimensions
of Infinite Cylinder
147-148 148-152
....
CHAPTER
152-154 154-156
156-158
IX.
SYSTEMS OF CONDUCTOES. 155-156.
Co-efficients of Potential
and Capacity
159-160
157-158.
Properties of Co-efficients of Potential
160-162
159-164.
Properties of Co-efficients of Capacity and Induction Comparison of similar Electrified Systems
162-164
165.
CHAPTER
165
X.
ELECTEICAL ENEKGY. 166-168. 169-170. 171-174.
The Intrinsic Energy of any Electrical System The Mechanical Action between Electrified Bodies The change in Energy consequent on the connexion .
.
166-170
.
.
170-171
of
Conductors, or the variation in size of Conductors 175.
176-181. 182.
.
Earnshaw's Theorem
A
System of Insulated Conductors without Charge fixed in a field of uniform force
Electrical Polarisation
CHAPTER
171-173
174-176 176-181 181
XI.
SPECIFIC INDUCTIVE CAPACITY. 183-185.
Specific Inductive Capacity
186-192.
Mathematical investigation of Faraday's theory of Specific
183-184
Induction 193.
194-195.
Lines, Tubes, and Fluxes of Force on this theory Application of the theory to special cases
195a-198. Extension 199-200.
of the theory to Anisotropic
Dielectric
.
184-194 195
196-200
Media
200-205
Determination of the Co-efficient of Induction in special
205-207
CONTENTS.
Xll
CHAPTER
XII.
THE ELECTRIC CURRENT. ART.
201-203. 204-206.
207-211. 212-215.
General description of the Electric Current Laws of the Steady Current in a single metal at Uniform
208-210
Temperature Determination of the Resistance in special cases Systems of Linear Conductors
210-213
.
........ ...... .... ......
219-221.
Generation of Heat in Electric Currents Electromotive Force of Contact
222-225.
Currents through Heterogeneous Conductors
216-218.
.
.
CHAPTER
.
.
214-218
.
.
218-221 221-224 224-226 226-228
.
XIII.
VOLTAIC AND THERMOELECTRIC CURRENTS. 226-229.
230-238.
The Voltaic Current. The Electromotive Force
229-232 of
any given Voltaic Circuit
232-236
.
Clausius's theory of Electrolysis Electrolytic Polarisation
236-238
242-243.
Thermoelectric Currents
239-240
244-248.
Laws of a Thermoelectric Circuit The Energy of a Thermoelectric Circuit
239-240. 241.
248-250. 251.
238
.
Systems of Linear
... .... .
Conductors with Wires
240-244
.
of
244-246
different
246-247
Metals and Temperatures
CHAPTER
XIV.
POLARISATION OP THE DIELECTRIC. 252-253.
248-254
Polarisation of the Dielectric
254-257
254.
Stresses in a Polarised Dielectric
255.
256-257.
Explanation of Superficial Electrification of Conductors The Relation between Force and Polarisation at each
258-262.
Theory of
263-264.
Displacement Currents All Electric Currents flow in Closed Circuits
.
point
265.
of.
257-258
258-259
a Dielectric
259-264
Electrical Displacement
264-266 .
,
266-268
CHAPTER
I.
GREEN'S THEOREM. ARTICLE
LET 8 be any
1.]
functions of
a?,
and
y,
#,
closed surface, u and w' any two which are continuous and single-valued
everywhere within 8. Then shall du du' du du' du du'
= /YV ds- /7YV |^= which the
ffw
?p4#_
f/Tw
taken throughout tfye space and the double by S, integrals over the surface, dv is an element of the normal to the surface inside of S but measured outwards in direction, and V 2 stands for in
triple integrals are
enclosed
9
+ (~dtf
For v lt x
let a line
and and a? #!
y, z
=
/**a / Xl
J Let and #2
.
w7
2
-
a?
2
=
,
2;.
,
/
/
dx
da?
4 )
2
one edge.
C X2 du du'
,du\
-ft*'--) v dx
^^ be the base of a prism is
&?)'
x cut the surface in the points Then y, integrating by parts between ir , we have 2 parallel to
/ ,du\ -^ = (^ \
tfc
+ "dy*
/^
A
_
3-^-^. cte
/
c?o;
1
of which the line between
xl
Then
.
dx dx
Now
if
1}
%,
drawn outwards
% be
the direction-cosines of the normal to
at the point
# 1}
y,
0,
and
by the prism, dydz =-~ll dSlt
area cut out at that point
VOL.
I.
B
if
d8l be the element
8 of
GREEN'S THEOREM.
2
[2.
and using corresponding notation at the point #2 dydz
l
2
dS2
,
y,
z,
.
Therefore r*i
dydz
^ dx
u
I
Jx-i *i
,
du.
.
~
,
C xz du du
du
xl and x2 are functions of y and and transposing, we obtain grating Therefore, noting that
z,
inte-
in which the triple integrals comprise the whole space within and the surface integrals comprise the whole surface of S. Similar equations, mutatis mutandis, hold for y and z.
S,
Therefore
~
=
U/
I
dudu
dudu
du'
u'
~I~^~~
u -=dS
1
1 1
u^u
dxdydz by symmetry.
We have
supposed the line through y z to cut S in two points It may cut it in any even number only, #! , y, z and #2 y, z. of points, but all the reasoning would apply to each pair so ,
to the point of ingress, and #2 of egress. The a? x relates equation will therefore hold equally where lines can be drawn cutting the surface in more than two points.
long as
Further, the proof evidently holds for the space between two St and S2 whereof S2 completely encloses St
surfaces,
On
.
,
the Application of the
Theorem
to the Infinite
External Space.
2.] Let us consider more closely the case of two and S2 of which $2 completely encloses S^ ,
surfaces,
JS^
GENERALISATION OF GREEN' S THEOREM.
3.]
3
Applying the theorem to the space between them we have
^^
/7T5
JJJ \dx dx
in
which the
second to
du^ dz dz
dy dy
first
S2 and ,
of the double integrals relates to /S^, and the the normals on S1 are measured inwards as
regards the space enclosed by S1 Now let S2 be removed to an infinite distance. .
/"*
case the surface integral
/ /
If in that
7
/*
u
f
-^-
dS2t extended over the
infinitely
S29 vanishes, the theorem is true for the infinite of S1 , as well as for the finite space within it, the outside space normal being in this case measured inwards as regards S^.
distant surface
'
In order that
dS2 should vanish, when extended over
II u'
infinitely distant surface, it is sufficient and necessary In the physical 1. that uu' should be of less degree than theorems with which we are concerned, this will generally be the case.
the
Generalisation of Green's Theorem. 3.]
Let
rrC JJJ
K be any continuous function of du du dx dx -j-
du dul
du du'
#, y, z.
Then
_
+ -r -r- + ~r j-[dxdydz dz dz 3 dy dy
The conpartial integration as before. dition for application of the theorem to the external space will f 1 in this case be that Kuu must be of less degree than
by the same process of
.
It will sometimes be convenient to denote the expression
d dx
*
\J*-
d
du =
)
dx'
(xL
$-
dy
d
du )
T-
ay
B 2
.
(A.
dz
du ~z~ )
dz
by
V 2K'U.
GENERALISATION OF GREEN
4 4.] x, y,
We
have assumed u and
and
be continuous functions of
u' to
discontinuous but
is
du efficients -=-
[4.
If at a certain surface within S, one of them,
z.
suppose Uj
THEOREM.
S
du
du -=-
>
da?
>
-7-
differential
its
co-
_
.
,
or one of them, are infinite, the theorem
>
d0
dy
and
finite,
.
requires modification as follows It still remains true, u being always finite, that :
d du\, u-r lv (K-=-)dne dx^
and from
dx'
.du\ = (uK-=-) .
dx' Xli
we may deduce
this
.
T
T
,du\ dx')Xl
(uK
C x* /
JXl
dudu' Kdx\ dx dx
Green's theorem in the form
du du
= ffuK ^-dSBut we cannot
fflu^u'dxdydz.
assert the truth of the
theorem in the
al-
ternative form .
du du dx
K(^dx
|-
&c.
) '
dxdydz y \udxdydz.
If u become infinite at any point within $, we cannot include in the integration the point at which the infinite value occurs. But we may describe a surface S' completely enclosing, and very
near to, that point, and apply the theorem to the space between S and /S", regarding u' and its differential coefficients as constant
throughout $'. For instance, let u become infinite at a point P within S. Let S' be a small sphere described about P, and let -=-2
ft'
and
dv surface of S'.
Kp
be the values of
ft',
-=-
av
,
and
K
in or on the
Then we obtain
(fudSf -
In this form the theorem can be made use of whenever the
two
surface integrals relating to
',
namely //^ J/S'and
// -g-dff,
THE CORRECTION FOR CYCLOSIS.
5.]
are
or
finite
distance of
For instance,
zero.
if
u
= -,
5
where r
is
the
any point from P,
sr=0
^=-
and
and the equation becomes
Correction
5.]
We
have assumed
functions of /
integral
#, y, z
-~- dl)
;
that
for
also that is,
Cyclosis.
u and
u' are single-valued
that for any such function the line
taken round any closed curve that can be
drawn within the space S to which Green's theorem is applied, is zero. The functions with which we shall have to deal in this treatise will generally satisfy this condition.
If however for any function u the condition
dl
not satisfied for certain closed curves drawn within
$,
=
be
the state-
ment
of Green's theorem requires modification in the manner pointed out by Helmholz and Sir W. Thomson. The reader will find the subject fully treated in
Maxwell's Electricity and
Magnetism, Second Edition, Arts. 96 b 96 d. It will be sufficient here to shew the modification required in a simple case. Suppose, for instance, S conof an anchor-ring, Fig. I, and that for any closed curve drawn within it, so as sist
to
embrace the
axis, as
OPQO,
I
-jj
dl= H,
but for closed curves not embracing the axis
/ do
dl
=
0.
Let us suppose u to be mea-
sured from a section
$
of the ring.
Let
be a point in the
THE CORRECTION FOR CYCLOSIS.
6
[5.
Then, if we start from 0, with u for the value of u will, on arriving again u, and proceed round the curve OPQO, at 0, have assumed by continuous variation the value U Q + H. Let Sl be any other section of the ring. Then $ and 8 section
.
f
divide the space within the ring into two parts, /S P/S i and No curve embracing the axis can be drawn wholly S1 QS .
within either
S PSl
Sl QS
or
be applied to either space.
.
Therefore Green's theorem
Applying
it
to
may
SQ PSlt we have
(1)
in which the
double integral relates to the surface of the
first
S ring, and the other two to the sections SQ and 1 respectively. <S and to the theorem regarding the 1 Q"S'o> Again, applying normals to S and S1 as measured in the same direction as in f
the former case, that question,
is
inwards as regards the space
now
in
we have du du -T-
(
+ &c. ) dxdydz duf_ ~d>>
(2)
If we now add the two equations (l) and (2) together, obtain for the whole space within the ring .du
we
du
JJ Jfffz(-
~ r r
Htnce
HI
I
fj
f
K^dS
Q
is
the correction for cyclosis in this
Its value depends on the section of the ring arbitrarily chosen as the starting-point from which u is measured. case.
DEDUCTIONS FKOM GREEN'S THEOREM.
7-]
Deductions
A
from
j > i Let u bee a constant.
T
-i
o.J
severally zero,
we obtain the
Green's Theorem.
mi
du'
Then, since
-=
0$
result that for
>
du' -y-
,
>
^
(a) There exists one function
u of x,
y,
^
any function
u,
8 and
the
and z which has
arbi-
the integrals being taken over any closed surface enclosed space. 7.]
du'
and -7- are
trarily assigned values at each point on a closed surface 8, and z at each point within S, satisfies the condition V K u being
=
K
everywhere positive.
For evidently an u
exist
satisfying
value at
within
number of forms of the function
infinite
the
each point of
condition S,
that
irrespective
u
has
the
of the value
assigned of V*K u
8.
For any function u
let the integral
throughout the space enclosed by 8 be denoted by Q^ This integral is necessarily positive, and cannot be zero
for
any of the functions in question, unless the assigned values are the same at every point of S in which case a function having that same constant value within S satisfies all the conditions 9
of the problem. If the assigned values of u be not the same at each point of S, then of all the functions which satisfy the surface conditions,
there
must be some
positive, is
one, or more, for which Q U) being necessarily not greater than for any of the others. Let u be
such function.
Let u + u' be any other function which satisfies the surface conf u at each point of S. Then also u + On'
ditions, so that satisfies
the surface conditions, if
whatever, positive or negative.
be any numerical quantity
DEDUCTIONS FROM GREEN' S THEOREM.
8
[7.
Then
du du
du du'
du
than
QU} and
by Green's theorem,
= Qu+0 because u' = Now
2
Q U '-2 on
Qu+eu
r
is
8.
by hypothesis not 0*
less
therefore
Quf-20 fffu' V^ u dxdydz
cannot be negative for any value of 0, or any value of u'. But unless V 2 be zero at each point within S, it is possible to assign such values to #', consistently with its being zero on S,
^
make
as to
/
differ
from
as to
to
zero.
uV K u dxdydz 2
Therefore,
it is
possible to assign such a value
make 0*
Qu>- 2
negative. It follows that
\\\u'
V\u dxdydz
=
at each point within 8, when u is a V^ function satisfying the surface conditions for which Q u is not greater than for any other function satisfying these conditions.
COROLLARY.
If u + u' be any other function satisfying the V ZK u' is not zero at all points
surface condition, but such that
within S, evidently (b)
The theorem can
side of
S with
also be applied to the infinite space outa certain modification, namely, There exists a y, and z which has arbitrarily assigned values at
function u of #, each point on S, and outside of S,
than
1.
VK u = that Ku 2 is
satisfies the condition
K being positive, and
such
z
at each point of lower degree
DEDUCTIONS FEOM GREEKS THEOREM.
9.]
For of
all
9
the functions which satisfy the surface conditions on
S and
the condition as to degree, there must be some one or more for which the integral Q u extended through the infinite exis not greater than for any of the others. be another function which is zero on S, and
ternal space
Let
11
satisfies
the condition as to degree. Then Green's theorem may be to the infinite external space with u arid n for functions. applied And it can be proved by the same process as used above that,
VK u = 2
at every point in the external space, some be given to u' which will make Q u+u less than Q u Therefore when Q u has its least possible value for all functions
unless
value
may
'
satisfying the conditions,
VK u Z
must be zero
The theorems can be extended
8.]
.
at all points outside
to the case
where V*KU,
instead of being zero, has any given value p, a function of or, y> z, at each point within the limits of the triple integral, i.e. within or outside of
For
let
VK V =
V
S
as the case
may
be.
be a function of the required degree which
satisfies
p at all points within the limits of the triple integral.
Such a function always
exists independently of the surface con-
ditions *.
Then
if
Art.
7,
W=
= W+
Then u has
at each point
the required value
on S the value
cr
V-\- 7,
that
is,
and
=P at each point within the limits of the triple integral. of u be given at each point on S 9.] If the value
9
and
V 2j^,
if the
whether zero or any other assigned value, be at each point within S, u has a single and determinate given
value of
value at each point within 8.
///
is
one such function.
DEDUCTIONS FROM GREENES THEOREM.
10
[lO.
For, let u and u' be two functions both satisfying the conditions.
Then u = u' and
uu'=
V^u-V^u''=
S
at each point on
=
or V*K (u-u')
0,
;
0,
and .......... (1)
Then
at each point within 8.
..
=
by
(2)
(1).
It follows that
du dx
_ du
du
dx
dy
_ du' dy
du
_ du
dz
dz
and therefore u and u', being equal on 8, have identical values at each point within S. It follows as a corollary that, as stated above, if u be constant z at each point on S and if V K u = everywhere within S, u has at each point within S,
9
the same constant value everywhere within 8. For the constant satisfies both the surface and internal conditions, and there can
be no other function which does satisfy them. The last theorem can be applied to the infinite space outside of 8 as well as to the space within it, if we add the condition is of a less degree than 1, without which Green's theorem could not be applied to deduce (2). 10.] There exists a function u of #, y^ and z which satisfies the
that Kit?
conditions following ; namely (1) u has values constant lut arbitrary over each of a series of closed surfaces 8lt 82 ... Sn , and given constant values over ,
each of a second series of closed surfaces $/, \. (2) u is of lower degree than
.
.
.
Sm'.
and
//; where
ely e2 ,
taken over
...
e n are given constants,
Slt S2
,
.
. .
8n
.
and
the double integrals are
DEDUCTIONS FROM GREEN'S THEOREM.
IO.]
(4)
V2 u =
11
at every point not within any of the surfaces
For, consider a function u which satisfies (l) and (2), and also satisfies
(a)
where
u 1 e1 + u 2 e<2 + ...+u n e n
= E,
or
E is
Sue
=. E,
u n are the constant any arbitrary constant, and u^ by u on S1 ... Sn Evidently there exists an infinite variety of such functions. For every such function %, the volume integral . .
.
values assumed
.
extended throughout all space not within the surfaces, cannot be be not zero, and is positive.
zero, if
E
There must therefore be some one or more of such functions for which Q u is not greater than for any other.
Let u be such function. Let u + u' be any other function satisfying (1), (2) and (a). Then u' has constant values, %', u2', &c. on the surfaces Slt ;
S2
. .
.
Sn which
the surfaces S-^
satisfy u^e^
Sm', and
+ n2'e2 + &c. =
0, is
zero on each of
of the required degree. These are its only conditions. Also u' being of less degree than i may be used with u in Green's theorem for external ...
is
space.
Let
Then
be any numerical quantity, positive or negative.
u + 6u' also satisfies
= Qu +
(l), (2),
2
Qu>
and
(a).
Then
+ &c. dxdydz
+2
uV since u' is constant
on each of the surfaces S1
on each of the surfaces S^
...
Sm'.
. . .
2
u dxdydz
Sn and ,
is
>
zero
DEDUCTIONS FROM GREEN' S THEOREM.
12
Now Q U+0U
>
cannot be
may
whatever
less
than
Qu
,
whatever
u'
[ll.
may
and
be,
be.
But unless the factor of 20 in the last expression be zero, than Q u there must be some value of 6 which makes Q U + 0U less for all zero be therefore must 20 The quantity multiplied by f conditions. its values of u consistently with V 2 & must therefore be zero at all points within the triple integral, and .
'
for all values of %',
# 2', &c. consistent with
ul'e 1 + u2'e2 + &c.
= 0.
we must have
Therefore
some constant, the same for all the surfaces u be found for any value of E, then jx function If the from (a), and is proportional to E.
where p
is
S-^
is
Sn
...
.
known
There must, therefore, be some value of E for which p is unity, and the function u determined for that value of E satisfies (1), (2), (3),
11.]
and
(4).
The theorem can be extended
to the case
where
V%,
instead of being zero at every point within the limits of the triple integral, has any assigned value p, a function of x, y, z.
V
For let be a function of the required degree which has Sn has constant but arbitrary values on each of the surfaces S1 the given constant values on S^ ... m', and satisfies V 2 p at .
.
.
,
F=
all
The existence of points external to both series of surfaces. is in 8. 7 so Art. determined, let proved being
such a function
Then, as we have proved, there exists a function required degree having, some constant values on S1 value zero on 8^ ... Sm', and satisfying S2
and
V /F= 2
=e
2
-e.;,
W of the .
. .
&c.,
at all points external to all the surfaces.
Sm
,
the
DEDUCTIONS FORM GREEN*S THEOREM.
12.]
u
Let
= W+
V.
Then u has some constant values on each of the
Sl
13
surfaces
...Sni and the given constant values on each of the surfaces
K...8.'. Also
Similarly,
= &c., V w= V JF+V F=p &c.
2
2
and
2
at all external points.
If u be a function which satisfies the conditions (l), (2), and for which V 2 u has any assigned value, zero
12.]
and
(3) of Art. 10,
or otherwise, at every point not within any of the surfaces, then u has single and determinate value at each point in external space.
For
u' be two functions, each of degree less than the surface conditions, so that u and u' are both J, satisfying constant on each surface, and let
u and
rr
J( 3 di* "
JJ and so on
for each of the surfaces.
V2 ^
Also
or
and
VV both
have the same given value at each and therefore V 2 (^ #') = at
point in the external space, every point in that space.
Then
-
V (u-u) dxdydz 2
fff(u- u')
+ ^-u, rrd(u-u) ,
.
=
)JJ
jjj 0.
-^-^dS.
2
u-V*u'} dxdydz
DEDUCTIONS FROM GREEN'S THEOREM.
14:
du jefo?
Therefore
and since
= du' T~
= w' at
?*
du T-
>
cfo
an
= du ;r~'
,
and
^
u
infinite distance,
du
:r "
[13.
= du T~'
^
= &' at every point not
within any of the surfaces. 13.]
We proved in Art. 10 that if
the surfaces
Slt S2
...
/ /
-7-
dS be given
Sn and u be constant on each
of degree lower than least value when
V% =
,
for each of
surface,
and
then the triple integral Q u has its at each point in external space.
\,
/
/ "j
We
can
now prove
dS
that given
as before for each
= p at each point not within any of the and given and u of degree lower than J, Q u has its least value when u is constant over each surface. For let u be the function which satisfies the four conditions of Art. 1 0, u' any other function
V%
surface
surfaces,
of degree less than J which satisfies conditions (3) and (4) of that Article, but is not constant on each of the surfaces 8l ... Sn .
Then
u'= u+u' and
if
Qu
spectively,
and
we
u, f
Q M denote the triple integral Q for u and u rehave, as in the preceding articles, /
which the double integral is understood to relate to each of the surfaces in succession. The second line is zero by the condiin
tions,
and therefore
The theorems
if u' differ
from
u,
of the last three articles can also be extended to
the more general case in which the value of
d {J^du d fv du J-(A dx')+-r- (K -j-) dx^ dy^ dy' instead of
and
V2 u
d
,
du.
Tr + T- (^ dz^ dz')> -
or
V*K u
are given within the limits of the triple integrals,
1
DEDUCTIONS FROM GREENES THEOREM.
4.]
K
where
is
and constant
positive
lower degree than
1
we have only
For,
for each surface,
15
and Ku* of
.
to replace
V2 u
by the more general
ex-
pression
d
.
Tr
du
.
dx^(K dx') and
d
+
d
du.
.
(K
dy^
)
dy'
.
du^
Tr
.
+ dz^(K dz'),
or
V*KU,
Q u by
and every step in the process applies as 14.] Again,
if
S
before.
be a closed surface, or series of closed surfaces o- be a function having arbitrarily
external to each other, and if
assigned values at each point on u satisfying the condition (1)
=
(2)
Vu= 2
u
(3)
is
a-
S,
at each point on
S
t
at each point in external space,
of lower degree than
For there must be an satisfy the conditions
J.
infinite variety of functions
(4)
/ /
UcrdS
trary quantity differing from zero,
than
there always exists a function
=
and
E
where
JE,
U
(5)
is
is
U
which
any
arbi-
of lower degree
J.
For any such function the integral Qu must be greater than There must therefore be some one or more of such functions for which this integral is not greater than for any other. Let u be any such function. Let u + u' be any other function satisfying zero.
(4)
and
(5),
Then -,-
oc
o-,
it
and
for
and
V2 u =
and that
preceding,
0.
may
at all points external to S, and that
we may make
QU + U
'
=
Qu + Q u
f >
=
-=
cr
V2 ^ =
and
by as
CuV
This theorem
be extended to the case in which
being zero, has assigned values at space.
=
can be shewn by the same process as in Art. 10 that
properly choosing E before,
which therefore Tlu'trdS
all
points
also, as in
y
2
u,
the
instead of
in the external
DEDUCTIONS FROM GREEN'S THEOREM.
16
[15.
Again, as there always exists a function u satisfying the conditions, so it can be shewn that it has single and determinate value at For,
all
external points. there be
if possible, let
two functions u and
u' of degree f 7
less
than
i,
at each point
both satisfying the conditions, so that
on
external points.
and
8,
V 2 u = V 2 */,
or
=
u',
-7-
dv
u) =
V 2 (u
sJ
=
-7
dv
at all
Then
= 0, and therefore an
-=-
=
da
-=, dx
&c.,
and u
since both vanish at
infinite distance.
We
can shew also by the same process that there exists a 15.] function u satisfying the condition that y 2 ^ at all points
=
in the internal space,
=
and
o-
(JvV
adS ffFor
that
if
condition
ua-dS=E might
=
were
at all points on 8, provided 0.
not
the
satisfied,
condition
be satisfied by making u a constant, in
which case Q u would not have a minimum value greater than In fact, if V 2 u zero, and the proof would fail. everywhere
dS and therefore -=- cannot be equal to ; dv JJ dv rr points on 8 unless / / a dS = 0.
within at all
If 8,
8, / / -r-
V 2 w,
instead of being zero,
the problem
maybe
is
to have the value p within
solved, provided //
=
fffpdxdydz,
as follows.
Let within
W S,
be a function such that
V 2 ZF=
and therefore that
dW Ji
dSt
p
at
every point
1
DEDUCTIONS FROM GREEN'S THEOREM.
6.]
Then there
dV at each point on S,
= V+ W.
(T
dv
V V 2
and
that
dW
=
dv u
V such
exists a function
at each point within S.
Let
Then du dv
at each point
on
S,
dV
dW _
dv
dv
and
V w = V F+V 2
2
2
JF
p
at each point within $. It can easily be
shown
also that if
u and
=^
both satisfying the conditions,
,
u'
be two functions
&c., at all points
within
and therefore u can only differ from u by a constant. 16.] Let p, g, r be any functions of #, y and z, each of degree less than f, satisfying the conditions
S,
lp
at every point
on
$,
where
without
is
a-
dp dx at all points
any
(1)
dr
dy
dz
and
S.
exists a function
u of degree
less
J satisfying the conditions
du dv at each point
on
S,
du
du
du
dx
dy
dz
~~
d du
at all external points.
du satisfies
(1)
It can
I.
and
now
~
_^
and
^u = dx
VOL.
arbitrary function,
dq
Then we know that there than
=
+ mq + nr
dx
d du 1-
-_
dy dy
d du 1
dz dz
=0
Therefore the system
_
du
(2).
be shewn that the integral
du
(3)
DEDUCTIONS FKOM GREEN'S THEOKEM.
18
extended throughout the space external to
p
=
dx
,
than when
&c.,
jp,
q,
du
du
has less value
when
and r are any other functions
f satisfying (l) and
of degree less than
S,
[17.
(2).
For
if
du
be any other three functions of the required degree satisfying
and
(1)
(2), a,
&
and y must la
at each point
on
S,
satisfy
+ mfi + ny =
Q
(4)
and (5)
dz at each point in external space.
Then
n ***** dx
By and
'
dy
integrating the last term by parts, and attending to (4) we prove it to be zero. Because na, ufi, and uy being (5),
of less degree than
2,
the double integrals
vanish for an infinitely distant surface.
is less
\\uadydz
&c.
Hence the integral
than
A
corresponding proposition can be proved for the space within S without restriction on the degree of p, q, r, and u. 17.] The propositions of Arts. 1 4 and 1 5 can be extended to the case in which
K-=dv
is
written for
dv
at the surface,
and
Via
1
7.]
for
DEDUCTIONS FROM GREENES THEOREM.
V 2 u at points in
space
;
and Art.
1
6
may
19
be similarly extended
to prove that 1
K has a
minimum
value
when du
du
du
dx*
dy*
dz
K
being in each case a given positive function of such that Kp &c. are of lower degree than f.
c 2
a?,
y,
and
0,
and
CHAPTEK
II.
SPHERICAL HARMONICS. If u be a homogeneous function 2 o, degree in as, y, and 2, satisfying the condition V u represents the operation
ARTICLE 18.] tb
of the
ft
where
V2
then u
is
Definition.
=
said to be a spherical harmonic function of the
in #, ^, and z. If % be any function of x, y,
V2 u =
o,
then >
z
satisfying
th
degree
the condition
coefficient
differential
every partial
ft
of
u,
as
will also satisfy the condition
^^ +
y..
dx*dydz
v
~ -Q
For since the order of partial differentiation follows that
is indifferent, it
dM^
***
= 0. be taken as origin of rectangular coordinates, and let the coordinates of
19.] Let any point
H
P
be
Let $
be any Let OH be any axis drawn from and designated by Ji, and let Q be any point in this axis, and let OQ = p. Ije ^ f> *7j C b e the coordinates of P x, y, z.
function of #, y,
referred parallel to the axes
the ratio
through
0.
to
Q
(#, y, z)
z.
as
origin,
with axes
Then the limiting value
of
1
21
DEFINITION.
9.]
as p is indefinitely diminished, is denoted
by
IJ#fe*;4 It is clear that
d
is itself
w d(j>
or
^<MW),
generally a function of x,y z\ and therefore 9
axis OH', denoted
by
h',
be drawn from 0, we
may
if
another
find
by a
similar process
d
and so on
for
any number of axes.
If u be any function
V2 u
=
0,
and
if
h^ ^2
,
of #, y,
^
...
z
satisfying
the condition
denote any number of axes drawn
from the origin, and the expression
A.A, '" JL. dl\ dh 2
dh{
be found according to the preceding definition, then
WU ^\'''~dhi For
by
let 119
m lt
>
% be the direction cosines of the axis h v
definition
du
But by hypothesis
,
du~
du dx
V
2
n 1\
i l
dy u
=
du ~J~
dz
0.
Therefore
vi*!, dx are severally equal to zero.
v -, 2
v
dy
2
dz
Therefore
and therefore by successive steps
v *^_*
J*
o.
Then
SPHERICAL HAEMONICS.
22 20.]
-
is
[20.
If
a spherical harmonic function of degree
d
1
.
,l
For
(
*1
Similarly
(I)
!* -_.! + 1
(-)
=-
d*
5
-^
+ -^-
*-"*^ 3z 1
1
and
2
whence
W + ^^ + *
J
d!
cte
2'
3
1_"
"
r3
r
Whatever be the
21.]
_ " __ 3
3(s
"*
r5
""
"*"
r
directions of the i-axes k lt ^ 2
function
JL
d
\
__
"
r3 ,
... /^,
the
M
~dh^""dh^ \ r
where
M
any constant,
(i+
degree
For
is
it is
is
a spherical harmonic function of
1).
evidently a homogeneous function of that degree,
and since
it
follows that
If
we
write this function in the form
1 1
of of
M, the direction cosines of the axes To fix the ideas we may conceive a
fi
-^ Y is a function
l}
>
&2
,
i
t
.*h it and those
sphere from the centre of which are drawn in arbitrarily given directions the ^-axes ... H^ Then OHi, Off2 ...OHi cutting the sphere l9 2, r.
mff
,
if 1^.
be any radius, at every point on OQ or OQ produced has a definite numerical value, being a function of the di-
Offly
rection cosines of
or
H
P
OQ
OP.
If h lt / 2
,
...
hi
OH^ and of OQ, and independent of r be the fixed axes of any harmonic, P any
...
SPHERICAL HARMONICS.
22.] variable point,
axes
h-fr
^2
Y
i
P
at
23
spoken of as the harmonic at
is
P
with
%i-
5
Since each axis requires for the determination of
its
direction
two independent quantities, Yi will be a function of the two variable magnitudes determining the direction of r and the 2i arbitrary constant magnitudes determining the directions of the 2- axes. also be expressed in terms of the ^-cosines t may
Y
fjiiy
H2
,
...
fa of the angles
- cosines of
the angles
2
and an expression
much
made by
for
Y
i
r with the 2-axes
and the
made by the axes with each form
in this
other,
be found without
may
difficulty.
22.] If
T
t
be a spherical harmonic function of degree
and if r = \/ 2 +^ 2 + ^ 2 , then monic function of degree i. For by differentiation
r 2i+l
V
i
(i+
I),
will be a spherical har-
= (2 * + 1) r '- x V + r"+1 2
1
i
Similar expressions hold for
Adding
these expressions, and remembering that
we obtain
V
1
2
(r**
V) t
=
(2i+
1) (2
1
+ 2) r
2 *'- 1
V
i
j
4 -t-
\
24
SPHERICAL HARMONICS.
and
=-+>". 5+^-5 = V V 2
[23.
0.
i
Therefore
= and
and
r* is
We
i+ ^
0,
a homogeneous function of as, y, z of degree therefore a spherical harmonic function of degree i. Fi is
i:
Y.
have seen that -~j , as above defined,
monic function of degree
(i
+
is
a spherical har-
1).
It follows then that or ri Y. is
a spherical harmonic function of degree
i.
23.] Every possible spherical harmonic function of integral positive degree, i, can be expressed in the form r Yi if suitable i
fi ^ 2 ... & t determining Yt l9 a of the i ih degree be function homogeneous Hi
directions be given to the axes
For
if
contains
-
-
2
of the degree
arbitrary constants.
2 contains
i
In order that
V 2 ^ may
-
-
leaving
^
'-
V 2 /^
being
arbitrary constants.
V 2^
all values of #, y, and z, must be separately zero.
- relations between the constants in
-^- -2
Therefore
L-
*-
it
*
be zero for
the coefficient of each term in
This involves
.
,
2i+l
or
2
H^
them independent.
of
Therefore every possible harmonic function of degree i is to be found by attributing proper values to these 2i-f 1 constants. But the directions of the ^-axes ^2 ... ^ involve 2 i arbitrary
^
constants,
,
,
making with the constant M, 2 i -f
therefore always possible to choose the e-axes the constant M, so as to make ....
r
M
d d d TnT'Twr -:/* dfi dhi r dh^ 2
.
'
or
r
Y
1
^
in ,
h2
It is
all. ,
. . .
li
i
and
SPHERICAL HARMONICS.
24-]
25
equal to any given spherical harmonic function of degree i. Therefore ri i is a perfectly general form of the spherical har-
Y
monic function of positive integral degree i. Again, every possible spherical harmonic function of negative
Y
integral degree
For
if
V
i
it
(i+l),
(i+l) can be expressed in the form -j~
be any spherical harmonic function of degree follows from Art. 22 that r 2i+l 7i is a spherical
harmonic function of degree follows
by the former part of
i. Hence, i being integral, it this proposition that r 2i+l 7i can
i always be expressed in the form r Yi by suitably choosing the axes of Yit and therefore that Vi may be expressed in the form
,
Therefore ri Yi and -~^ are the most general forms of the
harmonic functions of the integral degrees
spherical (i +
Y
i
1
is
where
i
)
i
and
respectively.
defined as the surface spherical harmonic of the order is
i
always positive and integral; r
the solid harmonics of the order 24.] If J^ and
Y
i
i,
'Y-
and
-^
are called
i.
Yj be any two surface spherical harmonics 0, and referred to the same or different
with the same origin axes,
and of orders
i
and j
respectively,
and
if
/ /
Y Yj dS i
be
found over the surface of any sphere with centre 0, then
YjdS=
0,
unless
i
= j.
Let H. and Hj be the solid spherical harmonics of degrees and j respectively corresponding to the surface harmonics Yi and J,, so that = r*Yit ffj = rJYj. t
i
H
Make U and
U' equal to fft and Hj respectively in the equation of Green's theorem taken for the space bounded by the aforesaid spherical surface, then
ZONAL SPHEKICAL HARMONICS.
26
rrH JJJ
(
because sphere
and
r
dx
^ d_Ei
^'
d
dx
dy
V 2^
and
V
2
dy
ffj are
[25.
dffjd dz d
each zero at every point within the
;
similarly,
being the radius of the sphere
that
;
is
or
j)
(i
therefore either or
i=j,
The points in which the axes h^ h^...h i 25.] Definition. drawn from any origin meet the spherical surface of radius unity round
When
as centre are called the poles of the axes h^ ^2 ... h it these poles coincide, the corresponding spherical har,
all
monics are called zonal spherical harmonics solid and superficial respectively, referred to the common axis, and the surface spherical
harmonic of order
i is
in this case written
Q
t
>
be the cosine of the angle between r and the common axis in the case of the surface zonal harmonic Q t of order i, then If
Qi
is
ju
the coefficient of
e
in ascending powers of
i
in the expansion of
e.
ZONAL SPHERICAL HARMONICS.
25.]
OA
Let
POA
be
and
axis,
OP
let
be r and the angle
0.
OA
In
common
be the
27
M
take a point
at the
V
distance p from 0. Then if be i the solid zonal harmonic of degree (i
+
1)
corresponding to the sur-
harmonic from definition that face zonal
Q i}
follows
it
o
F-/*vJL i
when
is
p
Let p
Fi e-3.
~dp PM
made equal
to zero after differentiation.
= er and let cos =
Then
F<
=
<
But ap
ju.
=L==
= ()*
with
and
constant
is
-
if
the coefficient of 1
T
Let
it
1
e
;
0.
therefore
.d* (-J-) X *
-
1 .
A
t
i
But
=
0.
by Maclaurin's theorem,
>
when e=0.
= -^A4
.
T'-pMii
Therefore
Q
=^ =
1
and
i
=A = ^1 = t
.
1
1
and therefore each
coefficient
Q
is
e.
is,
.
V
Therefore
whene
be expanded in ascending powers of
in the expansion
be denoted by
Hence Q
=
1
r
?
But
e
unity.
ju.
Also when
ju
=
1
28
ZONAL SPHERICAL HARMONICS.
It at
P
is
[26,
evident from definition that the zonal surface harmonic
OQ
to
referred
as
axis
equal to the zonal surface
is
harmonic
at
Q
referred
to
OP
as
axis.
26.] Let a be the radius of a spherical surface 8 described round as centre.
Fig- 4-
Let
P
8.
Let
be any point within or without
point
on
/.EOP
=
OP =/. the
E
And,
surface,
let
being any PE = D,
Then
6.
1
7
Va?7^1+
-;COS0
i
a
according or
-
1~
*
according as / or
a Therefore,
a1
if/
>
a,
But 2f
*
_ "
>
or
<
a.
as/
>
or
<
a.
ZONAL SPHERICAL HARMONICS.
27.]
29
Therefore
and similarly iff <
27.]
With
the same notation as before / /
and
a,
~=j f
3
=
/ / \j
%
when
when
J
we can prove
P is without
that
,
P is within S,
J
the integrations being taken over the surface S. Let EOP 0, and let $ be the angle between the plane of EOP and a fixed plane through OP then
=
;
d(r
=
a2
D =
2
2
Also
/* '
-2a/cos^+/
r^o-
2 ;
_ 27ra CdD
&~ ~TJ ~D*'
JJ
:
the limits on the right-hand side being
f
a and
af and
and TAZo'
5 77 s
=:
2wa
f
f+a when P is external, +/ when P is internal ;
1
1
)
T" i/r -7?^ I 1
when
p
1S
externa1 '
when
P
is
internal
,
or in the respective cases.
Hence and
f[
a
if--^[-i~-
;
ZONAL SPHEKICAL HARMONICS.
30 for external
and internal positions of
P
respectively,
[28.
and
for
both
cases I
ltf=aj
^
da-
= 4^a.
28.] In the last case let F(E) be any function of the position on the surface which does not vanish at the point in which OP cuts the surface, nor become infinite at any other point on
of
E
E
of order i, the surface, let Q be the surface zonal harmonic at be made to approach the the common axis being OP, then, if
P
surface, ultimately shall
For with the notation of the
last Article let
=// then when P approaches the is
and
/ is
indefinitely nearly every element of the integral vanishes except when is In this case indefinitely small. ultimately on the
equal to
D
surface
a,
P
and the integral has the same value as if F(E) were to equal F(P), its value at the point of S with which P ultisurface,
mately coincides, or
u
=
(P)-
da-
= F(P)
-dff
when/= a
Therefore
=
>naF(P) by the
last Article.
Suppose that/ is originally greater than
& i
and
i
#,
da;
then
ultimately.
ZONAL SPHERICAL HARMONICS.
2Q.]
31
And, by Art. 25,
i=f{&+.f +,+*} we get
Performing the differentiations and substituting,
29.]
and
if
If
Q
i
Y
be any surface spherical harmonic of the order
{
i,
be the zonal surface harmonic of the same order and
origin referred to any axis OP, and if da- be an element of a as spherical surface of radius a described round the origin centre, then
P
Y
where (7^ is the value of t at the pole of Q { tions being over the spherical surface dS. Substitute i for F{E) in the last proposition.
,
the integra-
Y
Then F(P)
7
>
(at
P)
the value of
is
Y
i
at
P
;
=
And, by Art.
24, each double integral vanishes except
or
if
(7^) denote the value of
By
putting
Y =
since,
by Art.
25,
t
Y
i
at P.
Qi we obtain
4
=
1
at the pole,
ZONAL SPHERICAL HARMONICS.
32 30.] If
[30.
F(E) be a
then, whether
spherical surface harmonic, i.e. be on the surface or outside of it,
P
F(N)
=
Y^
F.or
by Arts. 24 and 26,
where
Y
the value of (1^) denotes
harmonics, that
i
at the
common axis
of the zonal
along OP.
is,
Therefore
rr-pinZ
a
f+i
m-
;?>
31.] Considered as a function of
-
.
2
coefficient
the zonal harmonic Q*
of order
i,
and
We can prove the (a)
As proved
derived by the expansion of
//,
is
called the Legendre's
P
frequently written { following properties of the coefficients P. is
above, if
p
=
.
1,
l-e Hence,
if p =
1,
P = t
1
for all values of
1
Hence,
if
^
If u
<
1
Hence the
=
1,
P =+ i
==
'
series
i
;
if
IJL
=
1,
1
or is
1
according as
always
P + P2 +... 1
is
finite,
and
a convergent
i is
even or odd.
is finite if e
=
1
.
series.
It is evident from the formation of P^ as the coefficient of (V) i e in the expansion of
P
4
but can contain no than //, and no powers of which the index higher powers of differs from i by an odd number. Hence if i be even, Pi has the that
t
must contain jut
2
/u% ju*~
,
jut*"
,
&c.,
31.]
.
same value
ZONAL SPHERICAL HARMONICS.
for -f /x as for
and
p,
be odd the same value
if i
with opposite sign. Hence also // can be expressed in terms of 1
p Pj d =
Jf i
(c)
= For since ^
= cos 0, dfji
P
Also
-
:
if i
2t+l
=
=
it
f
2
,
&c
?.
sm0d0.
and P^ are both functions of
i
P P_
ifi=j,
.
fJ
i
33
p,
and therefore of
0.
Hence
PP< Pj dp =JoFpi Pj sm0d0
J-i
over the surface of a sphere of radius a
=
;
by Art.
24, unless
|a/ And if = /, i
ri (d)
/
J-i
PipSdp
For expanding
^
number of
into a
=
tf i
>j, or
if
J
i is
in terms of the P's, the integral
integrals of the form /
P
^-i
which (e)
i
=/, and
To
number Let
is
odd.
{
Pj dp,
resolved
is
in each of
therefore zero.
find the value of
/x'P^, where
/
^o
K is
any
positive
integral or fractional *. a i
P =
Then
K , '
r- , '
* See Todhunter's Functions of Laplace,
VOL.
I.
D
., ., if i be even.
if i
be odd.
Lam^, and Bessel, Art.
34, 35.
ZONAL SPHERICAL HARMONICS.
34 Let
i
Then if K has any member
be even.
or zero, the left-hand ri / Jo
K=
and therefore
I
pftpf *fft
of the values
[31. i
2,
J
4,
&c.,
ri
= i/ ^ *J-i = 0,
0.
It follows that ^T
Also A
= A.K.K
4...K
i
+ 2.
the coefficient of the highest power of K
is
P
= Hence,
2K
i (fji)
when
1
1.
K.K
Similarly, if /
=
therefore
be even,
if i
Jo
fj.
;
^
K
i
2
...
K
i
be odd,
P
i
du,=
If K be either an integer or a fraction whose denominator to its lowest terms is odd, then
when reduced -l
JO
= 0,
if ju*
P
i
if /x"P<
does not change sign with
does change sign with
Hence any
/u,
/x.
function, f(^)^ which can be expanded in a series of positive powers of ju,, whether integral or fractional, can be expanded in a series of the form
For we have
1
2
2i+T- * or
ZONAL SPHERICAL HARMONICS.
32.]
A
which determines
{
,
if
known
is
/(/n)
powers of JJL. It is perhaps necessary to
show that the
35
terms of positive
in
series
converges, if /(ju) can be expanded in a converging series of ascending powers of ju.
For
A
K
be any term in the expansion of f(^). derived from this term in ^ is
let cK
term in
i
Then the
fji
Ai+2
and the corresponding term in
is
2
from which
it is
easily seen
from the expressions
above obtained that, if i be large enough, A i+2 Now the series Pl + P2 ~f P3 converges. .
Hence
AQ + A P 1
1
-{-
for /
< A
.
^-i
{
.
.
.
A 2 P2 -f &c.
converges.
We
have hitherto regarded the coefficients 32.] functions of /x derived from the expansion of
Q
or
P
as
Vl
We may
however take for with the common coinciding
common
OH
axis
radius any line OC not and the direction of the
initial
axis,
of the zonal harmonics
be defined with
may
by the usual angular coordinates, namely, and a 6' and the Z.HOC, $' angle between the plane fixed plane through OC. In this case the angular coordinates
reference to this line
=
HOC
defining the direction of OP or r will be cosine of the angle will be
and
0,
and the
HOP
cos 6 cos tf
Now Q
t
is,
as
+
we have
sin
sin 0' cos (0
').
seen, a function of cos
HOP, and
is
therefore a function of cos
cos 6'
+
sin
It is evidently symmetrical
the value of
Q
{
at P,
sin 0' cos
(<
/).
with regard to
when O// is D 2
the
common
and
0'.
So that same
axis, is the
36
ZONAL SPHEKICAL HARMONICS.
as the value of
In this form 33.]
Of
harmonic
Q
Q
t
the
common
axis, see Art. 25.
the differential equation which a spherical surface
definition
If
we change >,
is
called a Laplace's coefficient.
is
satisfies.
By
r, 6,
OP
when
at H,
t
[33.
any spherical harmonic u
satisfies
the equation
the variables to the usual spherical coordinates
where
x
= r sin 6 cos
= r sin 6 sin
y
(f> t
r cos
z
,
Q,
the equation becomes
dzu
2
~d^
Let u
=
Now Y = i
1),
r
t
dz u
and
r2
is
2
sin 6
dfi
the above differential equation. i+l u is independent of r, therefore r i
u, r,
whence
1
d*u
+
2
du
dr
+
=
ri+1 dr*
y.
0,
ii
=
difierential equation
becomes
2
d u
Let us now change the variable from
=
.
T
1
cos 9
_
^~ 5^ ~
satisfies
where
-
cot# du
4
a spherical harmonic function of
(+ l)^'" ^ + 2 (i+ 1) r
and
Hence the
dz u
I
+
Then u
i+l
independent of
and
1
^d
-r^j
degree ~(i-tis
du
+ rd^ + 7
to
cos#, and
Then d
.
.
du
Substituting in the differential equation,
we
obtain
let
ZONAL SPHERICAL HARMONICS.
34-]
or restoring
This
-^
for &,
true for any spherical surface harmonic harmonic Q t as a particular case.
is
37
Y^ and
therefore
for the zonal
In the case of the zonal harmonic, if the common axis be taken for the initial line from which is measured, Qi * S as above J
mentioned, written
P
if
and
P
i
is
independent of
>.
Hence
P
t
the equation
satisfies
34.] If we differentiate equation (4) of last Article k times, obtain the equation
we
From
(4)
and
(5)
above
it
appears that
P
i
and
-~* respectively
satisfy the differential equations
We may also (6)
both
For
P
and
t
and integral in
finite
if in
prove that
A
and
A
If A'
=
0,
y
= AP
t
,
we
write
P^
for y,
we
u which gives on integration
are arbitrary constants
mences from some fixed limit
are the only solutions of
p.
the former of equations (6)
obtain a differential equation in
where
^
and the integral com-
;
an integral
finite solution in
38
ZONAL SPHERICAL HARMONICS.
[35.
If ^'=^0, the expression fory contains the term
A.,P-
T
d^
I
3
and therefore can be neither finite nor integral. Hence Pt or A P. is the only finite integral solution in /u of the former of equations (6). And in the same way it may be d k P-* .
proved that
is (I
the only
finite integral solution in
second of equations
T-l) For
if
we multiply the
2
(1
/m )*,
it
may
and changing
Jc
left-hand
side
...
into k
1
parts,
this
of (5) of Art.
becomes
we get
since the integrated terms vanish
;
and therefore
+!
/1
<1
we may
(i-k+l) when i =j.
be written
But integrating by
^
of the
(6).
By means
of equation (5) of the last Article the generalise proposition of Art. 24 by proving that 35.]
p.
JU.
-
and therefore by successive reductions,
-i
34 by
EXPANSION OF ZONAL SPHERICAL HARMONICS.
36.]
and therefore by Art. /-7&
/+!
M
1
(
r^
)
31,
Jlc ~p
~p
~~
d ju
'
To expand Q*
36.]
39
if i
~T^*
in a series
j
of cosines
of multiples
of
(*-*') Since {
it
Q
the coefficient of
is
f
2e (cos
1
cos 0'
+
e
in the expansion of
sin 0' cos (0
sin
follows that the term in
i
Q which f
<'))
+e
2
}"*,
involves cos
((/>
<')
must
fc
contain (sin 0) as a factor, or in other words, that the required expansion of Q t must be of the form
+q
g
l
cos
(<#>
>')
+ &c. + q k cos k ($
+ &c.,
')
where q k = (sin 0) k f (cos 0), and the function denoted by f is rational and integral. If we perform the requisite differentiations on Q iy substitute in (6) of last Article and equate to zero the coefficients of cos
$'),
(>
=
where y
And
we
cos
since
obtain the equation
0.
/ is
a finite integral function of
follows from
y, it
Art. 34 that
A k is independent of y or of 0. Now Qf is a symmetrical function
of
that
Ak
where
denote cos
where
P
i
6'
is
by
y' it will follow
the same function of y that
r
and if therefore we must be of the form ,
P
i
is
of
y,
and therefore
that ft
=
a P,
P/+ a
x
sin
-h
sin 0'
a^ sin
^ ^-
0^"
.
sin
/*
cos (0
^/tp d ---
-j-^
k
<J>')
P
+ &c.
/
-=-7^-
cos k
(<
^>')
+ &c.
40
EXPANSION OF ZONAL SPHERICAL HARMONICS.
For most of the applications of numerical coefficients a
,
Q
37.]
the actual values of the
a l5 a2 are not required,
however be determined without much
[37.
may
they
difficulty as follows.
To prove that 2
For k
it
p'
-
sin 0* sin 0'* cos
to Square both sides and integrate with respect to $ from 27T, remembering that the integrals of all terms containing
products of cosines of unequal multiples of <$>' are zero, and that the integrals of all quantities of the form (cosm($ are equal to
TT
and the integral ofafPfPj*
is
27ra 2
Again, integrate both sides with regard to y from
P P/ 2
2
i
1
;
to
+ 1,
remembering that
and we get 2
{
+&, sin^f + & c.
But
if
0=0' and
For in this case
=
<|>',
Q = (
1
;
(a)
;
^ becomes
The two expressions on the right-hand sides of (a) and (/3) cannot be equal for all values of 0' unless the corresponding terms are separately equal ;
EXPANSION OF ANY SPHEEICAL HAEMONIC.
39-]
'*
41
-
rfkp
/;
38:] If YI be any surface spherical then is a rational and integral {
harmonic of the
Y
cos
and
(b,
sin
_ d d -rr =l-jah ax
d d '~'''-
d
'
where
I?
where
p
l
" p
cty
x
also
d,
fl\P f
p,
/i,
d
+^3-; dz
ay
?
axis
m
\
m
Ts, and so of
m" and nT and ,
^.
o-
a
d
+m
means the product of
+ +T =
But
sin
dht
m, n be direction cosines of the
I,
0,
order,
...
dh^
dh. \i_
if
of cos
tb
for .
Also
function
i
(-) r'
dz
r cos
cos
=
^>,
,*
2
r2 "
=
i/
r sin
sin A
A sin
where
,
=
>,
cos 0^ cos
d)
v
r cos
; ^
sin
71
rf)
therefore J^ is of the form stated above. 39.1 Y, is of the form
2* where
a,K
and
^,. tC '
sin/t(/)
B
may assume A k
=
i
sin
A;()
we may assume and
Y
sin
3
7i:
-
are numerical constants.
It is clear that
in
dkP
a cos ^() +
to be
Ak
k
,
the coefficients of cosk
where
ft
^
6 to be determined.
Also we v to
be determined.
a A sin
k
v,
where a k
is
constant and
EXPANSION OF ANY SPHERICAL HAEMONIC.
42 But
Y.
and therefore the
every multiple of $
Now \/l
y
vsinflfc is to 2 ,
coefficient of the cosine
[39.
and sine of
the second of equations (6) of Art. 34
satisfies
;
be a rational and integral function of y and
which clearly cannot be attained so long as the second
term in v remains;
= a, sin
^ rk dy
Similarly
where a k and p, are numerical constants .'.
;
= 2 ac
Y.
the constants a^ and
depending upon the directions
(3 k
of
the
a'-axes.
It has already been proved, Art. 24, that
rzn
ri
J/
i
/
YiYj
./
unless i =/, aad we may now see that the same result follows from the general form of the function Y{ or Yjt
For and
Y = t
Yj
S^c
= 2 (a/ cos k $ + ft
7
sin
k <j>) sin 0*
multiply Yi by J} and integrate with regard to to 2 TT, all the terms will vanish except those in which the multiples of are the same, and the result therefore will be If
<
now we
from
<
of the form
^XLIFOR^
EXPANSION OF ANY SPHERICAL HARMONIC.
39.]
we again
If
integrate with regard to y from
1
43
+1, the
to
form
result will be of the
and by Art. 24 each of these terms
zero unless
is
i
=/.*
It does not follow that
rz*
/ii is
always
finite,
Y.Yj
/ Jo
inasmuch as the values of the ^'s may be such
that although each term in the integral be equal to zero. The values of the inclinations of the
when
two
is finite,
Aa 9
their
sum may
depend upon the Yi and J/, and
sets of 2-axes of the
these axes are so related that
PS*
/i
-i
Y.Yf
\
Jo
the two spherical harmonics are said to be conjugate. For example, take two spherical harmonics of the first order Yl and F/. If & and $' be the polar coordinates determining the axis of Ylt and 0" and <" those for the axis of 7/, then is zero,
Y may l
be easily seen to be cos 6 cos 'tf + sin
and similarly 7/
is
cos
cos 0" -f sin
sin 0' cos ($ sin
0" cos
$'),
(<$><$>"})
and 2
-l
[ "Y TfdydQ = JO 1
^
(cos 6' cos
0" + sin
+
if
^,
?#,
tf sin
sin 0' sin
6" cos
0" sin <'
# be direction cosines of the axis of Y,
l' t
In a similar manner the proposition of Art. 26
II-" may be deduced from
cos
sin
"
c/>")
m'y n' those
of Y'. *
'
the form of Yi proved in this article.
44
EXPANSION OF ANY SPHEEICAL HARMONIC.
[39.
If therefore these axes are at right angles to one another,
/
-
-i
or two spherical harmonics of the first order are conjugate their axes are perpendicular to each other.
For the second and higher orders there geometrical relation.
is
when
no such simple
CHAPTEE
III.
POTENTIAL. ARTICLE 40.]
IF the forces acting on a material system he such that the work done by them upon the system in its motion
from an
initial to a final position is,
whatever those positions
a function of the coordinates defining those positions only, and independent of the course taken between them, the system is said to be Conservative. The w6rk done by the forces
may
be,
on the system position which
ill
its
may
motion from any position S'to any given be chosen as a position of reference, is
defined to be the potential energy or shortly the potential, of the system in the position S in relation to the forces in question. If we denote by U the potential, and by T the kinetic, energy r
,
of the system, then, as
T+ U
shown
in treatises on dynamics, motion of the system under the
constant throughout any If q be any one of the influence of the forces in question. generalised coordinates defining the position of the system, it is
follows from definition that
-7- <>q is the
work done by the
dq forces
on the system as q becomes q +
tending to increase the coordinate q
<),
is
and
therefore the force
j-
If the system be a material particle of unit mass, situated at the point P, we may without inaccuracy speak of the potential as the potential of the forces at P. are in this chapter concerned only with 41.]
We
forces
of
attraction and repulsion to or from fixed centres, the force varying inversely as the square of the distance from the centre.
Now
if
distance,
the central force be any continuous function of the whether varying according to the law of the inverse
square or any other law, a potential exists.
j?
THE POTENTIAL.
46 For
be at
let there
[4'.
a particle of matter of mass
m
which
mass m' with the force mm'f(r\ where repels any other particle of function of r, the distance between' continuous (r) represents any
f
m and m' ; then it can be shown that if be fixed, the work done by the force upon m' as m moves from a point at the distance r from 0, to another at the distance r2 from 0, is a function of r^ and r2 the initial and final values of r, and of these ,
quantities only, and is independent of the form of the curve described by m' between these initial and final positions, and of the
in which the distances from For at any instant during the motion
directions
^
and
m
let
r2 are
measured.
be at P, and
let
Q be a point in the course indefinitely near to P. Let PQ ds, the angle
=
OPQ =
4>,
In the
OP =
limit,
r,
if
OQ = r + dr. Q
be taken near
enough to P, the force of repulsion may be considered constant, as m' moves Fig.
5-
from P to Q, and equal to mm'f(r)._ Therefore the work done by the force in moving the repelled particle
from
P
to
Q
is
mm'f(r) eoaQds, or mm'f(r)dr, and
independent of c/> if dr be given. Therefore the whole work done
from distance
r
to distance r
mm'
from
by the
force in the
is
motion
is
f(r\dr,
and depends upon rx and r2 and these quantities only. We have for simplicity considered m fixed at 0, but the proof evidently holds if both m and m' be moveable, and move from a ,
distance r^ to a distance r2 apart under the influence of the mutual repulsive force mm'f(r). If the mutual force had been attractive instead of repulsive, in other respects following the same law, the expression for the work done would be the same
as that for the repulsive force, but with reversed If in sign. case on the the for the any effecting integrations expression
work done prove
to be negative, this result must be interpreted as expressing the fact that positive work is done against, and not by, the force in the motion considered.
THE POTENTIAL.
42.]
In either
case,
the work done
47
whether the force be repulsive or
attractive,
proved to be a function of r- and r2 only, and independent of the course taken between the initial and final positions of
is
m.
We have thus shown that
if
f(r) be any continuous function of
the distance between the two particles m and m', a potential exists. At present, as above stated, we are concerned only with the case in
In that case the work done by the
which f(r) =-g
mutual force between
m
and
m', as their distance varies
from r
1
(*'2
^dr, that is
'-#
mm and
if
the force be attractive
mm,(1
1 }
<
J--
42.] We shall now consider two kinds of matter, such that two particles, both of the same kind, repel one another with a mutual force varying directly as the masses of the particles, and inversely as the square of the distance between them, and two particles of different kinds attract one another according to the
same law.
Then the work done by the mutual force between two particles m and m\ as they move from a distance r to a distance r2 apart,
is,
if
the masses be of the same kind, and therefore the
force repulsive,
and
if
.
.
mm' <
^
;
they be of different kinds, and the force attractive,
mm/u < t*i
If
^
now we
positive,
>
particles of one kind of matter as all particles of the other kind as negative, we can
agree to regard
and
n
r*>
all
combine both results under one formula
mm in
which
m
or m'
may have
i"^r either sign, expressing the
work
THE POTENTIAL.
48
done by the mutual force between
[43.
m and
m' in the motion from
distance r to r2 apart. Finally, we will take for the position of reference to which the position in which the two particles potential is measured, are at an infinite distance apart, that is, in which r2 is infinite-
Then we shall arrive at the following definition. The potential of two material particles m and m\ distant r from each other, is the work done by the force of mutual repulsion as r
/ when
r2
is
infinite,
that
is
-
j
and
is
zl dr, ~2 r
positive or negative
m and m' are of the same or different kinds of matter. In physics a body which is within the range of the action of another body is said to be in the field of that other body, and when it is so distant from that other body as to be sensibly out
according as
of the range of its action
The following
it is said
to be out of the field.
therefore equivalent to the one above two material particles distant r apart The potential of adopted. their mutual is the work done by repulsion as they move from the definition
is
distance r apart to such a distance as to be out of the field of one another's action^ attraction being included as negative repulsion. 4MU
Taking
m
1,
we
define
--
to be the potential of
m
at a
point distant r from m.
The
potential at any point of any mass occupying a finite of space is evidently the sum of the potentials at portion that point of all the particles of which the mass is composed. 43.]
If
m
be any particle of this mass, and r the distance of
the potential of the mass at
P
is
2 T
extends throughout the mass, or at #, y, z, the potential is
if p
,
m
from P,
where the summation
be the density of the mass
rrr Let
this potential be denoted
P
by
F.
of a mass at resolved in 44.] The repulsion at direction is the rate of diminution of the potential of the
any mass
THE POTENTIAL.
45-]
49
This is a particular case per unit of length in that direction. of the general theorem proved above, that the force tending to clV increase any coordinate a is If
dq be the potential of the particle m, and ds the given
V
direction,
__dV dr
_dV_~
dr
ds
m
ds
m
dr
=
the repulsion resolved in ds.
And
this proposition being true of every particle of
mass
is
evidently true of the whole mass. be the potential at of any mass
composed if
Hence,
V
P
pulsion of the mass in the direction indicated 45.] If
8
be
which the
is
any closed surface,
dS
repulsive force at
dS an
by ds
element of
M, the =-
is
ds
its area,
resolved along the normal to
re-
dS
N the
measured
outwards arising from a particle of matter of mass m placed at the point 0, then if the integration extend over the whole surface
ff NdS = 47rm, if m be within S; ffNdS = 0, if m be without S.
and Let a
line
drawn from
any direction cut the surface S at
in
P
the point distant r from 0, and let this line with the surface 8 at P.
make the angle $
Let a small cone with solid angle da> be described about from 8 in the neighbourhood of P the
as axis, cutting off
OP ele-
mentary surface dS.
The area of dS
m
from
is
,
is
equal to
+
;
in the
VOL.
two I.
,
also the repulsion at
and the resolved part
the direction of the normal to
according as without
Bm *
OP
is
in
S
at
sm
P in
.
(f>
or
passing out of
NdS = +mco>, cases respectively.
E
N
is
c/>,
S from or
of this repulsion in
drawn outwards from S
sm
P
within, or into
mdco
8 from
THE POTENTIAL.
50 But
if
as above it,
be within S, the line drawn from it in any direction must emerge from S one time more than it enters
and therefore the sum of
all
the values of
Taking the corresponding sum for all get the integral yy^V^-tf, and therefore since the
sum
lines
NdS
for this line
drawn from
we
of the solid angles about is 4 TT. S the line drawn from it in any direction in an even number of points, and therefore the
be without
If
must meet S
sum
[46.
NdS
of all the values of
for every such line
must be zero
;
therefore in this case
This proposition
is
true for any particle within or without
S
respectively.
any quantity of matter of mass manner within a closed surface S, and any if be the repulsive force of that matter at any point on S resolved in the direction of the normal at that point drawn Therefore
it
follows that if
M be distributed
in
N
outwards, then
And,
similarly, that if
and writing
in the
M be without
-j- for
8,
then
N, by Art. 44 we have
two
cases respectively. It follows from Art. 45, that if p, the density of matter, 46.] be finite in any portion of space, the first differential coefficients
of
V cannot
be discontinuous in that portion of space. For consider a cylinder whose axis is parallel to x and of
length I. Let the proposition be applied to this cylinder. If I be very small compared with the dimensions of the base, we may neglect that portion of the surface integral which relates to the
curved surface, and the proposition becomes
Jj ~fa
dydz
=-
EQUATIONS OF LAPLACE AND POISSON.
47-]
51
in which the surface integral is taken over the ends of the cylinder, and the triple integral throughout the interior space.
Also in the surface integral
dV is
-7
the rate of increase of
V
with
the normal measured outwards from the enclosed space, in the case of both ends of the cylinder. If it be measured in the same direction in space for both ends, the surface integral
may
be
written
Now if p
be
finite,
the triple integral ultimately vanishes when
and therefore the enclosed therefore the left-hand
become
space,
member
infinitely small;
also vanishes,
differ
finite
by any
quantity from (-7-)
Therefore also
continuous.
and (-j-) cannot V
JTT
777 ,
or
I,
and
-=
#'l cannot be dis-
^#'2 dx V cannot be discontinuous.
Equations of Poisson and Laplace.
of
47.] In the equation of Green's theorem let Fbe the potential any distribution of matter of which the density p is every-
where
finite,
tinuous, let
du
f
-7
dx
du' ,
But
-7
and therefore such that
8 be any ,
>
and
dV 7
dv
is
-=
are zero, the equation
And
becomes
the repulsive force of the
matter referred to
8 outwards from
the surface element
resolved in the normal to dS.
?
du'
dz
dy
closed surface,
and -7- are conj -7 -7 dz dx ay and let u'= unity. Since
therefore by Art. 45
-dS~\
**?***,.
Therefore also
= j
E 2
EQUATIONS OF LAPLACE AND POISSON.
52
[48.
Since this equation holde for every possible closed surface,
it
v F+47ip =
follows that
2
This is called Poisson's equation. at every point. At a point in free space p 0, and the equation
=
becomes
V F=0. 2
This
is
called Laplace's equation.
It follows as a corollary from Poisson's equation that if the potential of any material system at #, y, z,
r2
where
and the integral
=
(
x -x'y + (y-yj + (z-zj*
V be
;
throughout all space. can be deduced by direct differentiation equation 48.] Laplace's of For if the density of matter at #', /, / is p, the potential is
at #, y, z is
=
= Now
r/r JJJ y(^Z
(z-zj
rrr
if 0, or x, y, z,
be any point not within the mass, the
limits of the integration are not altered by any infinitely small 2 change of position of 0. Hence we may place the symbol V
under the integral sign, and obtain
VT = But
if
2 /YT/> V i dx'dy'dz'
be within the mass,
we
= 0.
cannot, in forming the triple at which the
integral for F, include in integration the point
element function - becomes
(
infinite.
It is necessary in this
* It may be proved by Green's theorem to be identically true for F) vanishing at infinity that
all
functions
the integration being extended over all space, and r being the distance from the point at which V is estimated to the element dxdydz; and this proposition may, of course, be made the foundation of an independent proof of Poisson's equation 0.
EQUATIONS OF LAPLACE AND POISSON.
49-]
53
case to take for the limits of integration some surface inclosing and infinitely near to it, and to form 7^ as the sum of two separate integrals, one
on each side of that
small change of position of the limits of integration, and
surface.
Hence any
infinitely
involves in this case a change in we are not at liberty in forming V^V
V2
under the sign of integration. This is the reason Laplace's why equation fails at a point occupied by matter. have hitherto supposed the matter with 49.] Definition.
to insert
We
which we have been concerned to be distributed in such a manner that the density p is finite, or in other words that the mass vanishes with the volume of the space in which it is According to this conception the mass of a small
contained.
volume dv of density p, is pdv, i. e. p is the limiting ratio of the mass to the containing volume when that volume is indefinitely diminished. At all parts of space for which this condition is satisfied
if
V
we have obtained the equation
be the potential of any distribution at the point at which
the density
is p.
It may, however, happen that p becomes indefinitely great at The distribution may be such that although the certain points.
volume becomes remain finite.
infinitely small the
mass comprised in
it
may
Suppose such a state of things to hold at all the points on a mass of matter comprised between any
certain surface $, so that the
portion of this surface, an adjacent surface S' infinitely near to it, and a cylindrical surface whose generating lines are the normals to
S along
taken to
S,
bounding curve, remains finite however close S' is then if the mass vanishes with the area of S, inclosed
its
bounding curve, we call the distribution superficial in distinction from the volume distribution hitherto considered. In this conception of superficial distribution we disregard the distance between S and S' altogether, and we say that the mass corresponding to an element of surface dS is crdS, where a- is the
by
this
obeing in other words defined as the limiting mass corresponding to, or as we say on, the surface
superficial density,
ratio of the
dS to
the area of dS,
when dS
is indefinitely small.
EQUATIONS OF LAPLACE AND POISSON.
54
[50.
be points for which not only p, but nalso, is infinite, and such that if a line I be drawn through these points, the mass of the superficially distributed matter Still further there
may
an adjacent indefinitely near and I at its extremities remains be taken to I. In such cases the distri-
comprised between this line parallel line finite,
/,
and perpendiculars to
I',
however near
I'
and neglecting as before the distance that the quantity of matter corresponding ^, say I ds of is the element on or \ds, where A. is the linear density to, bution
said to be linear
is
between
I
',
we
and
at ds.
On
50.]
of Poisson's equation at points of
the modification
superficial distribution
of matter.
Let dS be an element of the surface,, and let us form on dS a cylindrical surface like that mentioned in the definition of the last article.
Let
p be the
uniform density of matter within that cylindrical denote any element of that surface, including
dS1
surface.
If
its bases,
we have by
In the
limit,
when
Art. 45
the bases of that cylinder become infinitely member of this equation becomes
near each other, the right-hand
47T// vdS. either side of
And
dv, dv' be elements of the normal on
if
8 measured
in each case from $, the left-hand
t
member becomes
CCdV
dV
"//*>//'' dV or
dv
+
dV
- + 4770dv
= 0*.
* The cases of finite and infinite p have been considered separately, with the view to their physical interpretations. There is no exception in any case to the
equation
v F + 47rp = 2
discontinuous,
i.e.
0,
when
because,
v2 F
p is infinite.
becomes
infinite
whenever
-, &c. are
dx
THEOREMS CONCERNING THE POTENTIAL.
51.]
55
The mean value over the surface of any sphere of the potential due to any matter entirely without the sphere is equal to 51.]
the potential at the centre.
For
let a be the radius of the sphere, r the distance of any 2 in point space from the centre, a do) an element of the surface. Then denoting by V the mean value of over the sphere, we
V
have
=
/y
47T
~
/
t
JL^ 4-7T
dv
m
rr^F, rrdv,
= --i
da>
n n JTT
-=-a2 d
but
Hence
=
47rJJ dr
=
or
V
i
-
rrd rrdv
5 / / -jivtpJJ dr
=Q
by Art.
45.
of the radius of the
is
independent -jthe centre. sphere, and therefore equal to the potential at distributed matter of The uniformly any potential Corollary.
over the surface of a sphere, at any point outside of the sphere, is Hence the same as if such matter were collected at the centre. also the potential of a uniform solid sphere at any point outside of it is the same as if its mass were collected at the centre.
The mean value over the surface of any sphere of the within the sphere, is the same potential, due to any matter entirely
51
a.~\
as if such matter were collected at the centre. the For using the same notation as before, and denoting by in question, we have in this case algebraic sum of the matter
M
dfdr
=
rrdv ,
i
/ / -T-du 47TJJ dr
.
,
=
i
.
by Art. 45,
M
M
rrdv
; / / -jIncPJ J dr
THEOREMS CONCERNING THE POTENTIAL.
56
V=
and
t
[52.
V vanishes when
no constant being required, since
r is infinite.
52.]
The mean
potential over the surface of an infinite cylinder
due to a uniform distribution of matter along an infinite straight line parallel to the axis and outside of the cylinder is equal to the potential on the axis. the For let a be the radius of the cylinder, I its length, a the and a radius of fixed between plane through cylinder angle Let V be the axis, r the distance of any point from the axis.
the potential, a function of
V its
mean
value.
Evidently
T
7 ,
if r
be given,
is
only.
Then we have
F=
dV__"
dr
2irla
=
0,
by Art. 45
;
because that part of the normal attraction which relates to the ends of the cylinder may be neglected. It follows that 7 is independent of r, and is therefore equal to the potential on the axis. It follows also that the potential at any point outside of a cylinder of a uniform distribution of matter over the surface of
the cylinder, or throughout
its interior, is
the same as
if all
the matter were uniformly distributed along the axis, and therefore that the potential of such a uniform distribution at any /QO
point outside of
it
and distant r from the axis
is
J
/
where p Adx is the quantity of matter corresponding to a length dx of the cylinder. That is, pA (C 2 log r), where C is constant. .
.
53.]
The
potential of any distribution of matter can never or minimum at any point in a region not occu-
be a
maximum
pied
by any portion of that matter. maximum at any point
to
be a
For suppose the potential and describe a small
0,
THEOREMS CONCERNING THE POTENTIAL.
53-]
sphere about
Then
as centre.
dV ,
dv
57
the rate of increase of
V
per
unit of length of the normal to the sphere measured outwards, if the be small be must, sphere enough, negative at every point of
the surface.
Therefore -T-
therefore
or there is
p
dS
is
dxdydz
negative
is
;
positive
;
must be
positive matter within the sphere, and as this as any sphere, however small, described about there must be positive matter at 0. Similarly, if Fbe
true for
centre,
minimum
at 0, there
therefore never have a
must be negative matter
at 0.
V
can
maximum
value except at a point situated in positive matter, and never have a minimum value except at a point situated in negative matter.
53
If
V be
constant throughout any finite region free from attracting matter, it has the same constant value at every point of space which can be reached from that region without passing a.]
through attracting matter. For let the whole of space in which V is constant which can be so reached from the given region be comprised within the closed surface S.
Then on S, V either increases or diminishes continuously outwards. Let a small closed surface tf be described lying partly within S, and partly outside of it, and in the parts where V increases outwards from
S.
The normal
integral
-7- dS' applied to such
not zero, and therefore the interior space must be But there is no matter in the portion of occupied by matter. the small closed surface within S, therefore there must be surface
is
matter in the closed surface immediately outside of S. 53 $.] If two systems of matter have the same potential
throughout any finite portion of space bounded by a surface S, they have the same potential at all points in space which can be reached from that portion without passing through any matter of either system.
THEOREMS CONCEENING THE POTENTIAL.
58 For
let
V and
V be the potentials of the two systems,
V=V throughout the space enclosed greater than
V in some region
describe a closed surface
S',
so that
If possible let
contiguous to 8.
V be
Then we may
partly within and partly without S,
such that on the part without
dT* ^jdv
by
8.
[54.
dV 8, -=- is
everywhere greater than
It follows that for such surface
But
unless there be attracting matter belonging to either system within 8' both these quantities are zero, and they cannot therefore be unequal.
54.] The propositions of the last article can also be extended to the case where is given equal to 7' not throughout any finite portion of space, but only at all points in a finite straight
V
t
provided that both
line,
V
and
V
be symmetrical about that
line as axis.
For we must suppose that there exists some space about the We given line which contains no matter of either system.
may describe wholly within that space about the given line as axis a right cylinder of very small section. For that cylinder both the
dS and II ~r
~T- dS must be
zero,
V
differ
symmetry about the
axis
cannot
and therefore by from
V
at
any
V
And being proved equal point upon, or within, the cylinder. to at all points within the cylinder, the case is reduced to
V
that of Art. 533. 55.]
Let
7, instead of
solid harmonic,
origin.
and
let
Then by Art.
8
denoting a potential, be any spherical be any closed surface not enclosing the
6
the integrals being taken over and throughout 8 respectively. V 2 7, we obtain the result of Art. 4 5 as a Writing 4 irp for
Hence the propositions particular case of the general theorem. of Arts. 53 and 54 may be extended to the case in which For 7',
THEOEEMS CONCERNING THE POTENTIAL.
57-]
59
instead of being a potential, is any spherical solid harmonic. If, for instance, the potential of a given mass be proved equal to a
U
certain spherical solid harmonic at all points within a certain region, as the finite space S, or the given length of the axis of a it can be shewn that the potential is equal points which can be reached from the given region of equality without passing through any matter of the system. Further, U, instead of being a single spherical solid harmonic,
symmetrical system,
to
U at
may
all
be an infinite series of such harmonics, and the proposition which can be reached from the
will still be true for all space
given region of equality without passing through any matter of the system, or through any point where the series U ceases to be convergent. 56.] If the potential due to any distribution of matter on a closed surface S be constant at all points on S, the superficial density,
o-,
is
equal to
1
dV
47T
dv
at each point
on
the normal
S,
being measured from 8 on the outside of it. For since the potential V is constant at each point on S, and 2 at all points within S, it has by Art. 7 the satisfies V F
=
same constant value at all points within S. dV superficial equation =-; = 0, and therefore d>
dV -,
dv
+
Hence
in Poisson's
v
4 TT o-
= 0,
or
or
=
dV
I
477
But whether the potential be constant sum of the distribution over S is dv
dv
or not, the algebraic
dS, by Art. 45.
and only one, distribution 57.] It is always possible to form one, of matter over a closed surface 8, the potential of which shall have any arbitrarily given value at each point of that surface. For, as we have proved in Art. 7, there exists one determinate function u which has the given value at each point of S, and 2 satisfies V &=0 at each point in the infinite external space, and
vanishes at an infinite distance.
And
there exists one determinate function u' which has the
THEOKEMS CONCERNING THE POTENTIAL.
60
given value at each point of S and point within S.
Then a
distribution over S, whose density 1
(
47T
(
du dv
V #'= 2
satisfies
t
du'
)
dv'
\
[57. at each
is
the normals being measured from S, dv on the inside, and dv on the outside of the surface, is the required distribution. For let a small sphere S' be described about any external point, Q, as centre.
Let
- where
V
r is the distance of
any
point from Q.
Then, applying Green's theorem to the space outside of S and S', we have with the given meaning of dv,
ds
V=
Now V% = the triple
if
u
and V 2 integral, and
+
V
*r
at all points within the limits of
denote the value of u at O.
Also
\\V -^-dS dv
vanishes.
JJ Therefore the equation becomes
Again, applying Green's theorem to the space within have with the given meaning of dv
S,
we
ff or since both
Now ---
V 2 F=
=
if
-j-y
V 2 u'
'
and
Q
everywhere within
be not actually on
S,
S,
however near
to
THEOREMS CONCEENING THE POTENTIAL.
58.]
S it may we have
But
if
P
be,
and u
=
u on
be any point on
S.
Hence, subtracting
V at P =
S,
A
61 from
B
t
-
du
._
Therefore
Now
member
the right-hand
supposed distribution
the potential at
is
whose density
J_5^ 4:17
Q
of the
is
**M/ ]
dv
\dv
3
It follows that this potential is equal to UQ at every point S, however near to S ; and therefore, since the potential
outside of is
a continuous function, has the value of
,
or the given value,
at each point on S.
Similarly, if
Q
be an internal, instead of an external, point,
can prove that the distribution over _l_(du_
+
~4*t
/S
whose -density
we
is
<&/)
57)
has u' for potential at Q. And the functions u and u being both determinate, their differential coefficients -=-
dv
and
-7-7-
dv
are determinate
and of single
value.
58.] If $!
. . .
Sn be any
closed surfaces, there exists one
one distribution of matter over them whose potential u the following conditions, viz.
u u
= =
&c., &c.
Cl} constant, but C2 constant, but And ,
arbitrary at all points on arbitrary at all points on
"du
//dv //;dv
dS*
=
e,
over
dS<>
=e
over
and so on, and u vanishes at an
2
,
$,
infinite distance.
Slf S2
;
and only satisfies
THEOEEMS CONCERNING THE POTENTIAL.
62
[59.
For we have proved in Art. 10 that there exists one determinate function u satisfying the above conditions. It follows that
-=-
has a single and determinate value at each point of each
Then
of the surfaces. -
each point
if
-
4 7T -jUV
,
we
take for density of the distribution at
we can prove
exactly as before that the
potential of the distribution so formed at any point external to the surface is u, and therefore satisfies all the conditions. of Art. 57 may be extended to an unclosed 59.] The proposition surface thus. Let S be an unclosed surface, S' a similar and equal
surface so placed as that each point on
'
shall be very near to the corresponding point on 8. If we now connect the boundaries Let a of S and S' by a diaphragm we obtain a closed surface.
V
distribution be formed on this closed surface having potential on S, and at each point on S' the same potential as at the corre-
sponding point on S. Let
distribution
made
to coincide
with
S,
we
obtain cr-ft/ as the density of a
has potential V at each point on S. of If two matter, both within a closed surface systems 60.] have the same potential at each point on S, then distribution on
(a)
For
S which
S,
they have the same potential throughout all external space. T 7' be the potentials of the two systems respectively.
let
9
- V on S, V V = and V V =
Then 7
2
2
at all points in the external space,
V are both of lower degree than V cannot differ from V at any point in the Hence, by Art. V and
\.
9,
external space. (6)
The
algebraic
sum
to that of the other.
within S whose potential
of the matter of either system is equal For the algebraic sum of the matter is
V
is
-;r // dv 47TJJ the normal being measured outwards on the outside of at all points external to S, Art. 45. Now, since 7-=.
V dV_dT dv
dv
S,
by
THEOEEMS CONCERNING THE POTENTIAL.
6O.]
63
and therefore
The two systems have the same centre of
(c)
for the plane of y, z
taking Green's theorem to the
For
inertia.
and applying
any arbitrary plane, within any closed surface
space
$
we have
enclosing S,
V= V,
and since on S'
and
-=-
dv
= -5
dv
=
f ff
And
therefore if m, m' be the quantities of matter of the systems respectively within the element of volume dxdydz,
mxdxdydz which, as the direction of
a?
is
=
/ / /
two
m'xdxdydz,
arbitrary, proves the proposition. For principal axes.
The two systems have the same
(d)
and therefore / / /
xymdxdydz
=
/ /
/
xym'dxdydz;
the axis of z be a principal axis of one system, principal axis of the other system.
and
if
(e)
If A, B,
system, C' =.
those
C-K.
C of
it
is
a
be the principal moments of inertia of one A K, ' It K, the other are A'
=
=
For
-ffflVdxdydz,
THEOREMS CONCEENING THE POTENTIAL.
61
[6
1.
and therefore
-
ffftfmdxdydz
fffxz m'dxdydz +
2
f ff (V-V'}dxdydz.
2
jjhv-V^dxdydz.
Similarly
\ll ifmdxdydz
Similarly,
Definition.
it, is
z
y m'dxdydz
j
+
= (7-4 fff(V~ V'}dxdydz = C-K. B' = B-K, A' -A-K. A body which has the same potential at all C'
Therefore
outside of
=
itself,
mass were collected at a point its centre. and
as if its
a centrobaric body, from (c) that
It follows
if
a body be centrobaric,
its
points
within
centre
is
centre of inertia.
its
It follows from Art.
61.]
always exists over a surface
59 that a distribution of matter
S which has any given
point of S;
potential at each
constant
and therefore that any given
quantity of matter can be distributed over S in such a way as to have constant potential at each point of S. Such a distribution is
defined to be an equipotential distribution. If be the algebraic sum of a distribution of Definition.
M
matter over a closed surface whose potential has the constant value
V at
each point of that surface, -^
is
the capacity of the
surface.
The capacity
of a sphere
is
equal to
its radius.
For the sphere
being charged to potential F, the potential, being constant over the surface, must have the same constant value at the centre. 1
V
But
if
M
be the algebraic
at the centre
is
>
sum
where a
is
of the distribution, the potential
the radius.
a
We
have then
=
F,
or
-
= a.
^V
* t
THEOREMS CONCERNING THE POTENTIAL.
63.] If
$ be an
within of 8
is
it,
M -=
equipotential surface to a system of matter wholly be the potential of the system on S, the capacity
and j
V
where Jf
is
sum
the algebraic
of the matter of the
M
is also the enclosed system. For, by Art. 60, algebraic sum of a distribution over S which has potential at every point on it.
V
62.] If V be the potential of any distribution of matter over a closed surface S, and if
matter over
S which has
the same potential at each point on S
as that of unit of matter placed at
of the
the potential at
For
let
or
any point
0,
then
is
jrf'tr'dS
first distribution.
be the density of the
first
distribution,
7'
the
potential of the distribution, r the distance of any point from
Then on
0.
8,
-
1
(dV
+ dV)
_ dv
= the potential at
of the first distribution.
If S be a closed equipotential surface in any material f system, and if p, p denote densities of the matter of the system inside and outside of S respectively, and if R le the force due 63.]
to the whole system at any point on S in the direction of the normal measured outwards, then the potential at any external point due to the internal portion is equal to that of a disT>
tribution of matter over tial
at
VOL.
S whose
any internal point due I.
F
density is to
the
,
and
the poten-
external portion
differs
THEOREMS CONCERNING THE POTENTIAL.
66
[63. 7?
of a distribution over
that
from
S whose
density is
---
,
by
For if we take for origin any point the potential of the surface. be the potential of the entire system, we outside of 8 and if have by applying Green's theorem to the space inside of S, with
V
9
u
=
V>
where
and
V
t
is
u'
=
-
the constant value of
= 0, and
V2 -
>
is
since
on
S,
T--&8-**9 Jby Art. 9
45,
zero at all points within S.
The equation
therefore becomes
=-8,
But
which proves the Secondly,
V
if
first
and
part of the proposition. for origin a point
we take
inside
apply Green's theorem to the external space, with
u and
u',
we
obtain
snce
of S, and
V and -
for
SPHERICAL HARMONICS APPLIED TO POTENTIAL. 67
64.]
mal
V2 - =
and as before
in this case,
measured inwards from
is
S,
dV
-r=R, dv
Hence the
potential at
case, the nor-
and therefore
V F=-47rp'. 2
also
any
Also in this
0.
internal point of the distribution
T>
^
over
4
differs
by
a constant quantity from that of the
external portion M', and therefore the force due to the distribu-
--JP
tion
over
8
is
4lT
Hence
it
equal to that due to the external portion
follows that the force at
any external point due
to the
-n
internal portion is equal to that due to the distribution
4?r
over 8, and the force at any internal point due to the external
-73
is
portion
equal to that of the distribution
P
of any distribution 64.] To express the potential at any point of matter in a series of spherical solid harmonics.
Take as origin any point 0. Let OP = f. Let be any point in the distribution. referred to OP as axis, be r, 0, <, Let the coordinates of cos 0. Then sin Odd = dp, where is the angle POM. Let \L = is and an element of volume in the neighbourhood of
M
M
M
If p be the density of the given distribution in
r^dyidfydr. this
element of volume,
pr*dfJLd(bdr -
-yrjjp
its
potential at C 1
_
_
,, = pr*dnddr
+
P is r
..
,
.
j-
Q^
+ Qz
r*
+
) ...
J
or
The
P
potential at
rf r
rzir /
/
jo
ri
+
p-j-dudtidr
/
/iiJo
of the whole distribution
z
rzir I
J-iJo
+ &c.,
rs I
Jo
+
yi I
rzv r> I
J-iJo
^
QiP-^du.d
/
I
is
then
i
prdftdtydr
Jj
r^ ill ri
J-l.'o
r<
Jf
if
r
,
APPLICATION OF SPHERICAL HARMONICS
68
and since
du
/I-i
depends on p only, this r>2ir ^2 //
Q
f }
may be put
+
drp
d /
/
Jo
rf
rzir
f
(Jo /i d/iCil/
Jf r2ir
r*
/
drp~ +/ Jo 3*/ / //
d<j>
Jo
.1
drp
d<j)
Jo
J
in the form
/*
/"27r
(Jo
[65.
2
+ &C., known
in which the quantities within brackets are distribution
If
is
we denote
these quantities
by A
7= rdp{A +Q J-i in
if
the given
known.
1
,
A^ A 2
,
&c.,
A + Q,A 9 +...} 1
which the A's are generally functions of
we have
9
//.
To
find the density of a distribution of matter over a 65.] spherical surface, whose potential at any point on that surface shall be equal and opposite to that of a mass e, placed at an
external point. Let be the centre of the sphere, a its radius, outside of it, OC=f.
Let It
a-
is
C
the point
be the required density. evident that the density of this distribution on the
sphere must be symmetrical about OC, and must therefore be expressible in a series of zonal harmonics with
OC as
axis.
Let
this be
W
Let E be any point on the surface, any other point. Let us denote by Qf the zonal harmonic of order i referred to
OE as
Then
axis.
-
And
E due to the distribution
the potential at
VE
=
HA
'
"
Q<>
SS
Qt ds + Ai
because every term of the form that is,
being the value of Q+ at E.
/
'
ffQi Qi
is
ds+ &c
IQ Qj'dS, where i
i
'
=j,
is zero;
TO THE POTENTIAL.
66.]
69
But by hypothesis the potential at E of the distribution is to be the same as that of the mass e at C with reversed sign; that
is,
We
have therefore
and equating
coefficients of
Q
{
,
2t+l and
=
2A Q t
t
= CJE"
_?, where
(by Art. 26).
-L/
of matter over a 66.] If the density of any distribution a spherical surface spherical surface be equal to fj, where 7i is harmonic of order i, the potential at any point within or without
the sphere due to this distribution sponding spherical solid harmonic.
For
let
is
correproportional to the
be the centre of the sphere, a
ternal or internal point, Then at
OP =
and
r,
its radius,
P
2
= /T 7. 2 But
II
T Qj dS,
if
P be
Q. dS,
if
P
Y QjdS = i
internal,
be external.
0, unless i
= j,
and therefore
V=
-^
= -r
/TV, ft ^,
Y
i
Qi
ds
>
P
M a point on the
if
P
if
P be
be internal,
external.
any exsurface.
APPLICATION OF SPHERICAL HAEMONICS
70
But
[67.
Jf
where
Y
{
is
Y
the value of
t
at the point where OP, produced if
necessary, cuts the sphere.
_
And
therefore ri
external,
7
i
.
If
in the
is
Y
i
,
~Y*
or
<+ t
,
according as
the spherical solid harmonic at
we denote
this
by H
it
this,
is
internal or
corresponding to
we have
two cases respectively.
easily be deduced from
P
P
The following proposition may
but we prefer to prove
it
independ-
ently thus.
material system wholly within a 67.] If the potential of any S be given at each point of that surface in spherical surface
a series of spherical surface harmonics, then the potential of the same system at any point on the outside of the surface is found by substituting for each surface harmonic the corre-
sponding solid harmonic. For let the given potential be 2 of the superficial distribution of S is equal to 2 i Y^
A i Y^ and let p be the density on S whose potential at every point
A
Let
P
be any point distant
/ from
the centre on the outside
of 4.
P
Then the potential at of the given system of the surface distribution.
is
equal to that
But, as shewn in Art. 62, if p' be the density of a distribution over S whose potential at any point of S is equal to that of unit of matter situated at P, then
is is
P
the potential at of the superficial distribution whose potential 2AiI i9 and therefore of the given system.
Now P
-
TO THE POTENTIAL.
68.]
where a
is
where J^
The
the radius of the sphere.
is
71
Therefore
the value of J^ at P.
potential of the given system
is
also equal to
2
(
J
A Y
)
i
t
within the given spherical surface S. a \ i+1 Ai i both satisfy Laplace's equation )
for a certain distance
V
For
and
Y
external space, and are identical at all points outThey must therefore be identical throughout all space
throughout side of S.
2(
all
which can be reached from S without passing through attracting matter so long as
To express
2
(
A J i
)
i
is
a convergent
series.
harmonics the potential of any material system symmetrical about an axis. on the axis. Let / be the Let us take for origin any point 68.]
in zonal solid
distance from
of any point in space. Then we can first shew that the potential at any point P on the axis, if more distant from the origin than any point in the
system, can be expressed in the form
2B
i
-r-y
>
and
if
less
distant from the origin than any part of the system can be and A expressed in the form C'+S-^r*, where the functions
B
are determinate if the given system pendent of r.
For
let
in space at
density
Then
be the origin,
which there
P the
is
known, and are inde-
point on the axis,
M any point
is
matter belonging to the system of
is
symmetrical about the axis, we may volume the space between the two
p,
since the system
take for an element of
its
cones whose vertices are at
and semivertical angles cos" 1 ju and is between a and a + da.
cos" 1 (p + dp) and whose distance from
APPLICATION OF SPHEKICAL HARMONICS
72
If p be the density of matter within this element at
its
[68. potential
Pis 2ira*pdtJ.da.
that
,
is,
1
2ira*pdu,da.-\ r
(
or
Then
if
2
if
1}
a?p
dfJi
da -
< 1
.
+ &+ r
+Q
l
+
} if r
>
a,
<
a.
)
.
..
>
if
r
a 2 be the greatest and least distances from
any matter between the two cones, the potential of matter between them is
I* ftp ./"*>.
in which the
first
>
if r
Finally, the potential at
of
the
\j+Q^
integral will be omitted
second will be omitted
all
P
a
when
r
<
2,
and the
.
of the whole system
is
found by
I to integrating the above expression according to ^ from p and are func1, remembering that a^ and p 2 generally fji
=
tions of
/u.
Let a\ and a\ denote the greatest and least values of r for any point in the system. Then the result, if the integrations can be effected, must appear in the form
sn-Lif OP><, C+2A r* if OP<<;
and
t
and r
if
We
can
now
>
a\
<
a\.
find the potential of the
R
system at any point not in the axis and distant r from 0, by multiplying each term by the corresponding zonal harmonic referred to OP as axis.
For instance, suppose r > a\. Let V be the potential and let
68-]
Then
TO THE POTENTIAL. since on the axis
Q = t
1,
73
and
V and V are identical throughout a finite Now both V and V satisfy Laplace's
length of the axis.
equation at all points not occupied by matter belonging to the system. And therefore since they are identical throughout some finite length on the
and are symmetrical about the axis, they must by Arts. 53 and 54 be identical at all points in space which can be reached from that part of the axis without passing either through the system, or
axis,
through any part of space where 2
B
ri
i
_^l
does not converge.
Similarly, the potential at any point R' in space distant r from It' can be reached from 0, where r < a 2', is f Q^*, provided
C+ 2^
is less than a withthe part of the axis whose distance from the either or out passing, through system, through any part of does not converge. space where "SA t
Q^
CHAPTEE
IV.
,
i
DESCRIPTION OF PHENOMENA. Electrification ly Friction,
ARTICLE 69.] EXPERIMENT I*.
Let a piece of glass and a piece
of resin be rubbed together and then separated each other.
;
they will attract
If a second piece of glass and a second piece of resin be similarly treated and suspended in the neighbourhood of the
former pieces of glass and resin, (1) (2)
(3)
The two
it
may
be observed that
pieces of glass repel each other.
Each piece of glass attracts each piece of resin. The two pieces of resin repel each other.
These phenomena of attraction and repulsion are called elecphenomena, and the bodies which exhibit them are said to
trical
be
electrified or to
The
be charged with
electricity.
two pieces of glass are similar to each other but opposite to those of the two pieces of resin, the glass attracts what the resin repels, and repels what the resin electrical properties of the
attracts.
Bodies
may
be
electrified in
other ways as well as by
many
friction.
If a body electrified in any manner whatever behaves as the glass does in the experiment above described, that is, if it repels
the glass and attracts the resin, it is said to be vitreously elecand if it attracts the glass and repels the resin, it is said
trified,
to be resinously electrified. All electrified bodies are found to be either vitreously or resinously electrified. When the electrified state is produced by the friction of dis-
similar bodies, as above described, it *
The
description of these experiments
Electricity.
is
is
found that so long as the
taken almost verbatim from Maxwell's
DESCRIPTION OF PHENOMENA.
70.]
75
rubbed surfaces of the two excited bodies are in contact the combined mass does not exhibit electrical properties, but behaves towards other bodies in
had taken
friction
its
neighbourhood precisely as
if
no
place.
The exactly opposite properties of bodies vitreously and resinously electrified respectively, and the fact that they neutralise each other, has given rise to the terms 'positive' and 'negative* electrification, the term positive being by a perfectly arbitrary, but
now
universal convention
among men
of science, applied
and the term negative to the resinous
to the vitreous,
electri-
fication.
Electric actions similar to those above described
may
be ob-
served between a body electrified in any manner and another body not previously electrified when brought into the neigh-
bourhood of the electrified body, but in all such cases it will be found that the body so acted upon itself exhibits evidence of the electrification. This electrification is said to be produced by induction, a process
which will be
illustrated in the second ex-
periment.
No
force,
between an
either of attraction or repulsion, can be observed electrified
body and a body manifesting no signs
of electrification. Electrification by Induction.
70.]
with a
and
may
let
EXPEBIMENT
II.
Let a hollow
vessel of metal, furnished
suspended by white silk threads, a similar thread be attached to the lid, so that the vessel
close-fitting metal lid, be
be opened or closed without touching
the vessel and
lid are perfectly free
from
it
;
suppose also that
electrification.
Let the pieces of glass and resin of Experiment I be suspended in the same manner as the vessel and lid, and be electrified as before.
If then the electrified piece of glass be
hung up within the
sus-
and the pended vessel by its thread, without touching the vessel, be to found will be lid closed, the outside of the vessel vitreously electrified,
and
it
may
be shown that the
of the vessel, as indicated
by the
electrification outside
attractive or repulsive forces
on
DESCRIPTION OF PHENOMENA.
76
[71.
neighbourhood, is exactly the same in whatever part of the interior the glass be suspended. If the glass be now taken out of the vessel without touching
electrified bodies in its
it,
the electrification of the glass will be found to be the same as was put in, and that of the vessel will have disappeared.
before it
vessel, which depends on the glass and which vanishes when the glass is removed,
This electrification of the
being within
it,
is called Electrification
by
Induction.
If the piece of electrified resin of Experiment I were substituted for the glass within the vessel, exactly opposite effects If both the pieces of glass and resin, would be produced.
vessel,
Experiment I, were suspended within the whether in contact with each other or not, no electrical
effects
whatever would be manifested.
after the friction of
Similar effects would be produced if the glass were suspended near the vessel on the outside, but in that case we should find an
one part of the outside of the vessel Whereas, as has been just now part.
electrification vitreous in
and resinous in another mentioned,
when
the glass
is
inside the vessel the whole of the
In this case, as in the case vitreously of internal suspension, the electrification disappears on removal of the exciting body. outside
is
electrified.
Experiment proves that throughout the inside of the closed is an electrification of the opposite kind to that
vessel there
of the outside, that
is,
when
the
electrified
piece
of glass
is
suspended within the vessel, and the latter is therefore vitreously electrified on the outside, as just now explained, the inside will be resinously electrified, and vice versa when the resin is
substituted for the glass. Experiment proves also that the electrification on the outside
is
equal in quantity to that of the glass, and the electrification
on the inside equal and opposite to that of the
glass.
Electrification by Conduction.
71.] EXPERIMENT III. The metal vessel being electrified by induction, as in the last experiment, let a second metallic body be suspended by white silk threads near it, and let a metal wire
DESCRIPTION OF PHENOMENA.
72.]
77
similarly suspended be brought so as to touch simultaneously the electrified vessel and the second body. The second body will now be found to be vitreously electrified
and the vitreous
The
electrification of the vessel will
electrical condition has
to the second
have diminished.
been transferred from the vessel
body by means of the
The wire
wire.
a conductor of electricity, and the second body
is
is called
said to be
electrified by conduction.
Conductors and Insulators.
If a glass rod, a stick of resin or gutta-percha, or a white silk thread had been used instead of the metal wire, no transfer of electricity
would have taken
place.
Hence these
latter substances
A
are called non-conductors of electricity. non-conducting support or handle employed in electrical apparatus is called an Insulator, and the body thus supported is said to be insulated.
Thus the lid and vessel of Experiment II are insulated. The metals are good conductors ; air, glass, resins, guttabut all subpercha, vulcanite, paraffin, &c., are good insulators stances resist the passage of electricity, and all substances allow it to pass although in exceedingly different degrees. For the ;
present
we
shall, in
speaking of conductors or non-conductors,
imagine that the bodies spoken of possess these properties in perfection, a conception exactly similar to that of perfectly fluid or perfectly rigid bodies, although such conceptions cannot be realised in nature.
In Experiment II an
electrified
body produced
electrifica-
tion in the metal vessel while separated from it by air, a nonconducting medium. Such a medium, considered as transmitting
these electrical effects without conduction,
is called
medium, and the action which takes place through
a Dielectric
it is called,
as
has been said, Induction.
EXPEEIMENT IV. In Experiment III the electrified vessel produced electrification in the second metallic body through the medium of the wire. Let us suppose the wire removed and the 72.]
electrified piece of glass it
and removed
taken out of the vessel without touching The second body will
to a sufficient distance.
DESCRIPTION OF PHENOMENA.
78 still
is
[74.
exhibit vitreous electrification, but the vessel when the glass If we now bring will have resinous electrification.
removed
the wire into contact with both bodies, conduction will take place along the wire, and all electrification will disappear from
both bodies, from which we infer that the electrification of the
two bodies was equal and opposite. 73.] EXPERIMENT V. In Experiment II
it
was shown that
if
a piece of glass, electrified by rubbing it with resin, is hung up in an insulated metal vessel, the electrification observed outside does not depend upon the position of the glass.
we now
If
introduce the piece of resin with which the glass was rubbed into the same vessel without touching it or the vessel, it will
be found, as stated in Art. 70, that there the outside of the vessel.
From
this
is
no
electrification
we conclude
on
that the
and opposite to that in of the glass. number of electrified bodies, By putting any some vitreous and others resinous, and taking account of the
electrification of the resin is exactly equal
amount of
electrification of each,
we
shall find that the
electrification of the outside of the vessel is that
algebraic
sum
whole
due to the
of the electrifications of all the inserted bodies,
the signs being used in accordance with the convention already described. have thus a practical method of adding the
We
electrical effects trification of
74.]
of several bodies without altering the elec-
any of them.
EXPERIMENT VI. Let a second insulated metallic
vessel
B
be provided, and let the electrified piece of glass of Experiment I be placed in the first vessel A, and the electrified piece of resin in
Let the two vessels be then put in communication by the metal wire, as in Experiment III. All signs
the second vessel B.
of electrification will disappear. Next, let the wire be removed, and let the pieces of glass and resin be taken out of the vessels without touching them. It will
B
A is electrified resinously and vitreously. the glass and the vessel A be introduced together (the glass being no longer within A) into a larger insulated vessel , it will be found that there is no electrification on the outside of C.
be found that If
now
This shows that the electrification of
A
is
exactly equal and
DESCRIPTION OF PHENOMENA.
76.]
79
B
opposite to that of the piece of glass, and similarly that of may be shown to be equal and opposite to that of the piece of resin.
Thus the
A
has been charged with a quantity of elecand tricity exactly equal opposite to that of the electrified piece of glass without altering the electrification of the latter, and we
may
vessel
in this
way charge any number
of vessels with exactly we may take
equal quantities of electricity of either kind which as provisional units.
EXPERIMENT VII. Let the
75.]
vessel B, charged with a quan-
tity of positive electricity, which we shall call for the present unity, be introduced into the larger insulated vessel C without
touching
it.
outside of C.
It will produce a positive electrification on the let be made to touch the inside of C. No
B
Now
B
change of the external electrification of C will be observed. If be now taken out of C without again touching it and removed to a sufficient distance, it will be found that
charged, and that
C has become
B is
completely dis-
charged with a unit of positive
electricity.
We Let into
have thus a method of transferring the charge of B to C. B be now recharged with a unit of electricity, introduced
C
already charged,
removed.
It will be
made
found that
so that the charge of
C is
C
is
C and
doubled.
If this process be repeated
highly
to touch the inside of
B is again completely discharged, it
previously charged,
will
be found that however
and in whatever way
B
charged when it is first inclosed in (7, then made to touch and finally removed without touching C, the charge of B completely transferred to C,
and
B
is entirely free
from
is
C, is
elec-
trification.
This experiment indicates a method of charging a body with any number of units of electricity. The experiment is also an illustration of a general fact of great importance, namely, that no charge whatever can be maintained in the interior of any conducting mass. In what has hitherto been said it has been assumed that 76.]
we
possess the
means of testing the nature and measuring the
DESCRIPTION OF PHENOMENA.
80
[77.
amount of electrification on any body, or on any part of a body. This we can do with great accuracy by the aid of instruments called electroscopes or electrometers} whose modes of action will be more easily understood when the theory of the subject has been somewhat developed ; and which are fully described in practical treatises on electricity for our present purpose it will suffice to ;
describe one of these instruments in its simplest form, called the
gold-leaf electroscope. strip of gold-leaf hangs between two bodies A and B, charged one positively and the other negatively. If the gold-leaf be placed in conducting contact with the body
A
whose
electrification is to
be investigated,
it will itself
become a
part of that body for all electrical purposes, and it will incline towards A or according as its electrification, and therefore the electrification
body under
of the
investigation, is negative or
positive.
From
77.] (1)
The
the foregoing experiments
total electrification of a
we conclude that
body or system of bodies
remains always the same except in so far as it receives cation from, or gives electrification to, other bodies.
electrifi-
In all electrical experiments the electrification of bodies is found to change, but it is always found that this change is due to want of perfect insulation, and that with improved insulation the change diminishes. electrification of a
We may therefore
assert that the
placed in a perfectly insulating
body would remain perfectly constant. (2)
When
one body
electrification of the
one loses as
medium
much
electrifies another by conduction the total two bodies remains the same, that is, the
positive, or gains as
much
negative
electrifica-
tion, as the other gains of positive or loses of negative electrifi-
cation.
For
if
the two bodies are enclosed in the same hollow con-
ducting vessel no change of the total electrification on their being connected by a wire. (3)
other
When
electrification
known method,
is
is
observed
produced by friction or by any
equal quantities of positive and negative
electrification are produced.
DESCRIPTION OF PHENOMENA.
78.]
For the
electrification of the
whole system
81
may
be tested in
the hollow vessel, or the process of electrification may be carried on within the vessel itself, and however intense the electrification of the parts of the system
whole
may
be, the electrification of the
invariably zero.
is
The electrification of a body is therefore a physical quantity capable of measurement, and two or more electrifications may be combined experimentally with a result of the same kind as when two quantities are added algebraically. Let there be a needle suspended 78.] EXPERIMENT VIII. a fine vertical wire or fibre, so as to be capable of horizontally by about the vertical wire as an axis, and lefc vibrating horizontally a small pith ball A be attached to one end of the needle. Then the needle will rest in a certain position in which position, at in there are no forces work the supposing neighbourhood of ;
the apparatus except the force of gravity, the suspending wire Let or fibre will be perfectly free from any twist or torsion.
B
be situated at a certain point in the circumanother pith ball ference of the horizontal circle described by A.
B A
Now let the pith balls A and be each charged with one unit of positive electrification. repulsive action will arise between A and so that A will after certain oscillations come
B
to rest at a certain increased distance
from B, thus producing a
in the suspending wire. The opposite untwisting tenwire thus called into play depends upon the of the dency torsional rigidity of the wire and the angle through which
twist
the needle has been deflected, and can be estimated in any given apparatus with great accuracy. Hence the repulsive force
between also
A
and B, assumed to act in the
line joining
be determined with corresponding
accuracy:
them, can let it be
1
called/
.
Suppose now that the same experiment is made with another with a susapparatus equal to the former in all respects, but that in and torsional different wire of suppose rigidity pending ;
B
is the position taken up by A with respect to observed to be exactly the same as in the former case when the
this
case
number VOL.
of units of positive electrification of A
I.
G
is e,
and of
B is
e'.
DESCRIPTION OF PHENOMENA.
82
[79.
It will be found that the repulsive force between this case
A
and
B
is
in
ee'f.
Precisely the same positions would be taken cases respectively, if
A
up
in the
two
B had been each negatively electrified,
and
and to the same degree as before. By suitable adjustments of the two cases with opposite electrifications upon A and B each of the same number of units as before, it may be proved that, if the distances between are the
same as
and equal to /"and
Hence we
A
and
B in their positions
ee'f in the
of equilibrium
between them are attractive
before, the forces
two cases respectively. between two electrified
infer that the force
particles
by the them, upon regard being paid
any given distance apart is in product of the two electrifications
all
to the signs of the electrifications,
and the
at
respects represented
force being considered
when
the above-mentioned product is positive. repulsive EXPERIMENT IX. In the experiment of the last Article 79.]
B in the second apparatus as each one unit of positive electribe well as in the first apparatus It will be found that in the positions of equilibrium fication. the electrifications of
let
the distances between
A
A
and
and
B are
not the same in one apparatus
A
and as they are in the other. If, however, the forces between in these positions be estimated as before, and if the distances
B
between
A
and
B in
the two cases be r and
r', it
will
be found
that there are repulsive forces between them which are to each other in the ratio of / 2 to r2 or inversely as the squares of the distances between them in the two cases. ,
Combining the
we
and the preceding experiment law of action between two
results of this
arrive at the following general
electrified particles, viz. that if
cation of the particles be e
between them be
r,
the
and
then there
e'
is
number
of units of electrifi-
respectively,
a force
F such
and the distance that
'
'
F
ee -f
**?* where/* is the repulsive force between two particles each charged with unit of elect rification, and at the distance unity apart, regard being paid to the signs of e and /, and
F being considered positive
when the force
and / have the same signs.
is
repulsive,
i.e.
when
e
ELECTRICAL THEORY.
8o.]
83
In conducting Experiments VIII and IX care must be taken that the dimensions of A and B are small as compared with the distance between them, so that they may be regarded
They must also be suspended in air and at a considerable distance from any other bodies on which they might induce electrification (Art. 70), inasmuch as this induced as material points.
would
electrification
also
act
upon
A
and
B
and produce a
problem of great intricacy. Electrical Theory.
80.]
The most important
researches into the laws of electrical
to the present time have been based upon what It is conceived that all bodies as the two fluid theory.
phenomena up is
known
in nature, whether electrified or not, are charged with, or pervaded by, two fluids to which the names of positive and negaor vitreous and resinous, electricity are assigned. It is further supposed either that these fluids exist in all bodies in such quantities that no process yet discovered has ever de-
tive,
prived any body, however minute, of all the electricity of either kind, or that the changes in the proportion in which these fluids are combined, required to produce electrical phenomena, are It is further supposed that in unelectrified indefinitely small.
bodies these fluids exist in exactly equal quantities, but that it is possible by friction, as in Experiment I, or by othef means, to
cause one body to give up to another part of its positive or negative electricity, thus causing in either body an excess of one or other kind of electricity.
When
the quantity of either fluid
that body
said
to
be positively
is
in excess in
any body,
or
negatively electrified according to the sign of the predominant fluid, and the amount of electrification is measured by the quantity by which this is
predominant in
any
fluid exceeds the other.
electrified
body
The
fluid of either
kind
in excess of that of the opposite kind is
called the Free Electricity of the body,
and the remaining
fluids
of the body, consisting of equal amounts of fluids of opposite kinds, together constitute what is called the Latent, Combined or
Fixed Electricity of the body.
G
2,
ELECTRICAL THEORY.
84
it,
[80.
In the simplest form of the theory, although not essential to every process of electrification is supposed to consist of a
transference of a certain quantity of one of the fluids from any body as A to another as B, together with the transference of
an equal amount of the opposite
fluid
from
B
to A, so that the
and latent (without regard to and in every particle of every body cannot be every body sign) whatever. changed by any process It is further supposed that these fluids are not acted upon by gravitation or any of the forces of ordinary mechanics, nor, so far as our present knowledge goes, by ordinary molecular or chemical forces but they are supposed to exercise forces upon themselves and each other which are conceived to be proportional to the
amount of
total
electricity free
;
quantities of the mutually acting fluids, thus giving rise to the conception of electrical mass. And it is further supposed that the
between two particles of fluid of the same kind is repulsive, and proportional to the product of their masses directly, and to the square of the distance between them inversely, that between two particles of fluid of opposite kinds being attractive, but in
forces
other respects following the same law. According to this hypothesis the latent or fixed electricity in any body, consisting of equal quantities of opposite kinds, exerts zero force
on
all
electricity.
The
forces
of
attraction
and
repulsion
above mentioned manifest themselves only between the free electricities.
If
all
bodies be divided for the time into two classes, perfect
conductors and perfect insulators, it is conceived that either kind of electricity may pass with absolute and perfect freedom from point to point of the former, while the latter offer a complete and absolute bar to any such transference.
On
the hypothesis thus described we are able to explain many phenomena. It is of course merely an hypothesis, and
electrical
of value as supplying formally an explanation of facts in this respect being exactly on a par with the conception of the lumi;
niferous ether in the undulatory theory of light. The general mathematical treatment of this hypothesis is principally due to
Poisson and Green.
ELECTRICAL THEORY.
8l.]
another hypothesis, known as the one-fluid theory, equally successful as a basis of investigation, but it has
There
which
85
is
is
not been adopted and developed to the same extent as the twofluid theory.
We
shall confine our investigation to the two-fluid theory in
the form which
is
above enunciated.
81.] It is evident without
any further explanation that the two-fluid theory explains the qualitative results of Experiment I given above ; and we proceed now to shew that it also explains
VIII and IX. A and either two bodies conductors or nonB, For, suppose m m' units of mass and of positive eleccontain conductors, to tricity respectively, and n and n' units of negative electricity. Suppose that they are situated in an insulating medium, as air, and that their dimensions are very small as compared with the distance between them which we shall call r. the quantitative results of Experiments
of
Then, according to the two-fluid theory, the m positive units a repulsive force A exert upon the m' positive units of
B
which may be represented by
,
^
and the n and
n' units exert
a repulsive force upon each other, represented on the same scale
by
-3-
,
so that
on the whole there
the electrical fluids in
A
and
In the same way there electricities represented
is
electricities
^ that
-u
4.
is,
by
But m A, and
a repulsive force between '
B represented by
an attractive mn' + m'n
by
Altogether therefore there .
is
force
^
between the two
^
is
a repulsive force between the
_
A and B represented (mn)(m'n'} on
mm'+
mn'
nn'
by
m'n >
2
'-\
n
is
m'n'
the number of units of positive electrification on is the same for B, so that with the notation used
above the force between the presented by
ee' -o-,
and
is
electricities
repulsive
when
in
A
and
ee is positive.
B
is
re-
ELECTRICAL THEORY.
86 The unit
of electricity in this measurement
[8 is
1.
such that the
repulsive force between two units of positive electricity at the distance unity apart is unit force.
It appears therefore that the two-fluid theory involves the
two charged the law which has been experirespects following the mechanical to be forces between obeyed by mentally proved the bodies themselves. But the bodies are either non-conductors existence of a force between the electric fluids in
bodies in
all
conductors in an insulating medium, and on either hypothesis the fluids cannot move without the containing bodies or
else
accompanying them. Whatever force therefore is proved to exist between the fluids becomes phenomenally a corresponding We thus see that the results of Exforce between the bodies. periments I, III, and IV, and of Experiments VIII and IX are explained qualitatively and quantitatively by the two-fluid hypothesis.
The
application of the theory to the Induction Experiments II,
V, VI, and VII, is not so obvious, and can only be demonstrated after some further development.
CHAPTER
V.
ELECTRICAL THEORY.
WE
ARTICLE 82.] proceed now to develop the two-fluid theory as before enunciated, regarding for the present all substances as divided into two classes, namely, (i) perfect insulators, called generally dielectrics, throughout which there is an absolute bar to the motion of the fluids from one particle to another, and (2) perfect conductors, throughout which the fluids are free to move
with no resistance whatever from one particle to another. And it assumed for the present that the repulsion between two masses,
is
e
and /, of
ee'
electricity placed at distance r apart is
^
.
The
phenomena with which we have at present to deal are those of repulsion and attraction between particles at a distance according to the above law. The investigations of Chap. Ill are therefore applicable. It will be understood that
ence of the
fluids,
takes place. It of Electrostatics
is
we do not
assert the actual exist-
or that direct action at a distance actually proposed merely to show how the phenomena
may
be explained on this hypothesis.
In
like
manner the conception of space
as divided into perfect conductors and perfect insulators, will have to be materially modified hereafter. 83.] It follows from the above definition of a conductor, that electricities are in equilibrium, the resultant force is zero For if there be any force, at each point within the conductor.
when the
must tend to move one kind of electricity at the point in one direction, and the other in the opposite direction, and therefore it
to separate them. And since the substance of the conductor opposes no resistance to their motion, such separation will in fact take place until equilibrium is attained ; that is, until the
ELECTRICAL THEORY.
88
[84.
mutual attraction of the separated electricities, tending to reunite them, becomes equal and opposite to the force which tends to separate them, and so the resultant force becomes zero.
Now
it
follows from the reasoning- of Chap. Ill that in a field
of electric fluid distribution a potential function
dV
dV
?
,
dm of
a?,
y,
dV
are the
dz dy and z at any point.
component
And
V exists
such that
iorces parallel to the axes
since the resultant force is
zero, each of these components is zero at every point within the conductor, and therefore V has some constant value throughout
the substance of the conductor.
This
is
true whatever the law
of force, provided there be a potential. further from the law of the inverse square, that 84.] It follows free be no there can electricity within the substance of the con-
For whatever closed surface be described wholly within at every point of that surface is zero. the normal force
ductor. it,
N
Therefore
/ /
Nds
=
over the surface.
That
is,
by Art.
45, the
algebraic sum of all the free electricity within the surface is zero, and this being true for every closed surface that can be described within the substance of the conductor, it follows that there
can be no free
electricity, of either
volume or
superficial density,
within the substance of the conductor.
V
It follows that, in order to insure the constancy of throughis sufficient to make it constant at all
out the conductor, it points on the surface.
For we have seen that
if
V be
which
is
no attracting
at all points on a closed surface, within
matter, 85.]
it
constant
has the same constant value throughout the interior. free electricity is formed by the separation of
Whatever
the two kinds of electricity within the conductor, since it cannot exist within the substance of the conductor, and cannot penetrate the surrounding dielectric, must be found upon the surface in the form of a superficial distribution. And such superficial distribution must be in the aggregate zero for the whole surface because since the two kinds of ;
electricity are
supposed to exist in equal quantities at
for every quantity of positive electricity resulting
all points,
from their
ELECTRICAL THEORY.
87.]
must be an equal quantity of negative and each must be found somewhere on the surface.
separation, there tricity,
89 elec-
But
it is possible to place upon the conductor from external sources a quantity of electricity of either This also, for sign. the same reason, can only exist in the form of a superficial
distribution.
It follows then that electrification is
if
a conductor be in equilibrium its surface, and the algebraic sum of
wholly on the
the superficial distribution upon it is equal to that of the electricity placed upon it from external sources.
all
The algebraic sum of all the electricity on 86.] Definition. the surface of a conductor is called the charge on the conductor. If
dV
o-
be the density of the superficial distribution at any point,
the rate of increase of
-j
measured outwards in tribution,
V per
direction,
unit of length of the normal
immediately outside of the dis-
the same thing measured inwards in -^7 dv
direction,
immediately inside of the distribution, Poisson's equation gives
dV --h dV, dv
But ductor,
--j,} dv
is
dv
=
47TO-.
being the force within the substance of the con-
in this case zero.
We
have therefore at every point
of the surface
dv and the charge upon the conductor or
We
have seen that when an electrical system is in equi87.] librium, the potential must have a constant value throughout each Conversely, if the potential have a constant value each conductor, the electricity on fixed conductors is throughout in equilibrium. For the potential being constant throughout the conductor.
conductor, there can be no tangential or other force to move the superficial distribution along the surface or through the substance
90
ELECTRICAL THEORY.
of the conductor.
And
since
by the hypothesis
medium
nature of the dielectric
[88.
concerning- the there can be no motion of elec-
medium, all the electricity in the field must be at The constancy of the potential throughout each conductor
tricity in the rest. is
thus the sufficient and necessary condition of equilibrium. Hence can be established the following principle.
The Principle of Superposition. 88.] If
or
be the density of the superficial distribution on a con-
when
in equilibrium in presence of any electrified system which may include a charge on the conductor itself, and if be the density on the conductor when in equilibrium in presence
ductor E,
of the system E', then if
E and W both be
present, the conductor
the density is
when
it for
know
that the potential at any point is the sum of the two potentials, and density
E
constant for each conductor.
Therefore their
sum
is
and therefore the supposed instantaneous distribution librium and is permanent. It
follows
that
if
all
the volume,
or
superficial
constant,
is
in equi-
or linear
system in equilibrium be increased the system will remain in equilibrium, and
densities of electricity, in a
in
any given
ratio,
the potential at any point will be increased in the same ratio as the densities. 89.] Let us consider the simple case of a single conductor, and a outside of it having a fixed charge of positive electricity m. point
There will form on the surface of the conductor an induced
whose algebraic sum is zero, and of which the negative part is on the side of the conductor nearest to 0, and the positive part on the opposite side. The tendency of the charge at is to make the potential higher on the side of distribution of electricity
the conductor nearest to
than on the other
side.
The
surface
distribution has the opposite tendency. And the surface distribution must be such that these two tendencies shall exactly
ELECTRICAL THEORY.
90.]
91
neutralize one another, and the potential be the points of the conductor.
The
same at
all
actual solution of this problem consists in the determinathe potential of the system, to satisfy the
tion of a function
T
9
conditions (1)
F is
constant over
:r-dS=
(2)
0,
C;
taken over the surface of
G
;
F=
at all points in space external to (7, except where V (3) 2 the given external electricity is situated, and there 4-Trw. 2
V
We
always exists satisfying these conditions. F,
F=
have seen in Art. 10 that one determinate function
and therefore
side of C,
dV -=
,
it
V,
were determined,
would be known at each point on or out-
and a distribution over
satisfies all
If
C
whose density
1
dV
47T
dv
is
^~
the conditions.
If the external charge were at another point 0' instead of 0, the superficial distribution would assume a different form. If there be a charge both at
and
at (/, then,
by the
principle of
superposition above proved, the density of the distribution at any point on the conductor in this case is the sum of the densities due to the charges at and at 0' separately, and so on for any electrified system outside the conductor. In like manner
there be a charge on the conductor itself, that charge will so distribute itself as to give constant potential at all points on the conductor, and the density of this equipotential distribution if
together with that due to any external electrification will be the actual superficial density. inside of 90.] The case in which an electrified system is placed a closed conducting shell is of special importance. It will be
found that in this case we have two systems, separated by the shell, each of which would be in equilibrium separately if the other were removed.
For
let
C
be any such shell, and let there be any electrified outside of it. it, and any other electrified system
system within
The general reasoning shows
as before that the electrification of
92
ELECTRICAL THEORY.
[91.
is wholly on the surface, that is to say, in this on the inner and partly on the outer surface of the
the conductor case, partly
"We can then prove that the algebraic sum of the distrishell. bution on the inner surface, together with that of the enclosed For let a closed surface S be described system, is always zero. wholly within the substance of the conductor, and entirely The potential V is dividing the inner from the outer surface. constant at
points within the substance of the shell, and
all
dV therefore
is -j
zero, at every point of S,
and
//;dv rr,
But where
m
is
the algebraic
It follows that is
m
is zero.
sum of all the free electricity within S. But the only free electricity within 8
the distribution on the inner surface of the conductor C, and
If therefore the algebraic sum of that of the enclosed system. the electricity of the enclosed system be e, that of the distribution on the inner surface is
e.
It follows, that unless there be a charge on the conductor, the algebraic sum of the induced distribution on the outer surface is
+
,
since the whole surface distribution
on the conductor
is zero.
We can next prove
the following proposition. If a shell be in electrical equilibrium hollow conducting 91.] under the influence of any enclosed electrified system, and of any
V
external electrified system, then the potential of the enclosed and induced inner surface will of the distribution on the system
be zero at
all
the potential
points on or outside of the inner surface
;
and
V of the external electrification and of the induced
distribution on the outer surface will be constant at all points on or inside of the outer surface of the shell.
For
let
S be any
shell entirely
Then 8
is
closed surface within the substance of the
dividing the inner from the outer surface. an equipotential surface, and separates the enclosed
system with the induced distribution on the inner surface of the shell from the external system, and the induced distribution on
TJNIV38RSITY ELECTRICAL THEORY.
92.] the outer surface.
Therefore (by Art. 63) the enclosed system and the induced distribution on the inner surface have, at all points outside of S, the same potential as a distribution over S whose T>
density
is
)
where
R
is
the normal force on
S due
to the
4-7T
whole
electrification
;
that
is
R=
zero potential, because on >$'. distribution on the inner
Hence the enclosed system and the
surface have together zero potential at all points outside of S. Similarly the external system and induced distribution on the
V
outer surface have together constant potential at all points inside of S and since S may be made to coincide with either the ',
inner or the outer surface of the
shell, this proves the proposition. the enclosed system, together with the distribution on the inner surface, were both removed, or allowed
It
to
follows that
if
communicate and neutralise each
other, the distribution
on
the outer surface would remain in equilibrium. Its density is therefore independent of the position of the enclosed distribution
within the shell. It follows further that any charge placed on the conductor will assume a position of equilibrium on the outer surface without causing any electrification on the inner surface.
Again,
if
the external electrification and the distribution on
the outer surface were removed, that on the inner surface and the enclosed system would remain in equilibrium.
The agreement with experiment
of the
above proposition,
that a charge of electricity upon a hollow conducting shell causes no electrification on its inner surface or on a conductor
placed within it, has been employed, as we shall hereafter see, to establish the most conclusive proof of the law of the inverse square in electric action.
In Chap. IV it was shown that the qualitative results of Experiment I, and the qualitative and quantitative results of Experiments I, III, IV, VIII and IX, were completely explained by the two-fluid theory of electricity. We are now in a position to do the same with reference to the results of Experiments II, V, VI, and VII. For it has been proved (Art. 84), that there can be no free 92.]
electricity
within the substance of conducting bodies, but that
ELECTRICAL THEORY.
94
[93.
in the case of such bodies the charges, if any, are entirely superficial.
It has been also proved (in Arts. 90, 91) that in the case of the electrical equilibrium of a hollow conducting shell in the presence of any given electrical distributions, whether internal
or external,
a superficial electrical distribution on the inner surface of the shell equal in amount, but of opposite algebraic (1)
There
is
the algebraic sum of the given internal system. That the given internal system, with the last-mentioned a system superficial electrification of the inner surface, constitute
sign
to,
(2)
producing electrical equilibrium throughout the surface of the and that the given external shell and the whole of external space on the outer surface of electrification with any superficial system, ;
the shell, constitute a system producing electrical equilibrium throughout the shell and the whole of the internal space. It follows therefore that in the case of the closed insulated
metal vessel of Experiment
II,
containing an electrified piece of
glass as therein described, (1) There will be a superficial electrification on the inner surface, the total amount of which will be resinous, and equal to
the vitreous electricity of the glass, but the intensity of which at different points will depend upon the position of the glass. (2)
That inasmuch as the vessel
is
insulated,
and the
total
charge
the electrification must be superficial, there will zero, be a superficial distribution on the external surface equal in amount to, and of the same sign as, the vitreous electricity of the glass.
and as
all
Since however the external and internal distributions are in equilibrium separately by Art. 91, it follows that ttte intensity of the external superficial electrification at any point, unlike that of the corresponding internal electrification, will be entirely independent of the position of the glass, and will be determined by the given distributions in the field external to the vessel and
the shape of the vessel. 93.] vessels
In Experiment VI the external electrifications of the and B are equal and opposite before the introduction
A
of the wire.
When
the two vessels are connected by the wire,
ELECTRICAL THEORY.
95-]
95
the two equal and opposite distributions coalesce, producing evidently by that means external equilibrium. The effect on either
same as if, there being- no introduction of the wire, received an independent charge equal in amount to, and of the same sign as, that of the glass or resin in the other vessel, and
vessel is the it
therefore equal and opposite to that of the resin or glass within '
itself.
glass
These charges remain when the wire, and afterwards the and resin, are removed, as the experiment shows.
94.]
The
Experiment VII also follows at once from For the external superficial charge on C is the
result of
the same reasoning.
same in whatever part of its interior IB be situated, and is equal B in magnitude and of the same sign. If therefore B
to that of
be made to touch
C,
the external electrification of the latter will
B
after contact may be not be affected, but inasmuch as C and one as conducting body, the vessel C with regarded constituting in contact constitutes a metallic shell with a given internal
B
distribution zero.
must be
zero,
conductor 95.]
Hence the
internal superficial electrification
and there is no free electricity within the compound B, and therefore the whole of B is discharged.
C and
We
have hitherto considered cases of equilibrium in It is sometimes
which certain conductors have given charges.
required to determine the density of the induced distribution on a conductor or system of conductors placed in a known field of force ; as, for instance, when the force before the introduction of
the conductors
is
uniform throughout the
field,
such as
may
be
conceived to be due to an infinite quantity of electricity placed at an infinite distance from the conductors.
Another
of problems certain conductors are given. class
When two
conductors of
is
found when the potentials of
known
shapes are joined together by
any conducting connection, the conductors with their connection of course form one compound conductor, and must be treated as such. In the particular case however of the connection between them being a very thin wire, the total amount of elecmust be very small, and tricity on the surface of the wire generally
As
is
inappreciable in its effect upon the field. on the connection
far therefore as the electricity
is
con-
ELECTRICAL THEORY.
96
[96.
may be regarded as two separate and known form. the existence however of conductors independent that they are of the same potential. will ensure of the connection cerned, such conductors
;
If in the case last mentioned, of two conducting- bodies joined by a thin wire, one of them be removed to a great distance from the
field,
the charge upon the one so removed will at length any appreciable effect, and may be neglected.
cease to exercise If,
at the
same time, the potential of this removed conductor
be maintained at any given value, we may by this contrivance regard the remaining conductor as an insulated conductor at a given potential. In order to effect this object the charge upon In fact, the distant the conductor must be capable of variation. or other some connected with it, must be a conductor, body reservoir containing infinite quantities of either kind of electricity,
and
so large that the
withdrawal of electricity necessary
to maintain the given conductor at the required potential has
no appreciable
A very
effect
common
upon
it.
an arrangement occurs when one more of the conductors of the field are connected by a thin wire with the earth, for this latter is an infinite conductor always at the same potential*, which is taken as zero, the potentials of all bodies being measured by their excess or defect above or case of such
or
below that of the earth. is
A
conductor connected with the earth
said to be uninsulated.
98.]
It follows
from what has gone before that the most
general problem of electrical equilibrium, in such a dielectric medium as we have described, is reduced to that of given electrical distributions in the presence of given insulated conductors with given charges, or at given potentials, in a dielectric medium of infinite extent.
The solution of any such problem, that is, the determination of the electric density and potential at any point, involves the determination of a function F, the potential of the system, to satisfy the following conditions :
* The earth for any distances within the limits of any experiment is at the same But there may be differences in the potential of the earth between potential. distant points, as England and America.
ELECTRICAL THEORY.
97-] (1)
V has
the surfaces
S
97
some (not given) constant values over each of Sn bounding the conductors on which the
...
charges are given. (
-
2)
J/Jj
dsi
taken over
S
t
=\
;
&c.; i
F"
(3)
$/.
.
S'm
.
has given constant value over each of the surfaces bounding the conductors on which the potentials are
given.
V 2 F-f 4?rp =
(4)
at any point where there is fixed elecand of course, if such fixed electricity be superficial, this may be put in the form
tricity of density p.
what
called
is
dV dV _ + __ +4 ^=:0. (5)
It
and
Y vanishes at an infinite distance. was proved in Art. 1 that one such function always
if it
be
T
t
exists,
a distribution of electricity over the surfaces of
density
JL^E 4-7T
satisfies all
dv
the conditions of the problem.
dV dv
Then the equation
+47TO-=
determines the density of electricity at any point of the surface of any conductor, and the problem is completely solved. 97.] It was stated in Art. 91 that the fact of a charge of
on a hollow conducting shell causing no electrification on a conductor placed within it furnishes the most conclusive electricity
proof of the law of the inverse square in electric action. By hypothesis there is internal equilibrium when a distribution equilibrium is placed on the outer surface of the shell. Let the outer surface be a sphere. Then by symmetry this Let us take a- for the superficial distribution must be uniform.
itself in
VOL.
i.
H
ELECTRICAL THEORY.
98
density at any point, and since there let it
be
^
T'
at the distance r
[98.
must be a potential
from a
function,
particle of unit
r
elec-
tricity.
P be
any point within the shell at the distance p from the be the Let the radius of the shell be a, and let drawn the line to OP and from between any point Q on angle Let dS be an of the shell. the surface elementary area of that Let
centre 0.
surface in the neighbourhood of Q,
the whole charge at P.
and
let
V be
the potential of
Then
r^f/V/W = 2wf* Joo
= a 2 ap cos +> rdr = apsm6d9; r2
Also
2
2 ,
ra+p
=2TT
F= .'.
and the force
Hence the
V
is
to be constant for all values of p.
by p and
differentiating,
2'ncra\f(a+p)+f(a-p)}' >
0=f(a+p)-f(a-p)',
= -- - =
inverse square
dr
-
r2
must be the law
of force necessary
to satisfy the experimental data. 98.] It may be of interest to enquire within
accuracy the experiments which have been
what degrees of
made may be depended
upon.
Let there be an insulated conducting spherical shell within and concentric with the given spherical shell, and of radius I. If the law of force were that mentioned, the charge on the
ELECTRICAL THEORY.
98.]
99
smaller sphere would be accurately zero, even with the two spheres in conducting communication ; and, conversely, if the
charge were accurately zero, the law of force must be that of the inverse square. If, however, the law of force differed slightly from that of the inverse square, there might be a small charge on the inner shell,
and we propose to investigate the amount of this charge with any assumed small deviation from the above-mentioned law. Let the metallic communication between the surface of the inner sphere and the external surface of the outer sphere be made
by a very thin wire, then the electricity on this wire may be neglected, and therefore, by symmetry, the charges on the two And if the shells be spheres must be uniformly distributed. very thin, we may, whatever be the law of force, regard the charges as superficial. Let be that on the outer sphere, and
E
Lety(r)
=
C+m$(r)
where
m
is
W that on the
small;
i.e.
let
inner.
the law of
force be
where
m
and
this
small compared with C. the potential from the two charges will be point
is
At any
But
~
I
f(r)dr
+
-
J f(r)dr,
must be the same at the two ends of the
Therefore
That
P
is,
wire.
LINES OF FORCE.
100
[99.
and neglecting the products of the small
1
Therefore, substituting
magnitudes E' and m, we get a+b
For example, suppose the law of force to be -g+^, where q small.
is
Then
and
Therefore
=
/
C=
>
rr7-H^ logr m=
-
$ (r)
= log
r.
Substituting in the expression for E', and remembering that /
logrdr
= rlogr
r,
we get
is the theoretical basis of the experiment by which Cavendish demonstrated the law of the inverse square. The experiment is given in great detail in the second edition
This
of Maxwell's Electricity and Magnetism, appears, from what is there stated, that
pp.
76-82
we may
;
and
it
it
as
regard
absolutely demonstrated that the arithmetical value of q cannot
Lines of Force. 99.]
The
state of the electric field
under any given distribution
of charges and arrangement of conductors is completely known when the value of the potential at each point of the field has
been determined. It
is
obvious however that the direct subject of
experimental investigation in any case must be the magnitude and direction of the force at any point of the field, and hence has * See Senate
House
Questions, 1877.
LINES OF FORCE.
99-]
101
and fluxes of force, originally and suggested by Faraday developed by subsequent writers. Line of Force. Suppose a sphere of indefinitely small radius to be charged with unit mass of positive electricity and placed with its centre at any given point P in an electric field, and arisen the conception of lines, tubes>
suppose the electrical distribution of the rest of the field to be by the presence of this charged sphere, and suppose further the inertia of the sphere to be always neglected, then
unaffected
the centre of the small sphere would move through the field under the action of the electric forces of the field in a definite line, generally curved, this line field through P. tf
is
defined as the line of force in the
*/
When
the electricity of the field consists of an electrified mass and therefore all of very small volume, inclosing a point the the lines of force are clearly situated at sensibly point 0, if the charge at be positive, and straight lines radiating from if the charge at be negative. terminating in
If the point
were
at
called a
moved
off to
an
and the charge would become what is of force would be parallel
infinite distance,
infinitely increased, the field
uniform
field,
and the
lines
straight lines.
an infinite plane with a of its uniform over surface, the lines of force would charge density be parallel straight lines normal to the plane and proceeding from or towards that plane, according as the density thereon was
So
also if the distribution consisted of
positive or negative. If the distribution were that of uniform density on the surface of an infinite circular cylinder, the lines of force would be in parallel planes perpendicular to the axis of the cylinder, radiating
from or converging to the point in which that axis met each of these planes according as the electrification of the cylinder was positive or negative. For less simple cases of distribution the lines of force are not
are generally capable of any such immediate determination ; they curved lines, their direction at every point coinciding with the
normal to the equipotential surface through that point and prothat ceeding towards the region of lower potential. It follows
TUBES OF FORCE.
102 no
line of force can
tial,
and that
[lOO.
be drawn between points at the same potenof force in the immediate neighbourhood
all lines
of an electrical particle, i. e. a very small volume with a charge of infinite density, must radiate from or to the point with which that coincides, according as the density of the charge is positive or negative, because the potential in the immediate of such point is positive or negative infinity in the
volume sensibly neighbourhood
respective cases.
Tubes
100.]
A
of Force.
bounded
laterally
described,
Fig.
is
of space
region
by
in
the
called
a
tube
field
above
lines of force, as
See
of force.
6.
When
the transverse section of the region small it is called an elementary
is indefinitely
tube offorce.
Flux section
of Force. Suppose any transverse dS made through any point P in
the surface of an elementary tube of force, as in the figure, the angle between the normal to dS and the bounding lines of force If the intensity of the force at dS be i. and the area of the orthogonal section of
being
F
9
the tube at the point perpendicular to
be denoted by equal to Fig.
dS
Fn
FdScosi,
,
P
be
a,
the force resolved
will be .Fcos
the product or Fa,
and
i,
and
Fn dS
will be the
for every transverse section of the tube
7.
if this
will
be
same
in the
neighbourhood of dS. This product, from its analogy to the flux of a fluid flowing through a small tube with velocity M=F, is called the flux offorce across dS; the limiting value of the ratio of the flax offeree across any elementary area to the area is the intensity of the force in
the
that elementary area and perpendicular to it. the distribution arises from a so-called charged particle,
field at
When
the tubes of force are conical surfaces with their vertex at the particle;
by
when in a uniform field they are
surfaces limited laterally
parallel straight lines, and so forth. 101.] Let a charge of electricity of either kind, and with mass
1
FLUX OF FOKCE.
01.]
103
numerically equal to m, be situated at a given point 0. Let a as centre. Then the sphere of any radius be described about fluxes of force across all equal elementary areas of the sphere's surface will be equal to one another, and will take place from
within outwards, or from without inwards, according as the is positive or electricity at negative, the total flux over the
whole sphere being
4 irm.
as a source from which, or a
Faraday regarded the charge at
sink towards which, lines of force proceed symmetrically in all directions, and he further regarded the density of these lines of
number contained in each unit of solid angle at 0, as The number of lines of force therefore, which, proportional to m. in this view, traverse any surface, corresponds to the flux of force force, or the
and the force in any given direction at a point the limiting value of the ratio which the number of lines traversing a small plane at perpendicular to the given
across that surface,
P
in the field
is
P
bears to the area of that plane indefinitely diminished. direction
when
the latter
is
If the point were eccentric, the equality of flux over all equal areas would no longer be maintained, but the flux elementary
over the whole surface would, as we know from Art. 45, or as would result at once from the equality of flux over every transverse section at any point of an elementary tube of force,
proved in Art. 100, still remain equal to lirm. We know also from Art. 45, or we might prove at once from Art. 100, that the total flux across a closed surface of any form surrounding would be 4 urn. If there were any number of sources or sinks within the closed from each surface, the traversing flux across the whole surface such source or sink would be 4irm, where m is the numerical value of the charge at such source or sink, and the flux is outwards or
inwards according to the sign.
The
and 2n
<**> 4-7T Zn), where Sjt? (2jt? of the sources and sinks respectively,
be
would sums of the charges and would be outwards or
total flux in this case across the inclosing surface
are the
inwards according as 2/? was greater or less than 2#. If there were any number of sources or sinks in the
field
FLUX OF FORCE.
104
[lO2.
external to the aforesaid surface, their existence would not affect the value of the total flux across the whole surface.
that a tube of force, elementary or otherwise, in 102.] Suppose electric field, is limited by transverse surfaces S and S'} and
any
that
contains electrical distributions, such that the difference
it
of the sums of the masses of the positive and negative charges
is
m, then the flux offeree across the whole surface of the tube thus closed from within outwards will exceed that from without in-
wards by the quantity Ivm if the preponderating included flux will fall short of the electricity be positive, and the former
by 4?m
latter
if
the preponderating electricity be negative. But the flux of force across that portion of the tube's surface which contains the lines
of force
is
If therefore the
zero.
direction of the lines of force be
from S
f (see Fig. 8), the flux of force across S' will exceed or fall short of that across
to
S by
the quantity 4irm, according to the sign of the preponderating included electricity. and F' be the forces normal to S and If at any points in
F
them
&
respectively,
braical
sum
and
if
m
be
now taken
of the included electricity,
to represent the algethese statements are
expressed by the equation / /
The portions of any the same tube of force
FdS
surfaces in
= 4:7rm.
an
electric field intercepted by are called corresponding surfaces, and therefore in proceeding along any tube of force, finite or elementary, the fluxes across corresponding surfaces are continually increased
irm, where m is the algebraic sum of the elecincluded in the tube in its passage from any one surface to any other, such increase being a numerical decrease when m is And if there is no such included electricity, or if its negative.
by the quantity 4 tricities
algebraic sum is zero, then the fluxes across the corresponding surfaces are all equal to one another. 103.] Suppose that there is in the field a surface 8 charged
with
electricity, the density at
any point
P
being
a.
1
FLUX OF FORCE.
03-]
105
Let dS be an element of 8 about the point P, and conceive a small cylinder to be drawn with
its
generating lines passing through the contour of dS and perpendicular to that element. 1
The total flux across this cylinder must be equal to the included electricity, i.e. to
Fig. 9.
adS.
Also, if the length of the cylinder's axis be indefinitely diminished, the flux across the curved surface will become infinitely less
than either of the fluxes across the bounding planes, and must ultimately differ from one another by
these fluxes therefore
N
and N' be the forces in the field normal to dS and on opposite sides of it, we have N'dS-NdS=4.Tt
4770-^$, so that if
t
suddenly in value by the quantity 4 TTO- in passing from one side of the surface to the other and we may also prove that the ;
normal force upon the
electrified
element of the surface
itself is
the
arithmetic
mean
of the normal forces which would act on that
element
placed
first
if
on one side and then on the other of the
For, considering the elementary cylinder above mentioned, it is clear that the force arising from all the electricity in the field, besides that on the element dS, must be continuous surface.
throughout the cylinder, inasmuch as all the electricity from which it arises is without the cylinder, and therefore the normal force throughout the cylinder arising from that external electricity But the will be ultimately the same as it is at the surface.
from the charge on the included element vdS on points at any equal small distances from the surface and on opposite sides must be equal and opposite, and therefore the sum of the total normal forces on either side of the surface must
normal
force arising
be equal to twice the normal force of the external electricity
throughout the cylinder
;
or the normal force of the external elec-
mean of the total tricity at the surface must be the arithmetic normal forces on opposite sides of the surface ; and therefore the normal force on the elementary charge
FLUX OF FORCE.
106
[104.
what the normal forces on the same charge would be if placed on each of the two sides of the surface respectively, for the charge ad/S can exert no force upon itself. It is clear also that the charge crdScan exert no tangential force in one direction rather than another, and therefore the force resolved tangentially must be the same on either side of the If therefore F and F' be the forces on opposite sides of surface. the surface, and if i and i' be the angles between the lines of force
mean
of
and the surface normal, we have f
F' cos i
F' sin and therefore
tan
i
F cos + 4 = F sine; e
i'
= tan
i'
(1
7T0-,
+
.)
;..
or the lines of force on traversing a surface with superficial electric density a are deflected towards or
from the normal according as tive or negative
;
o-
is
posi-
see Fig. 10.
It appears also from the foregoing that the force exerted by an element
dS
of a surface of superficial density
a-
at points very close to dS is a normal force 27TO-, and repulsive or attractive
according as a j^ig. I0-
is
+
or
.
We
may now trace the possible 104.] course of an elementary tube of force
through an
electric field in equilibrium. axis of such a tube in passing through any point Pmust towards regions of continually diminishing poproceed from
The
P
tential.
It
may
then pass on to an infinite distance
if it
encounters no
free electricity.
Or
it
may
traverse a charged surface, in
which
case, if the
transit be oblique, it will be bent through a finite angle at the surface in the manner above explained.
If this charged surface be that of a conductor, the line, or rather elementary tube, of force will proceed no further, but it will be, so
I04-]
FLUX OF FORCE.
107
quenched in the sink afforded by the negative density of the surface at the point or element in which it meets it. Or it may traverse a region of finite volume density, in which case it suffers no abrupt refraction, and if the density of the region to speak,
be positive the tube emerges therefrom with augmented flux, if the density be negative the tube may be, as in the case of the conductor, quenched in the sink thus afforded and proceed no further. finite, has emerged from a positively and is quenched, as above described, surface charged conducting in another conducting surface without traversing any region of
If the tube, elementary or
then the positive charge on that portion of the surface of emersion contained within the tube must be equal in electric charge,
magnitude to the negative charge on the corresponding surface of the surface of reception; or, in the language of Faraday, the number of lines of force emanating from the source is equal to those quenched in the sink. In other words, the number of lines of force emanating from or converging to an elementary area of any conducting surface is
a measure of the positive or negative density of the electrification of that surface.
CHAPTEE
VI.
APPLICATION TO PARTICULAR CASES. is proved above that whatever be the given on a system of conductors, combined with any charges or potentials
ARTICLE 105.] IT
fixed distribution of electricity in space, there exists always one,
and only one, mode of distribution upon the conductors consistent with equilibrium.
But the
actual solution of the problem, the determination, of the actual density of electricity at a point of any given conductor, is one of great difficulty, and has only been achieved in a few simple and comparatively easy cases.
that
is,
Case of an infinite conducting plane and an electrified point. Let there be an infinite conducting plane, and a unit of positive above it. It is required to find electricity fixed at a point
the density at any point in the plane in order that the potential of the plane may be everywhere zero.
The
potential of the required distribution on the plane must at all points on the
be equal and opposite to that of the unit at
and therefore also at all points in space on the opposite side of the plane to 0, by Art. 60. If a unit of negative electricity were placed at 0', the optical
plane,
0, formed with respect to the plane as a mirror, its at potential any point of the plane would be equal and opposite to that of the unit at 0, and therefore equal to that of the
image of
It would therefore also be equal to that required distribution. of the required distribution at all points in space on the same side of the plane as 0.
Let
V be
unit at 0. tribution at
the potential of the required distribution, and of the
Then, by Poisson's equation, the density of the
any point
P
in the plane is 1
,dV
dis-
PARTICULAR CASES.
106.]
where dv
109
an element of the normal to the plane measured from the plane on the same side of the plane as 0, and dv the same thing on the same side as (/.
Now
is
the value of
plane as
V
at
any point
P
on the same side of the
is 1
JL_
OP
O'P*
side of the plane V is constant because it is constant over the plane, and there is no electrification on that side of the plane.
and on the opposite
JTT
Therefore
Also on the plane
-7-=
dV
and
0.
-=-,
dv
=
-,
=
-
-
d
--
OP
di>
d ,=___._; I
therefore
and
if
point
1
I
from the plane, r the distance of a from the intersection of 0(7 with the plane,
k be the distance of
P
in the plane
~
J_ 2irdh 1
h
27T
OP3
which determines the density
at
'
any point in the plane.
of Fmay be deter106.] In certain very simple cases the value mined by the integration of Laplace's equation. For instance Let the Two infinite conducting planes at given potentials.
planes be parallel to the plane of xy.
uniform throughout each plane,
T
Then, since the density is in this case
d2 V z only, and Laplace's equation becomes
-^
=
0,
is
a function of
from which
V
can be found with two arbitrary constants, and the constants are to be determined by the given conditions on the planes.
PARTICULAR CASES.
110 (1)
In
Two like
infinite
[107.
coaxal cylinders.
manner,
if
we have two
infinite coaxal
conducting
the density is uniform throughout cylinders at given potentials, and is a function of r, the distance of each the surface cylinder, from the axis. Laplace's equation is in this case
V
IdV _ ~dr*+~r~dr" which admits of integration. (2)
Two
Again,
concentric spheres.
if
there be two concentric conducting spheres at given is uniform throughout the surface of each
potentials, the density
sphere,
and
V
is
a function of
r,
the distance from the centre.
Laplace's equation becomes in this case
dW + 2^T_ = dr*
r dr
which admits of integration. In this problem, as in the
last, the two arbitrary constants in solving the differential equation must be determined with reference to the given conditions on the cylinders
which enter into
V
or spheres.
107.] Case of an insulated Conducting Sphere in a Field of Uniform Force.
Let us take the direction of the force a the radius of the sphere, must satisfy the conditions,
the
force,
for axis of #.
V the
Let
X be
Then
potential.
V
(3)
F is constant and = C on the surface of the sphere V 2 F = at all points outside of it ^r== Xx + C at a sufficiently great distance from the sphere
(4)
The
(1) (2)
;
j
total electrification
on the sphere
is
;
zero.
The function
where
/ is the distance of
these conditions.
any point from the
centre, satisfies all
PARTICULAR CASES.
I08.]
The density on the sphere It
is
-
Ill t
/
4?r
that ^-, dr
is,'
4-na
seen that
is easily
108.] Case of an uninsulated Conducting Sphere and another Sphere outside of it uniformly filled with electricity of density p.
This
the same problem as that treated in Chap. Ill, Art. 65.
is
We give Let C a the
another method of solution. be the centre, of the
radius,
conducting sphere ; and be the centre, b
let
the radius, of the other sphere.
Let
OC=f.
Let
V
be the potential of the whole system.
Fig.
n.
It is required to find the density of the induced distribution on the conducting sphere which gives zero potential on that sphere, and the general value of V in this case.
7
has to satisfy the conditions,
7
(1)
at all points
on the conducting sphere
;
V 7= at all points external to both spheres V F+ 4-7T/) = within the non-conducting sphere. az a point E in CO such that EC = -^ 2
(2)
;
2
(3)
Take
t/
Let Let e
OP =
VQ
EP =
/, where
P is
any point. be the potential of the charged sphere at P.
-
3
3
p,
r,
Then
if
or the total charge of electricity in the charged
sphere, the function
satisfies all
potential.
the conditions, and must therefore be the required
PARTICULAR CASES.
112
M
F =
ch For outside of the charged sphere
[108.
,
and therefore the
above equation becomes
r*j _-(. Now on
by a known property of the
its surface,
EP _
therefore
V F= 2
-
2
=
be
r ""/"
Therefore for a point on the conducting sphere Also for a point outside of both spheres
V
P
_a
r'
OP~
sphere, if the point
V
and
4=
2
F=
0.
-
;
0.
For a point inside of the charged sphere
V The density
at
2
= V -^ + 47r/3 = 0. 2
F+4irp
any point on the conducting sphere
In dv r
Also
where the angle from C\ also
and
z
PCO =
We
r
fr'
=f* + v*-2fvcos0, and v denotes the distance of a point
6,
in the expression for
differentiation.
(
is
-
cr,
made
v is to be
equal to a after
have therefore
dr dv
___ ==
afcoa0 ,
r
a? _
dv e
a
(afcosO
-^{
f
a --
f
/
cos 6
"
a but
Therefore
=
-P&
-Q - ^ 47T
_
ar3
j2 ,
as already J found.
PARTICULAR CASES.
IO9-]
From
113
the form of the potential function
F=fr
fr'
follows that the potential of the induced electricity on the conducting- sphere in the presence of the charge e at the external
it
ae
the same as that of the charge
is
point
The point
E
is
at the point E.
called the electrical image of
in the con-
ducting sphere.
of an
109.] Case
infinitely long Conducting Cylinder and a uniform distribution of Electricity throughout the substance of another infinitely long cylinder outside of the former one, and whose ,
axis is parallel to that of the former one.
In
this
(1)
problem
V has to
satisfy the following conditions, viz.
V = on the surface of the conducting V F = outside of both cylinders V F+4-n-p = inside of the charged
cylinder
;
2
(2)
;
2
(3)
density of the distribution within
cylinder, p being the
it.
Let a plane perpendicular to the axis cut the axis of the conducting cylinder in C, that of the charged cylinder in 0. (See Fig.
Let
1
of last example.)
OC =/. 2
In
OC take
a E such that fiC=~distance of any point P from
a point
Let r be the
cylinder through 0, axis
through E.
/
Then
its
=^
for every point in the section
made by the plane with the conducting Let
the axis of the
distance from a line parallel to the
cylinder.
R
be the quantity of electricity contained in unit length Then the potential of the charged of the charged cylinder. cylinder at any point outside of
it is
ro =C-2Elogr. It will be found that
ar VOL,
I.
I
ELECTRICAL IMAGES.
114:
[l IO.
the conditions, and is therefore the potential. in this case r is independent of z, and therefore satisfies all
d
Now d2
.
_log r=s -j^:^, V log r = 0.
Similarly
2
whence
On
For
the conducting cylinder r
=
a
/; and therefore
Outside of both cylinders
VF = 2
V
and
0,
2
log
^L
=o
ar
,
therefore
V V= 2
;
and within the charged cylinder
V The density
is
= V'F + 47r/> = 0.
any point on the conducting, cylinder c I d d = + R < -jr >, log r 27T ( dv dv log ) r2
where
i;
F+47r^
is
found
a after differentiation.
The
at
from
and
2
to be
2
2
=/ + v -2fv cos
made equal
to
(9,
/2_ a2
result is "~
110.]
Ow
Electric Images.
We
have seen in Art. 108 that
if
a sphere be at zero potential under the influence of a charged point outside of it, the induced distribution has at all external points the same potential as that due to a certain charge placed at a point within the sphere, and the last-mentioned charged point is defined to
be the image of the influencing point in the sphere.
An infinite plane is for this purpose a particular case of the sphere.
ELECTRICAL IMAGES.
III.]
115
Every electrical system outside of a sphere, inasmuch as it may be regarded as consisting of a number of charged points, is represented by a series of images in the sphere, and these together may be said to form the image of the external system. In like if the sphere be at zero potential under the influence of a charged point within it, the induced distribution has the same potential at all internal points as that due to a certain charge at
manner,
The external point
a certain point without the sphere. the image of the internal point.
electrified
Every
is
called
system within
the sphere has its image outside of the sphere. It can easily be shewn that no closed surface except a sphere or infinite plane generally gives rise to an image. For let 8 be any uninsulated closed surface, and let
E
be an
external point at which a charge e is placed. If the induced distribution on S have at all points on S the same potential as that of a charge / at a point .F within S that is, if be an image
F
9
of
E within
S,
we must have
EP _e FP ~?'
P
being any point on
S
8.
Thus the locus of
P
is
a sphere, that
a sphere. is, 111.] By the method of electric images many problems relating to the distribution of electricity on spherical or plane is
surfaces can be solved.
The case of two spheres cutting each other orthogonally (Maxwell's Electricity and Magnetism, p. 168). Let Clt C2 be the centres, a l} a 2 the radii of the spheres.
AB section, E Let
CL C2
represent the circle of interthe point in which the line intersects the plane of that circle.
Then C^AC^ C^BC^ and
are right angles,
f=a + >
Also
1
a
=
)
.
116 or
ELECTRICAL IMAGES.
E is
the image of
C2
the sphere
we
place at
Ea
and at
,
in the sphere
C1 and ,
the image of
2.
I
(7j
in
.
If therefore
quantity a2
C2
[l
Cl
a quantity of electricity a^
,
at
(?2
a
quantity
the potential at any point on either sphere will be unity, because the point be, for instance, on the sphere 2 the two charges,
if
,
^ at
Ci
-
and
Va*+a*
at E,
have together zero potential at each point on that sphere, while the charge a 2 at C2 has potential unity. 112.] Now let us consider the conductor FAGB, formed by the two external segments of the spheres. distribution
upon
points on
it, is
This then
is
The aggregate of a
which gives unit potential at
its surface,
the capacity of the conductor.'
Again, since the potential of that distribution is the all external points as that of the three at Clt C
same at z , and E,
charges
its
density
is
_ 4-7T
But 1
if
all
equal to
P be on
*
"-d+ C ^
the sphere
and therefore the density
By symmetry
2
a
dv
,
1
^Art
is
the density at a point on the sphere (,
.'
)
Cl
is
108
ELECTRICAL IMAGES.
I13-]
117
Instead of the figure formed by the two external segments, we take the lens formed by the two internal segments, or the
may
meniscus formed by one internal and one external segment, and calculate the superficial density in the same way. We shall consider this problem further Inversion.
Another interesting example
113.]
ing question
An
when we come is
to the Theory of
afforded
by the
follow-
:
ADEFB
uninsulated conductor
consists of an infinite plane
with a hemispherical projection DEF, the centre
C
plane
AB.
of the hemisphere being in the is mass of electricity
A
m
situated at the point m, in the radius
produced, where
CE
CE
perpendicular to and m{ be the plane. Then if the points taken on opposite sides of C such that is
%
,,__ ""
MI
CE* '
^Cm
and if m be taken onmC produced such that Cm' = Cm, the effect of the induced charge on the conductor under the influence of the
m
mass
at
m may
joint effect of
For
and
/
where
at m'\
Fig. 13.
-^,m^.
,
,
Wt
Wlj.
~^ +
771
Tfli-t
<~~7"
at all points on the hemispherical surface
and therefore Or
,
respectively.
r
*
L
g
P
._
Then
m =
be any point on the same side of the conductor as the distances of P from m, m l mf, and m be r, r
let
let
%
,
m
at MI and
and
be represented by the m 1 at m1 -f the masses
V=
m,
m
r
r
x
j
m /
m
i
'
.*
rt
we have
'
over that surface.
in other words, the induced charge on the composite conductor is equivalent image mi of the charge m at the electrical image of m in the hemisphere together with the image of m and TOJ in the plane.
to the
SUCCESSIVE IMAGES.
118 Similarly at
all
points of the plane's surface r
T=
and therefore
AgaiD
=/
^ = r^,
and
over that surface.
vi = vS = v
,
[114.
=
v=0
on the same side of the plane as m, where there is electrical density is zero, and at m electricity, or where p the
at all points
no
where p
is
the density within the small volume of m at m. all points on the aforesaid side of the plane
Therefore at
Therefore the function ficial
and
V
taken as above
satisfies
solid conditions of the potential of
m
the superand the
at m,
induced charge on the conductor, and must therefore be the potential of m and that induced charge.
In other words, the induced charge produces at all points on m the same effect as the charges m 1) m l and m at m the points 1 mf and m' respectively.
the side of
,
From
the equation 4.7TO-+
dv =0, -
we is
easily find that the superficial density a- of the induced charge everywhere negative, except at the circle of intersection of the
hemisphere and plane, where it is zero, and that at any point on the hemisphere
P
1
m'P3 and at any point proportional to
P
on the plane outside of the hemisphere
Cm3
^P 114.]
On
3
CE
o-
is
3
m^
Systems of Successive Images.
If we have given any two conducting spheres, including in that designation an infinite plane, at zero potential under the influence of an electrified point, the electrical distribution on
SUCCESSIVE IMAGES.
114,]
119
either sphere will be found to be equivalent in its effects to two series of images, the magnitudes or values of which
infinite
Hence the density of the
converge.
actual
distribution
on
either sphere generally admits of being calculated approximately. For instance, let us consider a sphere and infinite plane not
and an influencing point in the perpendicular from the centre of the sphere on the plane, between the centre and the plane. intersecting,
A
Let
be the centre of the sphere, c
E
its radius,
the influ-
encing point at which the charge
H
e is placed,
B and
the points in which AEAf cuts the sphere and
plane respectively.
HA'= Vtf-c\ The charge
at
E
Fig.
produces on the sphere a certain distribution,
which we may
the sphere, the effect of which at is
call
the primary distribution on
all
points outside of the sphere
the same as that of a charge
A
tween from
H
and E> whose distance from
is
Ji
-j^
i4.
-^e
placed at a pointjbe-
A
-7^, and
is
its
distance
That produces on the plane a distribution
whose density we may denote by bution over the plane at
all
^
;
and the
points on the
effect of this distri-
left side
of the plane rt
is
the same as that of
at a point distant k
Let
From
image, namely, a charge
its
r-=;
e
placed
to the right of the plane.
^
"
i
=
*~A;
this distribution, or its equivalent image,
we
derive in the
same way a second distribution on the sphere equivalent charge c
~~AE'
c '
to a
SUCCESSIVE IMAGES.
120
c
at a point distant
2
- from A, and from
-7
on the plane.
distribution of density p 2
We
this again a second
n-\- X-^
series of images to the right of the IT are #15 #2 &c. And from distances whose plane,
then have a
shall
,
c
and generally It
is
n?
_C
ip ne
>
The charges
and
the
>
_
_c
#n
,
c
c
_
)
and every
is
a?
less
than
c
c
AE \+*C AE between
two successive
charges
continually
-T-J>
A.A.
E
induces on the plane a primary disequivalent to the image of E in the plane.
Again, the charge at tribution
&c.
successi ve images continually approach A'. images are successively
ratio
approaches
-t n-rx n
at these
c
AE
=h
that # n+1
easily seen 2
w+1
2
which
is
This original image the distances from
is
at a point distant from
H
9
H of the derived images are
os\
HE, and
&c., &c.
which continually approach HA'. The charges at these images are successively e at the first image,
-
=
h + x^
and
e
at the second image,
so on.
Hence the density of the induced on the plane, where
HM = r
t
is
distribution at
any point
M
SUCCESSIVE IMAGES.
"50 c
e
yX
f
121
QGn
\
+ &C. Each differ
series
converges rapidly, and the terms soon cease to
sensibly from those of a geometric
series
/>
ratio is
Hence the actual density
-,
A.A.
to
any required degree of accuracy. integral charge on the plane
The
is
images irrespective of their position.
at
the
M can be calculated
sum
That
whose common
of both series of
is
AE -e\l + 115.] Another very interesting case is that of two concentric spheres and an electrified point placed between them, treated in
Maxwell's
Electricity.
In that case the distances of the images from the common centre are in geometrical progression. Also the charges are
in geometrical progression,
and their sum can be accurately determined.
Let
be the
common
centre, a the radius of the
inner sphere, b the radius the of the outer sphere,
E
point where the charge is
placed,
OE =
k\
all
Fig.
15-.
e
the images are in the line
We have then an image at
P
lt
where
OP =
an image at
Qlt
where
OQj_
an image at
P
z
,
where
OPZ =
an image at
Qt
,
where
002
1
6
OE produced.
2
T
= -^p = -TT
'
-^TJ
= -^ = UJTn
Tj-A.
122
We
SUCCESSIVE IMAGES. see
then that the distances from
[1-16.
of the successive images
derived from the primary distribution on the outer sphere are
and the charges at those images are o a* a e' e' &c
v
b
Again,
if
we
with the primary distribution on the inner
start
sphere, represented
-
by
-7
we
a^ Qi,
e
obtain a second series of
fa
images whose distances from
are
a
in
b*h
and whose values are a3
Hence the
total charge
images within
on the inner sphere, or the sum of the
it, is
a
f
VTb
ab
~"
~
a
h
~\
a)
b-a
h
and that on the outer sphere is a
e>
ba
116.] Another class of cases
'
ha
hb
images
)
a
b
or
h
v
zTTi
h(b
h is
ba
e.
that in which the
number
of
is finite.
For instance, let us consider two infinite conducting planes at
Y
right angles to each other, in presence of an electrified point.
Let the projections X'
X planes on the plane
^
'
o,
and
be
XEX\ YET,
an
electrified point.
of
the
of the paper
Ol
let
be
The image of Ol in XEX is Q. The image of (7X in YEY' is 2 The image of 2 in XEX' is C2 The image of <72 in YEY' is f
Y' Fig. 16.
.
.
.
1
1
The two planes
+e
at
for
C2
Ol and and
!
effect
123
SUCCESSIVE IMAGES.
7.]
2,
are at zero potential under the influence of e at If we now substitute and C2
Q
and
.
the distributions on
2
XEX
f
which have the same
with them on the lower side of the plane XEX', and for
the distribution representing it in YEY', the potential will be zero on the plane YEY' and on the part of the
EX
still
plane XEX'.
In
117.]
like
manner instead
we may have n
right angles,
of two planes intersecting at
planes intersecting at angles
f
(Maxwell's Electricity, p. 165). Taking an electrified point Ol3 and forming successive images in the planes, we shall have a series of positive points Olt 2 ... On and a series of negative points C1 ,
,
,
C2 ,... Cn placed symmetrically round E, the projection of the
The If
common
potential
YEY
f
is
section on the plane of the paper.
zero on every plane.
and SES' be the two planes between
which the point Ol lies, we may substitute for all the points on the left of YEY' their corresponding distribution on YEY', and for all the remaining points except bution on SES'.
the system
is
itself their
Then the
unaffected,
corresponding
potential on the portion
and remains
zero.
Fig. 17.
distri-
Y'ES
of
CHAPTEE
VII.
THE THEORY OF INVERSION AS APPLIED TO ELECTRICAL PROBLEMS. ARTICLE 118.]
THE
some
of
solution
electrical
problems
involving spherical surfaces, or portions of spherical surfaces inoluding planes, can be effected by the method of inversion.
This application of
inversion
is
due
originally
to
Sir
W.
Thomson.
Taking for origin any point 0, and for coordinates the usual spherical coordinates r, d, <, let us suppose we have found the solution of a given electrical problem, that is, we have found the single function, P, of
r, 0,
conductor of the system, and
which
$,
satisfies
is
constant within each
the characteristic equations
at all external points, and vanishes at an infinite distance, and hence we have found the density at every point on any conductor.
For then invert the geometrical system as follows we will take a point any point P of the system distant r from
"We
will
P' in the
:
line
OP, and distant
/
from
K is a
_
Q .
0,
where
(
and
called
is
the centre of inversion. If P and Q be any two points f in the original system, and
P
Q
f
the points corresponding to
__
_ OP.OQ
P'Q'~
,
constant line called the
in the inverted system, the triangles similar, and therefore
1
K2
T
radius of inversion, and
them
_
/=
1
2L
K2
_
PQ
POQ, Q'QF
are
1
GENERAL THEORY.
INVERSION
19.] if
Again,
Q
be very near P, and
if
OP =
125
r,
Therefore any linear element of length dv in the original K2
system acquires the length follows that K4
comes
-j-
dA
dv in the inverted system.
dA
any element of area
It
in the original system be-
in the inverted system ; and the element of volume
dv in the original system becomes
$ dv in the inverted system.
119.] Every sphere in the original system becomes another sphere in the inverted system. be taken in the line joining with For let the point
E
the centre
C of the
sphere, such that
a2
where a
is
the sphere, -~~ Therefore
Let
the radius.
if
constant, and
is
E' and
corresponding to
OC =f.
if
Q
be a point on
=
be the points in the inverted system
Q,'
E and
Then,
Q,
according as the centre of inversion original sphere, and in either case the centre of a new sphere.
is
is
without or within the
constant.
Therefore
If the centre of inversion be without the sphere, and
E
f
is
if
**=f-a\ the sphere
is
in position.
unchanged
For in that
case,
That is, the centre of the new sphere coincides with , the point which was the centre of the original sphere, and the radius of ..
the
new
,
sphere,
K
-^
2
a =-
,
= a.
126
GENERAL THEORY.
INVERSION
[l2O.
Again, a plane in the original system whose perpendicular distance from the centre of inversion is p } becomes a sphere of passing through the centre of inversion, and whose
diameter centre
is
in p.
Again, a sphere of radius a in the original system passing through the centre of inversion becomes when inverted an infinite plane at right angles to the diameter through the centre of inK2
version,
from that centre. Again, since two inter-
and distant
secting spheres becomes spheres when inverted, their common Hence every circle on the original section becomes a circle* sphere becomes a circle on the inverted sphere.
Again, since the triangles
POQ, Q'OP'
are similar, if
be the
angle made by the
radius r with any elementary line at its the angle in the new system is TT 0. corresponding extremity, in the which original system is within any closed Every point surface S, not enclosing the centre of inversion, will in the inverted system be within the corresponding closed surface S'. And every point without S will in the inverted system be
without
S'.
But
if
S
enclose the centre, all points within
S
correspond to points without /S", and vice versa. Evidently every conductor in the given electrical system will be represented in the inverted system by a certain closed surface. 120.] electrical
We
will
now
construct upon the inverted system a
new
system as follows, viz. 2
If
pr sin0d0ddr
be the quantity of electricity in the space-element of the original system, we will place in the corresponding space element of the inverted system the quantity r
Since as
we have
seen the element of volume dv in the original K
system becomes
6
r
dv in the inverted system,
the volume density J in the
new system J
r
5
is
r
K
6
p
=
follows that
it
K5 .
r
5
o.
GENERAL THEORY.
INVERSION
122.]
127
In like manner
if
surface element
/c
r
This gives a surface density
ag-
in the
new
system.
We have
thus constructed a new electrical system, in which S> of the conductor every original system is represented geoa S' in surface the new system, and every quantity metrically by of electricity in the original system is represented by a corre-
sponding quantity in the new system. 121.]
We now proceed to find the relation between the potential
any point Q of the original system and that at the corresponding point Q'j due to the electricity which we have supposed placed on the inverted system. Let s denote an element of volume at P in the original system, at
ps the quantity of electricity in the element
is
v
s\
=
In the inverted system, ps at its
potential at Q'
Then the
it.
potential at
Q
of
a
P
becomes
s -^p P
is-
v 1
but
P'Q' whence
,
v
=
K
=_
OP.
OQ
~^"
1
PQ
5
ps OQ = OQ^- -77= K K fty
P
and P', it holds As this is independent of the position of true for the whole potential of the original system at Q, and denote the That is, if 7 and of the inverted system at Q'.
V
potentials at
Q
and
Q',
conductor whose bound122.] It follows that if Tbe zero for any 7' is zero throughout the ing surface is S in the original system,
GENERAL THEORY.
INVERSION
128
[l22.
corresponding surface S' in the inverted system. Therefore if the space S' be occupied by a conductor, the assumed distribution of electricity throughout the inverted system will, as regards such conductor, be in equilibrium with zero potential. And if
any
electrical
system consists of conductors
all at
zero poten-
in presence of fixed charges of electricity, the inverted system will also be in equilibrium with all its conductors at zero potial
tential.
Again, let the original system be one in which the potential of a distribution over a closed surface S is equal at each point on S to that of any electrification enclosed within S. Then if invert with respect to an external point, and S becomes S', the potential of the corresponding surface distribution over S'
we
will be equal at each point of S' to that of the corresponding enclosed electrification. If, for instance, the distribution on S
have the same potential in all external space as if it were collected at a point C within S, that is, if the original system be a centrobaric shell, the surface distribution over S' will have the
same potential in
all space outside of &, as if the (f, point corresponding to C that is, the a centrobaric shell too. ;
it
were collected at
new system
will be
V
is not If in any system V be not zero for the conductor S, generally constant over ', and the inverted system will not be
&
in equilibrium with
for
a conductor.
But, as
we have
seen,
F
F' =
If therefore we place at the centre of inversion a charge * F, the potential of this charge, together with that of the inverted system, will be zero at each point on &. If therefore
we have given a conductor
S,
and know the
density at every point on its surface of an equipotential distribution giving potential F, we can, by inverting the conductor so electrified with
for centre
any point
and
K for radius of
inversion, find the density of the distribution over S' required to K at ; namely, if give zero potential in presence of a charge
V
a be the density at
any point
P
on the original conductor, the
>
INVERSION
124.]
PARTICULAR CASES.
density at the corresponding point J? of the distribution giving zero potential
^
is
where
/= OP'.
A
conducting sphere of radius a, uniformly 123.] Example. coated with electricity of density o-, has constant potential 47r#
at
each
Let us invert it with point of the surface. from the centre, with K for radius
respect to a point 0, distant
f
The sphere becomes another sphere
of inversion. K2
-^
a2
f
of radius
K2
a
>
if C>
t> e
external, or
a^
-^-
a if
be internal.
And
/
according to the general result above proved, the distribution on the new sphere will be such as together with a charge K V> or will give zero potential at each point of the liTKaa, at inverted sphere. But if dA be an elementary area of the original sphere distant r from 0, ad A is the charge upon it in the original system.
the
new
The charge upon the corresponding
adA
sphere will be
9
dA
and
area in
becomes -^dA. There-
fore the density at the corresponding point of the new sphere is r3 K3 adA or -^- o-dA, that is, it varies inversely as the cube of K*
7*
the distance from 0. 2 2 a2 being without the original sphere, let Again, then the sphere does not change its position. Let the charge /c
at 0, or
4
TT
Kaa
=
e,
,
or o-
Then the density
=f
=
-
.-
.
47TKa
for zero potential is
7T 6
^2 __
ft
'*
or
we have
already found by different methods. of the original sphere there be 124.] Again, if at the centre 477& 2
uniform distribution of density the surface or in external space VOL. i.
a-,
the potential at any point on
is zero.
K
130 If
PARTICULAR CASES.
INVERSION we
[125.
invert the system with respect to an external point
2 a 2 for radius of inversion, distant/" from the centre, with \/f the sphere is unaltered in position, but the original centre C
47r# 2 o- was placed becomes a point
upon which the charge a?
distant 4tna?(T
from the centre in the
becomes
line
C",
OC, and the charge
K
-.
47r
at 6
,
and the distribution on the inverted sphere whose density
is
^3 -7-3
a gives,, in conjunction with that charge at
(?,
zero potential
at each point on the inverted sphere and in external space, without there being any charge at 0. That which was a centrobaric shell with centre of gravity ' shell with centre of gravity C
C
has become a centrobaric
.
125.] Again, we can sometimes position to that of Art. 122. If,
make
use of the converse pro-
namely, we have given any system in which
all
the con-
ductors are at zero potential under certain electrifications, and if part of the given electrification consist of an electrified point at which a given charge is placed, we can, by inverting the
system with
new
for centre* obtain a
electrified
system in
which the conductors have unit potential. K be the charge at 0. For let Let us take for centre of inversion a point distant x from 0, and K for radius of inversion. Then all the conductors when inverted remain at zero potential,
and the charge K
point distant
K
becomes a charge
K at
-
2
- at a
os
2
from the centre of inversion.
x
Now
let
x be
indefinitely diminished.
If
all
the conductors,
when
inverted, are of finite magnitude, K2
K2
the infinite charge diminished, that
at distance
X is,
when
will,
when x is
iC
is
indefinitely
taken for centre of inversion,
at each point on the conductors. The reelectrification of the inverted system will therefore
have potential
maining have potential
1
-f 1
throughout the conductors.
INVERSION
126.]
As an example
126.]
XEX', YEY'
PARTICULAR CASES.
of this process, let us take two infinite
at
right angles to each other, and four C19 2 , C2 as in the points 19
planes
131
9
,
forming a rectangle whose
figure,
*
diameters intersect in E.
Ol and K,
and
+ K,
each
2
having charges, each
X*-
Cl9 C2 having
charges, the potential is zero at
each point on either plane. Let y be the distance irrespective of sign of any one of pig. I9> the four points from the plane XEX', x the distance of any one of them from the plane YEY'.
we
If
invert the system with
Ox
for centre
and
K for radius,
the two infinite planes become two orthogonally intersecting The common section of the planes becomes the circle spheres. of intersection of the spheres and passes through Olt The
plane
XEX' becomes
corresponding to
a sphere whose centre
C and whose
radius
is
a2
=
/, the point
is
K
2 -
%/
YEY C2 and whose radius is a = Similarly the plane
becomes a sphere whose centre
'
K
is
2
The portion XEY' of the two infinite planes becomes on inversion the figure formed of the two outer segments of the spheres. Similarly X'EY becomes the lens formed of the two inner segments, and XEY, or X'EY', becomes a meniscus formed of the outer segment of one and the inner segment of the other sphere.
(See Fig. 19.)
The charge
K2
at <7/
/
a2 and the charge at ,
that
is,
K
or a^
is
2
is
,
and the charge at C{
is
2
or
132
We
PARTICULAR CASES.
INVERSION
[127.
obtain therefore the system already treated in Art. 112. Further, if before inversion we substitute
x
for the charges at C^
lent
and
on
distributions
their equiva2 the plane XEX',
C2 its equivalent distributions on densities on XEX' and YEY' these YEY',
and
for
will in the inverted tential
.Fig. 20.
system give unit poon XEY' and are the same which we ',
found by a different method in Art. 112. instead of two infinite planes, let there be 2n Again, 127.] infinite planes, having a common section E and making with each other the angle
n
Let there be n negative points : ... O n each having charge K, and n positive points C ... Cn each having charge -+-K, all at the same distance from E and placed alternately, so that negative point is the image of the next positive point Then all the planes are at zero in the plane between them.
each
potential.
Let YEY' and SES' be two adjacent planes. Let the n points on the left of YEY' be replaced by the corresponding distributions on YEY, and the points on the right of YEY', that
n\
Fig. 21.
Fig. 22.
the points on that side except be replaced by their 1 corresponding distribution on SES'. Then the portions STEY
is, all
,
are at zero potential.
When we
invert the system with respect to
Olt KEY' becomes
INVERSION
I2Q.]
PARTICULAR CASES.
133
the figure formed by the outer segments of two spheres intersecting at the angle
The density
n
at P',
any point on the outer segment of the
sphere corresponding to YEY', or
is
OP
is
C P3
3
that -^-30-, >
is,
the density at P, the corresponding point
distribution on
^K f to P
o-,
,
where of the
YEY', which can be determined without much
difficulty.
128.] Returning to the conductor XET'O of Art. 126, with its surface distribution above determined in Art. 112, let us invert
the system, taking for centre of inversion a point on the internal segment of the sphere Cr
The sphere C2 becomes then another an
infinite plane cutting
sphere,
the inverted sphere
and the sphere
C2
orthogonally, a diametral plane, and the external segment OXE becomes the portion of that infinite plane which lies within the C-L
that
is,
new sphere C2 So that the figure XEY'O becomes on inversion the closed surface formed by a hemisphere and its diametral .
P be a
point on the outer segment of the sphere C19 P' the point on the diametral plane which corresponds to P. And o- being the density above found for P, namely plane.
Let
7 the density at P required to give zero potential under the within the hemisphere is influence of a charge unity at a point 2 (T.
OP* 3 129.]
The construction
for finding
o-
in
terms of known
will be as follows.
quantities on the hemisphere Let C be the centre of the hemisphere, a its radius,
point where the unit charge
system with respect to
0,
we
is
placed.
the
Then by inverting the
shall reconstruct the original figure
which by its inversion of the hemisphere and plane. gave rise to the existing system referred to the existing sphere. Let 2 be the image of of
two orthogonally intersecting
spheres,
PARTICULAR CASES.
INVERSION
134
Then the point corresponding
to
C2
[130.
becomes on inversion the
Inasmuch as the centre of one of the two original spheres. absolute value of K affects only the scale,
^ ^ ---
v^
and not the proportions, of the inverted
we may take OC2 = K. Then C2 becomes the actual centre of
figure,
the
new
sphere.
Its radius
is
PC,*
The
radius of the other orthogonally intersecting sphere, that namely into which the
infinite plane is converted, is
Let P' be a point on the diametral plane, and on inversion become P on the sphere. Then
let
OP* Then
Thus
C,P all
2
=
^
C,P'\
the quantities in the expression for
namely
4'7ra 1
are
known
in terms of given dimensions in the hemisphere,
therefore the density at
P is
and
known.
In like manner we might find the density of the same distribution at a point on the hemispherical surface. 130.] By inversion of the system of sphere and infinite plane, or two concentric spheres, Arts. 114, 115, with the point at which the charge is placed for centre of inversion, we obtain two spheres
external to each other at unit potential, and the density at any point on either sphere required to produce this result can be calculated approximately. well's Electricity,
131.]
We
Chap.
The
shall here give only
taken from Sir
W.
subject is fully treated in
Max-
XL one more example of the method
Thomson's papers.
INVERSION
131.]
It is proved
in
between two
shell
PAKTICULAR CASES. on attraction that an
treatises
similar,
if its
equation be **
+
a2
ellipsoidal
and similarly situated points on its surface, and
concentric,
ellipsoids has constant potential at all
that
135
b
*
+
2
-
"?
the external equipotential surfaces are the confocal ellipsoids
whose equation
is
shell when the ellipsoids nearly coinproportional to p, the perpendicular from the centre on the tangent plane at the point considered. It follows that the density of electricity on the surface of a conducting ellipsoid
The thickness of such a
cide
is
which gives constant potential at all points on that surface, in the absence of any other electrification, is proportional to p. If the axes a and b of the ellipsoid are equal, and if c be diminished without limit, the ellipsoid becomes ultimately a flat
And
therefore the density of the equipotential distribution of electricity at any point on the surface of such a circular disc.
disc is proportional to the limiting value of p for that point.
Now
generally a?
1
7 7 ==
"
y h
6
+
*
and the limiting value of p
is
tf
2
z* "
4
h
1
7 zz
r therefore proportional to 1
Let
C be
the centre of the disc, (7
Then
the density at
P
is A.
8
-/a
where X
is
a constant.
P
any point on
it.
Let
136
INVERSION
To determine
A,
we note that the potential r \dr = ^ = 7T2 A, 2lT Jo
and
if
PARTICULAR CASES.
W-r
the potential be unity X
at the centre
I
32
-
is
2
=
.
bution which gives unit potential
[
The density of the
distri-
therefore
is
1
and the whole charge on the disc
is
in this case
2a
Therefore -
-
is
the capacity of the disc.
(a)
the disc so charged at any point 132.] The potential of for which its axis of figure, ^, is
M
in
CM =
2irrdr
~~
__ "~
2
an
*
_x a
CM
/3 be the angle A, where A is a point in the circumference of the
if
M
disc.
Again, the equipotential surfaces to the disc are the confocal ellip-
C
soids
whose equation
is
Fig. 24.
and since the potential of the disc at the point in the axis of z distant h from the is origin k
TT
it
follows that the potential of the disc at the point #, y, z a 2, 1 -
tan-
ir
where h
is
the positive root of
,
h
is
INVERSION
I33-]
To find the potential at distant r from the centre
P
potential at
is
PARTICULAR CASES. any point
P
we make
z
9
-tan- 1
137
in the plane of the disc
=
0,
and therefore the
--^,. vV a
(c)
2
JT
in 133.] Let us now invert the disc with respect to a point the axis with K for radius of inversion. The infinite plane now becomes a sphere passing through 0, and the disc a spherical
bowl, whose rim is a circle at right angles to the axis. The colatitude of that rim measured from the point where the axis cuts the bowl is the vertical angle of the cone at 0. Let it be a.
method of inversion, that which would be assumed by the bowl as a conductor in
The density on the bowl, according
is
to the
And we K at 0. presence of a charge in can find the potential at any point
M
the axis due to a spherical bowl under influence of a charge at the extremity of the diameter thus
For
simplicity
let
:
K,
the
radius
of
=
to OA, the distance from inversion, the rim of the bowl. Then the rim of
the bowl coincides with the circumference
of the disc which the bowl was before
Find M', the point in the uninverted system corresponding to If, that inversion.
OM- OM' The
potential at
M'
= OA
is,
let
2 .
2
due to the disc was
-
-8
,
where
the disc at M'. angle of the cone subtended by potential at
Now ft
=
M
is
the triangles
0AM,
if
OM<
OA
2/8 is the
And
the
23
AOM and
M'OA
the diameter, or
are similar. it
0AM
if
Therefore
OM > the
M due to the disc under the influence
diameter, and the potential at is of a charge OA at QA
2
/Q
_r,
(d)
138
PARTICULAR CASES.
INVERSION
[134.
Again, the potential of the bowl so influenced at any point on the remaining segment of the sphere whose colatitude is 9
OA F,
7
where
point in
its
is
7= IT
2
.
134.]
And
-tan
NA
l
2~ circular disc,
is
the potential of the uninverted disc at the
plane which on inversion becomes P.
We
P
tan
2
have thus dealt with the case of a conducting which may be regarded as part of an infinite plane
of which the infinite external part is non-conducting. will now take the converse problem, namely, that of a
We
circular non-conducting disc of radius a, the infinite external
Let it be portion of the plane of the disc being a conductor. to find the at a on the required density point conducting plane when that plane is at zero potential under the influence of a charge at a point in the non-conducting disc. In order to solve problem we will invert the conducting disc when at unit in itself potential as before determined, with respect to a point this
and with Va 2 f 2 for radius of inThe disc then becomes the infinite external plane, and version. the infinite plane becomes a disc, the boundary between the two distant /from
Cihe
centre,
after inversion a,
being a
circle of radius
and whose centre Cf is distant f on the opposite side to C. Let P be a point in the plane
from
outside
of the
new
the
point within the original conducting disc which on inversion becomes P.
Fig. 26.
at P, when the infinite plane influence of a charge the K at 0, is under potential
Then the density
K*
Let
P
f
circle,
C'P
!
- r,
LPG'O
=
0.
is
at zero
INVERSION
I3S-]
Then
a
P
and the density at
PARTICULAR CASES.
^ is
a 2 -/ 2
~
The density 1
at
P due
>/^7*
t '
from
1 '
2
C',
/.../-f df
is
d0
(*"
_f.df
yV-a Vo
V*
1
r
to a uniform ring of electricity of density
in the plane of the disc distant 2
139
r2
2
+/ -2r/cos0 >/a2 -/ 2
_ 2
/rff
The aggregate of the electricity
m in
distribution over the plane due to any the disc distant /"from the centre is
m
T 00
.
2-nrdr :
135.]
If the whole non-conducting disc
electricity of density
plane
when
-
m.
(h)
be covered with
P
the density at in the surrounding at zero potential under that influence is 1,
2
vV-a
TT
or
;
a
2
C
w
(./ r2_ a 2
2
Now let the entire plane, including the non-conducting disc, be covered with a uniform stratum of density + 1 There will then be zero density on the non-conducting disc, which may .
and the conducting have constant potential, and the density at any point distant r from the centre of the disc is
therefore be regarded as a circular aperture,
plane will
upon
it
This
differs
from the constant density + 2 f a a
- tan
it
we have an
1
by j
~^^r
conducting plane with a circular must be placed upon the plane in the that aperture, total^charge If therefore
infinite
140
PARTICULAK CASES.
INVERSION
[
I
3^-
order to bring it to the same potential as a complete plane would have when coated with density -f 1, is / C> ardr a 4 ^tan-1 2 2 ^a2 4 (t) /-,z / j a yr ^/a a2 Vr a
== =
^~
,
same quantity which would be removed from the infinite making the aperture. will We now proceed to Sir W. Thomson's problem, to 136.]
or the
plane so coated in the act of
find the density of electricity
sphere cut off
by a
circle,
on a spherical bowl, or portion of a when at unit potential under the influence of its
own charge
In order most easily
alone.
to effect this, let us recur to the non-conducting disc
and
infinite external con-
ducting plane, and instead of the density of electricity
on the let
P be
the density at
Q
f
Ji
being uniform,
a point in the axis
is
from the centre, and from the centre. Then
of the disc distant
the disc distant
where
>
3
disc
P
any point within
1
at any point in the conducting plane, when at zero under the influence of this distribution, is potential
The density
W
2 TI
Let
/ = h cot
>
Vr a
1/1
2 ,
= K cot
r
2
2
_-" "2*
-7T
=
(3
Q
cot
1
_^ 2~2
77
(
2~2'
are the angles subtended at
Ti
.
by/,
#,
and
r respectively.
integral then becomes cot
cosec
2
9
-
A/
~
c t
cot 2
2
^)
cosec
cot
3
-
2
The
PARTICULAR CASES.
INVERSION
13 7.]
This
141
be put in the form
may
2
T
sin a A/COS
sm
cos a
/3
cos a
cos
*""
2
v
Let
Then when a =?r,
=
a;
= x, =x sin a da =
cos
(3
cos a
cos
/3
cos a
\/cos
/3
+ 1,
cos 6
J/3
,
and when a = +1 f
sm A/COS ^ - cos ^ _ [da.
Then
z
g
cos a
Jo
^
(3,
x
= 0. ^
cos0
cosft
+ x* _
.
^
cos ^
Hence the density 2
?'
^3 J
is
// cos/3+1
(
.
sm 83
7S
cos
-JA
tan" 1
A //
cos
+1
)
>
(j)
137.] Let us now again invert the system, taking Q for centre of inversion, and k for radius. The infinite plane becomes a
The infinite conducting sphere whose diameter is Q(7, or h. a outside of the disc becomes plane spherical bowl, cut off by a circle at right
the pole
Q
is
angles to QC, and whose colatitude measured from
(See Fig. 27.)
/3.
The density on the remaining segment of the before inversion was
sphere,
which
7,2
*+*) on the
disc, is constant,
bowl remains zero ,
Sin
where p
is
:
and equal
to
-7
and the density upon
The it
potential of the
at colatitude 6
is
the density at the corresponding point of the
|
on a spherical bowl, Hence the density at colatitude in zero the makes which presence of a uniform charge potential
plane.
of density
-7
on the remaining portion of the sphere, cos/3+1
_!
cos/3+1
is
142
PARTICULAR CASES.
INVERSION
[138.
Let us now place round the sphere a concentric and
138.]
nearly equal sphere with a uniform density of electricity +-7
upon it. When the two spheres ultimately coincide, the potential The at any point on the bowl, which was zero, is now 2ir. density upon
ment
it is
of the sphere
Therefore
a-
+ jw
,
is
and the density on the remaining segT h,
+
-7
>
or zero.
fi
of the distribu-
tion on a spherical bowl which gives potential 2
of any other electrification, and therefore
77
in the absence
-- --r
is
\-
the
27T
density which gives unit potential under the like circumstances. 139.] The capacity of the bowl is therefore
^W
re 2
~4jo
/ COS/3+T
( /
7rA(
cos
Y'
cos
k^
/
/
\/
cos/3+1
T
-
+
=
h 27T
(/3
)
cos /3)
cos r>p
sin0d50
\
+ sin/3).
The capacity of the bowl formed by the other segment of the sphere
is
h
Hence we see that if a sphere be divided by a plane into any two parts, the sum of the capacities of the two parts exceeds the capacity of the sphere by the capacity of a circular disc coinciding with the intercepted plane.
If the bowl be hemispherical the capacity
is
|
a being
>
the radius, or the arithmetic mean between the capacities of the sphere and disc of the same radius. 140.] Recurring to Art. 138, let us next place at the centre of the sphere of which the bowl forms part, a charge will reduce the potential of the
bowl to
zero.
.
2
This
PARTICULAE CASES.
INVERSION
141.]
143
We
may now again invert the system so formed, taking for centre of inversion any point whether in the spherical surface from the centre. In that case, if or not, distant be not on
/
the surface, the sphere becomes a new sphere, and the bowl becomes a new bowl. In the particular case of being on the original spherical surface the new sphere the new bowl a circular disc upon it.
The the
is
an
infinite plane,
centre of the original sphere becomes (7, the image of
new
sphere; and the charge
f K
in
at the centre becomes a
J)
at the image.
-
charge
and
2i
If r be the distance from of a point P on the new bowl, othe density of the equipotential distribution, as found above, at K3
the corresponding point of the original bowl, then density at
P
of the distribution on the /
zero potential in presence of
5-
a-
new bowl which
is
the
gives
7
- at
(7.
K
We
can therefore give the following rule for finding the density at any point on a spherical bowl under the influence of
an
not on the surface of the sphere.
electrified point
find (7, the
image of
in the sphere.
First,
Secondly, suppose the
system inverted with respect to (7, and a new bowl so formed, and rim of the supposed new bowl, and /3 be the colatitude of the
let
be the colatitude of P', the point on the supposed bowl corresponding to P on the given bowl. Then we know cr, the
let
density at P' of the equipotential distribution on the supposed bowl, as a function of (5 and 6. And if r be the distance of
P
from
(7,
141.]
the density at
On
P is proportional to
the effect of malting a small hole
in a spherical
or
infinite plane conductor.
The above results enable us to estimate some of the effects of making a small circular aperture in a conductor otherwise spherical.
144
PARTICULAR CASES.
INVERSION
[141.
For instance, let /3 = TT y, where y is a very small angle. spherical bowl becomes then a spherical conductor of radius
The >
with a small aperture whose radius
-
is
y.
Its capacity is
h
.
n
.
-(/3 + sm/3),
,
h y y --.-.ZZ; sin
Ti
.
.
that
is
The capacity of the complete sphere
-
is
We
.
see then that
2t
making an
the effect of
centre the small angle
aperture, whose radius subtends at the to diminish the capacity
y, is
by
h
sin
y
'
~^~
2
If
A
y
be the area of the aperture, we
3 higher powers than y
A% y sin y '= x -'
h
may
write, neglecting
,
where A
2
=
'
Again, the conductor being charged to unit potential, the density at a point whose colatitude is (less than /3) is COS /3
_t COS0-COS/3
Now
-
is
=-
271%
the uniform density which would give unit
potential on the complete sphere. 5 J_ 2 A(
-7r
/
/
V
cos/3+1
"
The term
/
^ "V
cos^-cos^
cos/3+1 cos
)
^- cos /3)
expresses the density due to the existence of the aperture. The total quantity of the distribution due to the aperture on a
and
ring between the parallels of
g sm0d0 "
2
Now
unless
(
/
IV
-f
_a
cos/3+1 cos
d&
6- cos ft
be very nearly equal to
TT,
cos
/
V
cos/3+1 cos
is
/3
cos
/3+1
J
cos0-cos/3f
not only does
INVERSION
141.] itself
become very
PARTICULAR CASES.
small, but also
equality with
/
tan" 1
Hence
if O l
tends to vanish in a ratio of
cos
A /
V
it
^+
cos
be a value of
145
1
cos/3
which
is,
and
2
a value which
is
not, nearly equal to (3, it is easily seen that the quantity of the distribution due to the aperture on the ring between and 1
1
very small compared with that on the ring between /3 and distribution due to the aperture has therefore the same
is
2
The
.
were For instance
effect as if it
C* J.
and
is
W
/
C
on the aperture.
all collected
cos/3
+1
/
V
- tan
COS0-COB/3
independent of
cos
+1
coBtf-eoB
****
0.
The system is therefore equivalent to a complete sphere charged to unit potential, that is, having a uniform density on 2
7f
n
its surface,
on the aperture.
together with the additional charge
h y
siny
2
TT
This quantity h y siny
-2
~ x A% .
or
'
*
i?'
be called the abnormal charge, since it constitutes the difference between the capacities of the perfect and the im-
shall
perfect sphere.
Let
P
be any external point distant r from the centre of the / from the centre of the aperture. Then the po-
sphere, and tential at
P
of the charged sphere
h
or
is
h
y
is
siny
1
the potential of the perfect sphere, together with that of
the abnormal charge Ti
~2 placed on the aperture. VOL. i.
y
siny 7T~~
^
INVERSION
146
PARTICULAK CASES.
[142.
142.] Let us now invert the charged conductor, taking for centre of inversion a point 0.
On
the imperfect sphere when charged to unit potential, such not being very near the aperture. The sphere becomes then point an infinite plane with a circular aperture, at zero potential under
And the potential at any the influence of unit charge at 0. side of the on the plane to 0, instead of being point opposite that is due a small to zero, positive charge upon the aperture.
P
These results, which are accurately true in the limit as the aperture vanishes, are approximately true for a sphere whenever the aperture subtends a very small angle at the influencing point.
To
find the effect of a large aperture it
would be necessary to
any point due to a spherical bowl charged to unit potential, when /3 is not nearly equal to TT. This might be done approximately by the method of Art. 61, or otherwise. find the potential at
CHAPTEE
VIII.
CONJUGATE FUNCTIONS AND ELECTRICAL SYSTEMS IN
TWO DIMENSIONS.
ARTICLE 143.] Let there be an infinite cylinder whose axis is of 2, and whose section is the element of
parallel to the axis
cutting the plane of xy in the point #, y. charged with electricity of uniform density so that p is independent of z, but is a function of as and y. In
area
das dy,
Let p,
like
this cylinder be
manner we may conceive an
axis
parallel to that of
is
electricity,
so that
z,
infinite cylindrical surface
having
a-
whose
for surface density of
constant along any infinite line parallel to the axis, is independent of z, and a function of x and y only.
If an electrical system be
made up
of such cylinders, the potential, V, is evidently independent of z, and a function of x and y only. Poisson's equation then becomes at a point in the plane
of
x,
y
and at a curve in the plane of xy coated with
electricity
we have
as usual
dV
dV
and dv the element of the normal to the }
We may treat
surface, is in the plane
such a system as in two dimensions only. In in the plane of xy the present chapter the charge at a point will be understood to mean a uniform distribution of electricity
along an
P
infinite line or cylinder of small section
And axis of z. parallel to the the in line a on plane of point
through
P
in like manner, the density at a
xy means the density per unit area of a cylindrical surface parallel to the axis of z drawn in question. through an element of the line at the point L 2
CONJUGATE FUNCTIONS.
148 In
like
f(x, y)
=
manner, a conductor the equation to whose surface means an infinite cylindrical conductor whose axis
parallel to the axis of
xy
is
[144.
/(#, y)
=
2,
is
is
and whose section with the plane of
0.
P and
in the plane of #, y, the potential at Q Q, be points due to a uniform distribution of electricity of density p along an infinite line parallel to z drawn through P } is evidently of the form
If
p
and does not vanish
But
\C-2logPQ}, be removed to an infinite distance.
Q
if
there be another parallel line through R, a point in the and Q, on which there is a distribution same plane of xy with if
P
of density
and vanishes
/>,
if
the potential at
Q
Q
is
be removed to an infinite distance.
It will be
understood in this chapter that the potential does so vanish, and therefore that the algebraic sum of all the electricity in the
system of which we treat is zero. 144.] Let us suppose then that in such a system there are certain conductors
whose equations are
/i (*, y)
=
0,
f2
(x, y)
=
0,
&c.,
and given charges are placed upon them and also certain fixed charges on given points or lines of the system. Let us further suppose that we have by any method obtained the solution of ;
this
electrical
function, V, of
and
ductors,
that problem x and y, which
satisfies
:
is,
is
we have found constant within
the all
single
the con-
Poisson's equations
at every point in free space,
and
dV -- dV t
5 dv
H ~T~f
dv
=
47TO-
and by consequence we have determined the density at any point on any of the con-
at every curve charged with electricity;
ductors.
The
solution so found contains implicitly the solution of a
CONJUGATE FUNCTIONS.
I45-] class of
149
all those problems namely that can be formed from the given one by substituting f (oc,y) and *}(%, y] for x and y, and 77, being functions of x and (x,y) and 77 (x,y}> or shortly ;
a certain property.
y having
145.] For
f and
let
be so chosen that
rj
d ~T~
~ drj ~T~
dy
__. dx'
ax
and
rj
'
ay
are then defined to be
CONJUGATE TO x AND
y.
It follows immediately that
__
dx
(Za?
By
dy dy
the ordinary formula for change of independent variables that
we know
_
dv
dx
__
dy_
d~ddrj
-ir 7
dy
drj
and =, dx
= (-^--r1 /
?
dy
_
d dy dx dr]
dr]
-^ dy dx
^
dx
d
d
dy dx
dy dx
dfdrj
Also
~~
dgdrj'
dx dy
__
dx
=-
in this case,
dy
dx
and
Again,
if
F
ddr\
}j?dxdy.
be any given function of x and y,
we have by
ordinary differentiations,
d*V_d*V
dj[~
dtf~~~d'l( dx>
d*V
d*V.d*
'
^dj[d^ 3W *n**V_ ^!f,^^, dx* dx* (
ddi}dxdx* d*V ddri
dif'^dx'
o*V
fy.dV +
dr *' dy {
l
^df
)
dr)
d^ * dVd^n
df df
d^
df
CONJUGATE FUNCTIONS.
150 Therefore,
[146.
remembering that
&
and that
^dx'
_o
cZf ofy
cZf
dx dx
dy dy
+ <|fy ^dy'
= $)' +
^dx'
d2 V
2 cZ
?)'
= S,
dy'
F"
now take two planes, one in which the position as P, is determined by the values of the rectangular any point, coordinates os and ^, and the other in which the position of a 146.] Let us
of
7
point P is determined by the values of the rectangular conductors of and y' ; and when x' and y are connected with x and y
by the equations where k
is
make
-7
to
such a power of the unit of length as
and
j
linear, let the point
called the corresponding point to
Then
P
may be
required
P' in the second plane be
in the first plane.
to every curve in the first plane of the
=
form
/ (#, y) =
in the second there will be a corresponding curve /(, 17) plane, and if the former curve be closed, so also will be the
and if any point out the closed curve
latter,
f
P in (x,
the former plane be within or with7 in 0, the corresponding point y)
=
P
the latter plane will also be within or without the closed curve
/(f,~o. It follows from the equation
d_dri_ dx' dx'
=
+ d_
drj
__ ~~
dij dy
=
that the curves f b in the second plane intersect each. 0, 17 other at right angles these curves may be regarded as a species of curvilinear coordinates, the case in which they are linear being ;
P
7
that in which the point is always so taken in the second plane that its coordinates referred to axes inclined to those of so', y' are 1
The quantity
cases,
fc will be omitted until we come to the application to special none of the general results obtained in the next few Articles being affected
by regarding
Jc
as unity.
CONJUGATE FUNCTIONS.
151
respectively equal to the coordinates of the corresponding point P, viz. x and y in the first plane. It follows also that
any two
F(x,
curves, as
y) = 0,
f(x, y)
= 0,
in the original plane intersect each other at the as the corresponding curves
F(, in the
new
= 0,
rj)
/(,
TJ)
same angle
=
plane.
For the tangent of the angle which the tangent to F(x, y) = at any point P makes with the axis of x is
d
and
.
this is equal to
from the relations between
and
as
y and
f,
rj.
d_
-$
But
is
=
to
P
-r> in the curve
F(^
in the second plane,
between the curve through
P
7 ,
F(,
77)
at the point
rj)
i.e.
=
since the curves
77
P corresponding
the tangent of the angle
it
is
at
P' and the curve
=
const.,,
77
f = const.,
=
const.
intersect
everywhere at right angles. Also, if dA be any elementary area dxdy in the original plane, we have dA = dxdy = ddrj. from But the last article
and therefore
if
corresponding to
dA' be the elementary area in the second plane dA in the first plane, we have
dA
ELECTRICAL SYSTEMS IN
152
And
TWO DIMENSIONS.
manner the length of any elementary
in like
[147.
line in the
second plane corresponding to the element dv in the first plane
may
be proved to be
we have any given electrical system 147.] Suppose now that of two dimensions in equilibrium in the original plane of oo, y^ with conductors whose bounding equations are given by closed curves of the form
/ (x, y)
0,
sum
the algebraic
Construct in the
of all the
new
plane of #', y' a being and curves for every linear of 0, system corresponding /(, 77) or superficial charge in the original plane of #,y, place the same linear or superficial charge upon the corresponding lines and zero.
electricity
=
new plane of of, y'\ then the electrical system so formed in the plane of x', y' will be a system of two dimensions in equilibrium with conductors bounded by the corresponding areas in the
And closed curves to the original curves in the plane of os,y. in the old plane of as, y will be the potential V at any point
P
equal to the potential new plane of #', y'.
V
P
at the corresponding point
For since the total charges on corresponding are the same, but the areas themselves are in the
f
in the
superficial areas
ratio of
\j?
to
1
,
follows that if p be the surface density at any point in the old plane, and p' that at the corresponding point in the new plane, it
then p
f
=
2 /u
densities
/o,
and similarly
if
a-
and
let V be the same function Then from the equations
Again,
andy.
it
follows
that since
V
is
in the plane of at, y, closed curves /(, 17) sponding
=
d
z
V
dx*
d
z
of
and
r?
constant over the
f (#, y} =
if
be corresponding linear
= per.
V
is
that
closed
in the plane of gft
dy*
P in the
of x
curves
constant over the corre-
V
p be the electrical density at
V is
plane of
#,
if.
ELECTRICAL SYSTEMS IN TWO DIMENSIONS.
148.]
where
= -4
+
Therefore
153
= -4./,
MV
the electrical density at P' in the plane of af, y'. dv and dv be elements of the normal to a curve, as
p' is
if
Again,
^
=
in the plane of #,y, and and dn' be the elements of the normal at the corresponding point to the corresponding curve /(f, rj) in the plane of #',./, we have, since V =. and
/(#, y)
V
=
c
^ dv by what
is
~~
^~ dn
if
o-
be the linear density at
dV dV' -dn
therefore
o-'
/
(
/x'
proved above,
dT_-^ dV dn dv' But
1
-j-?
dT_ dri
P
~ M dV_
dS
on the curve/(#,y)
17)
=
4 77 wr
=
cZn
P
linear density at the point on the curve in the plane of af, /. see then that the function F'9 formed as above
=
148.]
0,
dV
being the ,
=
We
stated, is constant over each of the conductors f(^rf)
new plane, and
of that plane. point of the #, y system, the function tial at the corresponding point P' in the
V=
V will be
a?',
equilibrium, and we have only to eliminate
V in terms of
=
in the
the characteristic equations at each point If therefore the function V be the potential at any satisfies
the poten-
y' system
and
17
when
in
and express
y' to obtain a complete solution of the problem of an electrical system of two dimensions bounded by conductors
of the
form/ (
a?',
77)
=
0.
course the procedure might be reversed, and if we had x a function of # , y with conductors bounded by curves found
Of
V
/
(
17)
=
and take
0, F",
to obtain the solution for conductors
f(*,y)
=
-
V
we have only to express the same function of x and
y
in terms of
that
V
is
and
of f and
?;,
17,
bounded by the curves
ELECTRICAL SYSTEMS IN TWO DIMENSIONS.
154
149.] Further,
a and
R
ft,
then also a and
da
da -
M
.
dx
be conjugate to two other functions
rj
are conjugate to
ft
da
da
d
da
drj
dx
d
dx
dr]
dx
da
__ d{3
4.
and therefore
-,
and
if
__
d/3
= dft -f-.
dr]i
dr]
dy
da
,
and similarly
d
P"
_ dr\
dp d dg dy
x and
=
y, for
d^
_
dr\
dp + -^.-^=-^-
dy If therefore
[149.
}
dy
dp f-
dx
we take
a third plane of', y" and determine a and such that a (#", if') 17, f and p (of', y")
=
on it point the points P, P' and
=
,
P" be called corresponding points on the three planes, the solution for a distribution of electricity for conductors bounded by the curves /(*, y)
= 0,
/(
T?)
= 0,
/(a, 0)
= 0,
on any one of these planes respectively, leads at once to the corresponding distribution on the two others, and similarly for
any number of planes and systems of conductors. 150.] Further, as is easily seen, if the problem were to determine, not the potential due to given charges, but the charge on any conductor necessary to produce given potentials, the solutions for the several members of the class would be connected by the same law as in the case we have already considered. That is if obe the required density in the in the
new
151.]
known problem,
jxor
is
the density
one.
We
proceed to illustrate
the
above
process
by an
example.
Let there be a conductor in the form of an infinite cylinder of and radius having its axis parallel to z and meeting the plane of a?, y in the point C. Let a charge e per unit of length be uniformly distributed along an infinite straight circular section
,
t
and parallel to the through an external point axis of the cylinder, then, as proved in Art. 109, the density at any point of the cylinder of the charge necessary to reduce the
line passing
ELECTRICAL SYSTEMS IN
151.]
TWO DIMENSIONS.
155
potential of the cylinder to a constant value in presence of the
charge on the line
is ,
p*
where /is the distance of the axis of the cylinder, p that of the Also the algebraic sum of this point in question, from 0. distribution over the whole circle is
e.
We now
proceed to transform this problem. conditions in polar coordinates log r and 6 with
CO
Expressing the for origin and
C
we have in the original system a conductor whose equation is log r a constant, and a charge e at the log point whose coordinates are for fixed line,
,
= log/
logr
where
any integer. x and y are conjugate to log
therefore K
log
charge
to
log r
where
a,
= log a
x
we
r
and have
shall
K is unit of length.
And
0.
Corresponding
the
infinite
line
corresponding to the
the point
e at
logr
we
Sin,
i is
Now x
=
and
= log/, 0=
have a charge e at each of a /clog/ distant from each other
shall
the axis of
x.
on the
x
line
2iv, series of points in the line
+
2-rrK,
whereof one
is
in
of a distribution of electricity The density will K log a which give constant potential on a-
=
that line in presence of the infinite series of charged points on
the line x
=
pression for
o-
K
log/
found by substituting - for
is
found above, and 1
e
is
in the ex-
therefore
f-a*
remembered that to obtain the actual physical conditions we must understand by the infinite line x = K log a an infinite conducting plane whose equation is x = K log a, and It will be
series of charged points a charge uniformly distributed along an infinite line through the point parallel to the axis of z.
by any one of the
156
ELECTRICAL SYSTEMS IN
TWO
DIMENSIONS.
[152.
152.] The general problem, the solution of which is derivable from the given problem by the substitution of x and y for log r
and
found by choosing for origin, instead of C, the centre of the circle, any point in the plane. Suppose we choose a point 0, is
D
such that
CD =
c,
and
LGDO =
a,
DO =p.
log r and 6 with
circle referred to polar coordinates
and
DC for fixed
tricity
to the
D for origin
line is
log r
We
The equation
= log \c cos B
Va?
c
2
2
sin 0}.
obtain then by the transformation the density of elec-
on the curve x
=
K log
jc
+
cos^
A/ a
2
2
c sin
in presence of a charge on a series of points situated in the line K log p at equal distances 2 TT K apart, of which one is distant K a from the axis of x.
=
x
This includes
the problems the solution of which can be
all
derived from that of the given one by the use of the particular functions x and y as conjugate to log r and 0. But we may obtain others by the use of different functions.
For instance, x 2 y 2 and 2xy are conjugate to x and y, and therefore to log r and 0. If therefore, taking C again for origin, /jj2
we write
2
2
y 2 y
2
a?
O
njii
~-
fYt'JJ
for log
r,
and
|-
for
in the above problem,
we
solution for the density on the hyperbola in loga presence of charges placed on the hyperbola K 2 log/ at the intersections of that hyperbola with the
obtain the
shall a?
__
= =
K
2
hyperbolas xy
=
ir*
2 ,
sey
=
2
TTK*,
&c.
It is of course understood that the hyperbolas represent infinite cylindrical surfaces parallel to z whose intersections with the
plane of xy are the hyperbolas in question. the points represent infinite straight lines.
As
in the former case,
we can
And
in like
manner
generalise this solution
by
taking for origin any point in the plane of the circle. particular case of transformation by conjugate func153.]
A
K
tions is that of inversion in
and
are conjugate to log r
two dimensions. and
0.
Evidently log
2
1
54.]
ELECTRICAL SYSTEMS IN TWO DIMENSIONS.
from the theory of conjugate functions, in equilibrium, and in which V
It follows therefore,
that
if
157
we transform a system
vanishes at an infinite distance, by substituting the coordinates K
2
and
6 for r
and
0,
and placing on corresponding elements
the same charges, the transformed system will be in equilibrium. will now prove the same result by a method analo154.]
We
gous In
to that of Chap. VII. the plane of as, y let us take
inversion, and
P
and
for centre
K for radius of
be any two points in the plane.
P, Q Q' be corresponding points to let
P and
Q,
Then
if
we have
In the present case the potential at P of a charge p at Q means the potential at P of a uniform distribution of linear density p along an infinite line drawn through Q parallel to the The potential is therefore axis of z. v
= p\C-2logPQ\.
If in the inverted system there be the same quantity of matter placed at Q', according to the method of transformation used with conjugate functions, the potential at P' of the charge at Q' is
=v and
therefore, if F,
K 2/o log
OPOQ
V denote the
potentials at
P
and P' of the
whole system,
V=
7-2 ffp log OP'dxdy + 2 ffp log OQdxdy
= 7+2 log dxd + 2 OF/fp y ffp Since we assume the potential we must have in this system
Hence
to vanish at
l
S OQdxdy.
an
V = 7+ 2 Tjp log OQ dxdy,
infinite distance,
ELECTRICAL SYSTEMS IN TWO DIMENSIONS.
158
V
[154.
V by the potential at the origin of the that is, by a constant for all positions of P. original system, 7 is constant over every conductor in the since Therefore is constant over every conductor in the new original system, that
is,
exceeds
V
system. The new system is therefore in equilibrium. If the original system consist of a conductor potential under the influence of an
opposite side of zero, that
S
S
at
zero
point on the to the origin, the potential at the origin is electrified
is
V
at every point of the new conductor. the original system be an infinite cylinder uniformly coated with electricity of density or, and if there be a distribution of density 2 TITO- along the axis, the potential
and therefore For instance,
is zero,
if
and we might by inverting the system with respect to
an external point obtain the Chap. VI, Art, 109.
result obtained synthetically in
CHAPTER
IX.
ON SYSTEMS OF CONDUCTORS.
WE
ARTICLE 155.] proceed to consider further the properties of a system of insulated conductors external to one another, and each charged in any manner. And we will suppose that there is no electrification in the field except the charges
Let placed
on the conductors.
Clf C2 ... Cn be the conductors. First let a charge on C19 the other conductors being uncharged,
el
be
Let the potentials of the several conductors be denoted by
V1 'V2
'
>
T*' r
By the principle of superposition, if el were increased in any Vn would be increased in the same ratio. It ratio, Tlt V% follows that we may express Flt F2 Vn in terms of e1 in the . . .
.
form
V
1
= An e
1,
r
a
.
.
= Aa e
lt
&Q.;
where A n , A 12 &c. are coefficients depending only on the forms and positions of the conductors. In like manner if C2 received a charge e2 all the others being uncharged, we should have ,
,
Y '
i
_
V r 2
A t> "*H e2'
<
A
e2> 8ra *XlUt t>
22
>
the coefficients being again dependent on the forms and position of the conductors.
By the principle of superposition, if at the same time Cl receive a charge elt and C2 a charge 2 the others remaining uncharged, we shall have ,
==
&c.= 'n ==
-
e
"*"
^
&c., -"in ei
"t"
-"
e
160
SYSTEMS OF CONDUCTOKS.
And, generally, if the conductors all receive charges e lt ez the potentials will be expressed by the linear equations
...
e
.
&c.=
A are called the coefficients of potential. there exist algebraic values of V corresponding Evidently 156.] to any assigned values of e lf e2 ... eni though we do not assert The
coefficients
that
it is
practically possible to charge the conductors without
limit.
solving the above linear equations we should obtain a new set expressing the charges in terms of the potentials, namely,
By
&C.= &C, e n =B ln V + B, n V,+ ..+B nn Vn l
" > -,
B
which the coefficients are functions depending only on the forms and positions of the conductors. Since the equations (B) must give possible and determinate in
V
V
values of e for any assigned values of lt F"2 ... n it follows that must exist a set of charges corresponding algebraically to ,
there
any assigned
set of values of the potentials.
The
capacity of a conductor in presence of any other conductors the charge upon it required to raise it to unit potential, when all the other conductors have potential zero. Thus, if F2 ... Vn is
are all zero,
we have from
equation (B),
4-4*?,; and
if Fj
The
=
1, el
= Bu
coefficients j# n
coefficients
,
,
so that
.Z?
of capacity.
22
,
The
Bn
is
the capacity of
&c., with repeated coefficients
B12
,
,.
suffixes, are called
-Z?
21
,
&c., with dis-
tinct suffixes, are called coefficients of induction.
Properties of the Coefficients of Potential.
An
157.]
For
let
F!,
V^
...
Tn
= An
.
be the potentials of the
conductors,
SYSTEMS OF CONDUCTOKS.
158.]
when C has
the charge e and
l
161
the others are uncharged,
all
V the general value of the potential in this case. Then V^ A^e. Let U19 U2 ... Un be the potentials of the conductors, when (?2 has the charge e, and all the others are uncharged, U the Then U = A 2l general value of the potential in this case. ,
e.
l
Green's theorem applied to the infinite space external to
By all
V 2 V and V 2 U
the conductors, in which
we have
are everywhere zero,
#* 7
in
which
cr //
dSl denotes integration over C19 and
other conductors.
M~dS
U
dS
so
on
for the
rr/JV
But
dS
/ /
is
the charge on the conductor
is
T.
C
in the system
whose potential
Hence
and
all
and
all
the other integrals in the
member
first
vanish.
Similarly
dv the other integrals in the second equation therefore becomes
or or
Ae* = A
21
=A
lz
lz
J
Cl
In other words, the potential of
,
member
.
due to unit charge on
presence of any conductors, is equal to the potential of to unit charge on Ct under the same circumstances. 158.]
The
with distinct
coefficients of potential are all positive, suffixes, as
J lr An
,
is
The
vanish.
greater than the
2
2 ,
in
due
and no one
coefficient
with
or A rr . either suffix repeated, as in as Art. 53, the potential can never be a maxiFor, proved
mum
minimum
at any point unoccupied by free electricity. be If therefore there any positive potentials, the highest positive or
VOL.
i.
M
PROPERTIES OF THE COEFFICIENTS
162
[159.
and if there be any must be on some conductor must be on some conductor. If negative, the lowest negative
potential
;
the potentials be all greater or all less than zero, then zero, the potential at an infinite distance, is the least, or the greatest, potential as the case may be. If any conductor has zero charge, the density of the disupon it must be positive on some parts of its surface, on other parts. Where the density is positive the lines negative
tribution
of force proceed from the surface, and there must be some neighbouring part of space in which the potential is less than Where the density is negative, there that of the conductor.
must be some neighbouring part where
it is
greater than that of
the conductor. Therefore, neither the greatest nor the least potential in the can be the potential of a conductor with zero charge,
field
neither can
it
be in free space.
Such greatest or least value must be that of a conductor having an actual charge, and the density on such conductor must be of the same sign throughout its surface, and must be positive for the highest positive potential, negative for the lowest negative potential. Therefore, if all the conductors
be uncharged, and C-^ have positive charge el9 Flt i.e. the potential of C\, must be the greatest potential in the field,
C2 ...C'n
and zero must be the distance.
We have Vi
71
least,
namely the potential
then from equations
= An e
F and 72
= 4M
i>
Vn
at
an
infinite
A i.
&c.;
between 7l and zero, and are therefore all positive. Hence also. A u is greater than A 12 ... or A ln and each of these latter is positive. in which
is positive,
.
.
.
lie
,
159.]
For
let
Properties of the Coefficients of Capacity
V denote
and Induction,
the general value of the potential, when C^ is all the other conductors are
charged and has potential K, and uninsulated.
when C2
is
And
let
U denote
the general value of the potential all the other
charged and has potential K, and
OF CAPACITY AND INDUCTION.
162.]
163
The equation of Art. 157 then
conductors are uninsulated.
becomes
or
=
.
21
12
In to
,
Bn = B other words, the charge on C if uninsulated when C2 is raised unit potential is equal to the charge on C2 if uninsulated or
lt
.
when
CL is raised to unit potential, all the other conductors in either case uninsulated.
160.]
Each of the
coefficients of capacity is positive.
For as we have seen
possible so to charge the conductors Then l must be either the
it is
make 72 ...7n each
as to
being
V
zero.
greatest or least potential in the
field,
the greatest
viz.
if e l
be positive, the least if e-^ be negative. Therefore, we have in this case
% = J&u F,p and since
and F1 are of the same
e^
161.] Each If
F2 ...7n
sign,
of the coefficients of induction are all zero
C2
the conductors
.
. .
and
Cn must
el
n
_Z?
is
is positive.
negative.
not zero, the density on each of
be of the same sign throughout its for let e 1 be positive, then if
surface, viz. opposite to that of e1
;
the density at any point on any other conductor were positive, there would be a less potential than zero, that is, less than that of
any of the conductors, at some point
But the charge on C2
is
*2
and since V^
in free space.
in this case
positive and
=A
2
^>
negative, I> 12 is negative. 162.] The sum of the coefficient of capacity and all the coefficients of induction relating to the same conductor is positive.
For
let
is
at the
same
positive, for if it
were
the conductors be so charged as to be
if
all
Then
potential V.
Now
e%
V
be positive,
e1
must be
than negative, there would be a greater potential in free space. Therefore
Bn + B^ +
. . .
-f
Bnl
is
M
positive.
2
V somewhere
PROPERTIES OF CAPACITY AND INDUCTION.
164
[163.
(7 be connected 163.] If two conductors 15 2 originally separate, wire so as to form one new thin a conductor, very together by ,
the capacity of the new conductor is less than the sum of the For let e 1 be the capacities of the two original conductors.
charge on C^ when potential, that
on
C2
is,
it
el is
is
at unit,
and
the capacity of
C
all
the others at zero
Let
.
/2
be the charge
in this case.
Similarly,
when C2
potential, let e\
and
e2
is
at
unit,
and
all
be the charges on
C?t
the others at zero
and C2
respectively.
e\ and e'2 are negative, ^ and e2 positive. If C^ have the charge e l + e\ and G the charge e2 -f e'2 and every other conductor have the sum of its charges in the two
Then
,
,
cases, Ci and C2 will both be at unit, and all the other conductors at zero potential, and if the connexion between Ct and C2 be now made, no alteration takes place in the distribution of
electricity.
The charge upon the new conductor, that which is less than e l + e2 2 t -f a + e^ + e?'
is,
its capacity, is
.
,
164.] Any conductor of given bounding surface solid or a hollow shell, and all the coefficients
may
be either
of potential, or to or whether induction, that, capacity, relating any other connot same in either and are affected are the case, ductor, by the introduction of any conductors whatever inside a hollow conductor.
For
let
Cr
be a hollow conductor.
Then, as proved in Art. 9 1
,
there be any electrification whatever the algebraic sum of which is zero within Crt that electrification together with the induced distribution on the inner face of Cr have zero potential
if
at each point in the substance
of,
or external to,
Cr9
and
may
be removed without affecting the distribution on the outer face of Cr or anywhere external to it. It follows that the potential
any other conductor due to a distribution on the outer surface C is unaffected by the presence or absence of such electrification within Cr and depends only on the forms and positions of the surfaces bounding Cr and the other conductors. Therefore the coefficients A, and therefore also the coefficients J3, depend only on these forms and positions. of
of
r
,
SYSTEMS OF CONDUCTORS,
165.]
On
165.]
165
the Comparison of similar Electrified Systems.
Let there be given two electrical systems similar in all respects but of different linear dimensions. Let the linear dimensions be denoted by A, so that A has different values in the two systems respectively. If the quantity of electricity per unit of volume be the same in both systems,
the potential will vary as X2
and in
this case p is
.
For
of the form
it is
independent of
A,
and
dan
/ / /
,
dy dz and
r
each
Evidently the force at corresponding points, proportional to A. being the variation of V per unit of length, varies in this case as A.
on the other hand, the quantity of electricity in homologous portions of space, instead of in unit of volume, be given constant, If,
p will vary as
^
,
and Fas - , and the force as
^
If the system consist entirely of conductors, and the superficial density, that is the quantity of electricity per unit of surface, be constant,
V
will vary as
A,
and the force
will be
invariable. If,
on the other hand, the quantity of
electricity
on homolo-
and the gous portions of surface be invariable, V will vary as -, A i
force as
A It follows
from these considerations that, as between similar
systems of conductors efficients
A
of different linear dimensions, the
vary inversely, and the
the linear dimensions.
coefficients
B
co-
directly, as
CHAPTEE
X.
ENERGY.
On
the Intrinsic
Energy of an Electrical System.
ARTICLE 166.] If V be the potential of any electrified system, the work done in constructing the system against the repulsion of its own parts
is
taken throughout the system, where e dxdydz is the quantity of free electricity that exists within the volume element dxdydz.
For we may suppose the charges in all the volume elements to be originally zero, and to be gradually increased, always preserving the same proportion to one another, till they attain their values in the actual system. The potentials at any instant during this process will be proportional to the charges at that instant.
Further,
we may suppose
the process to take place
uniformly throughout any time r. Then if t be the time that has elapsed since the beginning of the process, the charge in any element of volume may be represented by A t, and the potential
by pt where A and p are constants for the same element. The final values of e and Fwill be Ar and /ur. The charges which t
will be
added in the small interval of time
dt,
or
t
-\-dt-
1,
will
be \dt, and the work done in bringing these charges to the then existing potentials, represented by //,, will be fji\tdt
dxdydz.
The whole work done from beginning
['/[fa
i
dt
dxdydz
This quantity
=
Ifff^ by
E.
end of the process
T* dxdydz
is called the intrinsic
shall generally denote it
to
is
= i /TTve dxdydz.
energy of the system.
We
1
ENERGY OF AN ELECTRICAL SYSTEM.
67.]
167
If the system consist only of conductors on which the charges are e lt all
&c
e.,
,
we have
E=
\*2.Ve,
2 denoting summation
for
the conductors.
In like manner if there be two distinct electrical systems and the charges and potentials in one be denoted by e and T and in the other by e' and F', the work done in constructing the first 9
against the repulsion of the second, supposed existing independently, is
1 1
lY'edxdydz, and this must be equal to the work
done in constructing the second against the repulsion of the first,
that
is
fffvedxdydz or for a
=
f f f V'edxdydz,
system of conductors
sFy=sr, 167.] To prove that
the integral being taken throughout all space.
Let us consider an the whole electric
within
S,
-//
8 enclosing Green's theorem to the space Applying
infinitely distant closed surface
field.
we have
'-
which the first double integral relates to the infinitely distant S, and the second to the surfaces $', if any, within S on which there is superficial electrification.
in
surface
and the two remaining terms on the right-hand side of the equation are together equal to
/// 1-n.Vedxdydz. Hence
ENERGY OF AN ELECTRICAL SYSTEM.
168
[167.
was proved in Art. 13 that if V be one of the functions which satisfy the following conditions, viz.
class of
It
(0
/ / TJJ dv
=
dS* taken over the surface
dS taken
over the surface
S2
=e =
&c.
e,
2
,
,
&c.,
V V has given value at every point outside of all the surfaces, V vanishes at an infinite distance, 2
(2) (3)
then the integral
all space outside of all the surfaces has constant over each surface.
throughout
V is
We now an
see the physical
electrified system.
its least
meaning of the theorem
when
as applied to
V being the potential,
For
value
/ /
-r-dS,
taken over any surface, is the charge upon that surface, whether the charge be so distributed over it as to make V constant or not.
Now
in whatever
way
the charges be distributed over the
surfaces, consistently with the whole charge on each surface
being given,
-
/ / /
Vedxdydz
is
the energy of the system.
integral on the right-hand side relating to the space outside, and the second to the space inside of the surfaces.
the
first
And has
the proposition shows that the
its least
as to
make
possible value
V constant over
first integral,
when the charges each surface.
or
Q
are so distributed
And
in that case
V is
constant throughout the space inside of the surfaces, and therefore the second integral is zero.
1
ENERGY OF AN ELECTRICAL SYSTEM.
68.]
the distribution actually assumed on each the electricity is free to distribute itself, that is, the surface is a conductor, is the one which makes the
It follows that
surface
when
when
intrinsic
energy the
least
possible,
given the charges on the
conductors respectively. 168.] The potential of any conductor, as Cr due to a quantity of electricity in the volume element dxdydz } is evidently of the form A rs e> where A rs is a coefficient depending, like the coefficients ,
A
already investigated, on the forms and positions of the conductors and the position of the element in question, and the suffix
s relates to
that element, and r to the conductor.
Similarly the potential at the element due to a charge e on Cr will be A sr e, where A sr depends only on the form and position of the conductors and the position of the element.
A
Also the equality of
A sr
and
rs
follows from that of
2 Ve'
and S V'e above proved.
Thus the systems of equations (A) and (B) of Arts. 155, 156 can be extended to any electrified system, whether consisting exclusively of conductors of finite size or not. J 2 Ve we express every V Evidently if in the equation will be a means of the of in terms charges by equations (A),
E
E
quadratic function of the charges with coefficients depending on In this form we the forms and positions of the conductors. shall write it
E
e
.
Similarly if we express every e in terms of the potentials by will be a quadratic function of the means of equations (B),
E
potentials.
In this form we
It follows
shall write it
from the equality of A J2 and d
**- rK" '
~ _l-y
'
d^
Ey.
A zl
,
&c., that
&c
de,
or generally,
= V.
=
(JL&
For as from
dE
E=
\
2 Ve we have
dE
dE dV ~
~
)
and
-
=
\V
+
--
dE dV, ~' & ()
MECHANICAL ACTION BETWEEN
170
t
therefore
171
-^
-I
dE
=
Similarly
0#
169.]
We system
/$
-I
-I
= -V, + -A^e^ -A
12 e 2
+ ...&c.
e^ &c.
Mechanical Action between Electrified Bodies.
have seen that the intrinsic energy of any electrified is of the form 4 Ve> where e is any quantity of elec-
forming part of the system, where that quantity is situated. tricity
V
the potential at the point
If q be any generalised coordinate defining the position of the system, the force tending to produce in the system the displace-
ment dq
is
--dE 7
dq
Now
if
tentials
V
or
-Id 2 dq
(2 x
Ve).
the charges e are invariable in magnitude, the poare functions of the coordinates q, and therefore the
force is
in
which every Fis a function of q in respect of the coefficients A. on the other hand, the potentials be maintained constant
If,
notwithstanding displacement, by proper variation of the charges. the force is
in
which every
coefficients B.
in these
two
e is a function of q as it is involved in the It remains to find the relation between the forces
cases.
170.] If q be any one of the generalised coordinates defining the denote the force tending to inposition of the conductors, and if / crease q when the charges are invariable, and be that force when
R
R
all the potentials are maintained constant, then
For let e, V)
J
and q
all
vary.
R + R' =
0.
ELECTRIFIED BODIES.
171.]
171
Then we have
dE
dE H
de
dV
dq
But
dq
*jl.= de
and
dV 2-r
hence
2
and
?8i
=^-
61
?g g+
therefore ofy
8?
~ ^T
Now
is
the rate of variation of
are constant, and
is
dE v
i
bimilarly
dq hence
E with
q
when the charges
therefore the force tending to diminish q
imder those circumstances; that i
= 0.
efy
7
7
o-
7,
72
is, it is
R.
=R, ,
+ ^=0.
171.] If any group of conductors previously insulated from one another become connected by very thin wires, so as to form one
conductor, the energy of the system is thereby diminished ; and the energy lost by it is equal to that of an electrical system in which the superficial density at any point is the difference
of the densities at the same point before and after the conthat is, is equal to the energy of the system nection is made :
which must be added
to the original in order to produce the
new
system.
For
let
nection
is
V
denote the potential of the system after the conthe original potential. Then V and T' made,
V+V
are both constant throughout each conductor of the system. The charge on any conductor which retains its insulation, or -=- dS. remains unaltered.
/ / dv 4*w
ENERGY OF AN ELECTRICAL SYSTEM.
172
[172.
Any group of conductors which become connected form one combined conducting- surface on which the aggregate charge, or
--
1 1 -j- dS, is unaltered. 4t-nJJ dv
V is then a function which satisfies the conditions at
all
(3)
outside
points
of
all
has given values
-j-
the
V2 V =
(i)
connected
conductors,
over each of them, (3)
V
is
F+ V
satisfies conditions (i) constant over each of them, while and (2), but is not constant over each of the connected con-
ductors.
Therefore,
by Art.
Qv+v
=
1 3,
Qr + Qv,
where
throughout all space outside of the surfaces, and have corresponding values.
Now -8
77
Qy+v'
is
made, and
Qp
O7T
>
-- Q v O7T
the original energy,
after the connection is
Q v+ v and Q y
is
is
the energy
the energy of the
system in which the potential is the difference of the two potentials, and therefore, by the principle of superposition, in which the density at any point is the difference between the
and
densities before
after the connection is
made.
any portion of space S, previously not a conductor, become a conductor, or which is the same thing, if a conductor 172.] If
be brought from outside of the field and made to occupy the space S within the field, the energy of the system is thereby diminished and the energy lost by it is equal to that of the system in which ;
the superficial density at any point on any conductor or on the surface of S is the .difference of the densities at the same point before and after
S became
a conductor
;
that
is
to the energy of
the system of densities which must be combined with the original to produce the new system.
For
let
V be
conductor, V-\-
the potential of the system
V the potential when S
is
when
the space
S
is
a
non-conducting space.
1
ENERGY OF AN ELECTRICAL SYSTEM.
74.]
V is constant
Then
V+ V
is
Let Sir
8
7T
173
Q v+ v
>
throughout S and throughout each conductor, constant over each other conductor but not over S.
Qy be
the energy of the system in the former case,
the energy in the latter case.
It can then be shewn, as in the last case, that
throughout all space outside of the original conductors whether within or without S. But the integral of the second term is the
energy of the system in which the potential is T' that is of the system which must be combined with the original to produce t
the
new
system.
It follows that a conductor without charge is always attracted by any electrical system if at a sufficient distance from it.
Hence
any number of .uncharged conductors
also
in a field
of constant force generally attract each other. 173.] It follows as a corollary to the two last propositions that if a conductor increase in size, the energy of the system is
thereby diminished. if C be a conductor,
8 an
adjoining space, if S became the (7, energy would be diminished, conductor were then connected with C so as
For
a conductor insulated from
and if the new form one conductor with
to
diminished.
Hence
of a conductor
is,
if
it,
the energy would be further
the resultant force on an element of surface
the charges be constant, in the direction of
the normal outwards.
S be a surface completely enclosing a conductor (?, 8 were itself a conductor, its capacity would be greater
174.] If
then
if
than that of
For
let
C.
the conductor
C
be charged to potential unity,
all
the
M
be its capacity, Let other conductors being at zero potential. that is, the charge upon it under these circumstances. Since unity is in this case the highest potential in the Let the potential on S is less than unity at every point.
field, it
be
EARNSHAW'S THEOREM.
174
[175.
denoted by V. If
/VVdS =
M
by Art. 60,
and
7 <
since
1,
therefore
J
fFcrdS < M.
M
Next let the same quantity of electricity be so distributed S as to have constant potential in presence of the other
V
over
uninsulated conductors.
Let
/YVvdS' <
Then
Therefore, a fortiori,
V'M <
its
ffr*48
/T V'
also
be
density in that case.
ty Art.
1
67,
V ff
M or V <
charge would be necessary to bring
S
1,
and therefore a larger
to unit potential.
Earnshaivs Theorem. 175.] If an electrified system A be in mechanical equilibrium in presence of another electrified system J3, and be capable of
movement without touching the
latter,
the equilibrium cannot
be stable.
For
let
in space,
B
is fixed us suppose first that all the electricity in all the electricity in A fixed in the system, so that
and
the system A is capable of us further suppose first that
And let as a rig-id body. of translation of motion capable
movement it is
only.
In
this case the position of
A
is
completely determined by the
in it, and is therefore a position in space of any single point referred to any system of axes function of the coordinates of fixed in space.
coincides with a Let the position of equilibrium be when certain point C in space. Let be the potential of the system S. Let pdxdydz be the
V
1
EARNSHAW'S THEOREM.
75.]
quantity of electricity of the system
A
in the
175 volume element
Then
dxdydz.
V dxdydz throughout
A
is
the intrinsic energy of the mutual action of the is, that part of the whole energy which varies
two systems, that
with the position of 0. Let
dv d*E dx*
and
as similar equations hold for
= Now
=
so long as 0,
z,
1 /YY pV*Vdxd,ydz.
no part of
and therefore
y and
V2 E =
A
coincides with
any part of
,
0.
A is capable of motion of translation without any of it part coinciding with any part of B, it must be possible to describe a small closed surface 8 about C, the position of equiAgain, since
librium, such that, if
V E= 2
But
be anywhere within S,
VZ V
'
0,
and
0.
E
is
a function of
#, y, z,
by Green's theorem applied
= Either therefore
E
the coordinates of 0.
to the surface
S and the
Therefore, function E,
0.
must be constant
for all positions of
in
the neighbourhood of C which are consistent with A not touching B, in which case the equilibrium is neutral ; or there must be on that portion, be some part of the surface S such that, if
E
is at C, that is less than in the position is less than when Therefore any small displacement of A in this of equilibrium. direction will bring it into a field of force tending to move it
further in the same direction, and the equilibrium fore unstable.
still
is
there-
A SYSTEM OF INSULATED CONDUCTORS
176
[176.
If the system has any more degrees of freedom, as for instance freedom to rotate about an axis, or for the electricity on A or
B
instead of being- fixed as we assumed it to change will be capable of becoming less in the the a energy be, fortiori and the equilibrium is unstable. therefore displaced position, its position,
to
On a
176.]
system of insulated conductors fixed in a field of
without charge. uniform force and Let us first suppose the uniform force
the axis of x.
is
unit force parallel to origin of
Then we may always choose the
coordinates, so that the potential of the force at
any point may x. be denoted by The induced distributions on the conductors must be such
them
that the potential due to shall
be
<7-f #,
where
C
is
at any point in any conductor constant over each conductor but has
generally a different value for different conductors. Let the density of this induced distribution at any point be Then X(f> x is the density of the or shortly (f) x <$> x (os, y, z), .
induced distribution which would be formed
And CX+xX.
X.
if
the potential due to the distribution
Definition.
the force were
X$ x
is
The function \\x $ x dS taken over the
therefore
surface of
every conductor of the system is the electric polarisation of the system in direction x due to a unit force in direction x acting at every point of the system. For position of the origin.
dS
=
It
is
In
evidently independent of the
"
since 1 1 $*$$ is the algebraic
and
is
sum
of the induced distribution
therefore zero.
like
manner we may
define
to be the electric polarisation of the system in the directions of
y and z respectively due to unit force in direction x. In like manner if there be forces T and Z parallel
to the other
A FIELD OF UNIFORM FORCE.
IN
1 7 7.]
177
coordinate axes, they will produce in the system induced disYfa and Z
tributions whose densities are
functions
will denote the polarisation in the directions y, x,
ively due to unit force in direction y.
and z respect-
Similarly,
denote polarisation due to unit force in direction of the quantities
y(f) x
dS, &c.
llzfadS,&e.
And
z.
independent of the position of
is
the origin if the direction of the axes be given. If *l> C denote the total polarisation parallel to respectively, due to the three forces, we shall have
y fa
dS+
dS+ 177.]
We
can
now prove
Yy $ dS+ zy y
Y^fz
<j>
C+
v
dS + zffz
#, y,
z
j. dS,
z
dS.
that
For the potential of the system of
we have
each
densities denoted
by
x is,
as
C
being, as before mentioned, a constant for each conductor, but having generally different values for difTherefore ferent conductors. seen,
a?,
the work done in constructing the distribution whose density is
ductor the algebraic
and
C is
sum
of the induced distribution fa
constant,
jfcfa VOL.
I.
dS
N
=
0,
is zero,
A SYSTEM OF INSULATED CONDUCTORS
178 and
In
#(f) v
like
system
dS
expresses the
manner (j>
x
y<j> x
dS
is
[178.
amount of work above mentioned. the work done in constructing the
against the repulsion of the system <$) y previously these two quan-
But by the conservation of energy existing. Therefore tities of work must be equal.
ffx
Similarly,
=ff* **
178.] If the direction cosines of
any
line
referred
to
any
system of rectangle axes be a, (3, y, the polarisation parallel to this line expressed in terms of these coordinates becomes
that
is,
that
is,
af + jSq + yf
quantities,
The
polarisations
and admit of composition and
Evidently, if
a, fij
y be proportional to
,
rj,
fare in
fact vector
resolution. ,
f = ap, t] = pp, =yp, and the polarisation in any direction a', /3',
and
r?,
then
that
is
and
is
y', is
(a
therefore zero for
any
direction of the resultant of
,
direction at right angles to the 77,
f
179.] Let it now be required to find a direction relative to the system such that if the resultant force be in that direction,
the resultant polarisation shall be in the
same
direction.
I
IN A FIELD OF
79.]
In that case
f,
UNIFORM FORCE.
must be proportional
77,
(= where
e is
179
to X, Y, Z.
That
is,
*X,
a quantity to be determined.
Hence we have
.
x+
ds(ffyy
These equations are of the same form as those employed in Tait's Natural Philosophy 2nd Edition, p. 127, for
Thomson and
,
determining the principal axes of a system leads to a cubic equation in
As
strain.
of which,
,
there shown, the
when
the three roots are always real. The equation is the same as that treated of in Todhunter's Theory of Equations, 2nd Edition, As shown by Thomson and Tait, each of the three p. 108. values of e corresponds to a fixed line in the system, and the three lines corresponding to the three roots are mutually at right angles.
There
exist,
therefore, for every
system of conductors three
directions at right angles to each other, and fixed with reference to the system, such that a uniform force in any one of these
We
directions produces no polarisation in either of the others. might define these directions as the principal axes of polarisation
of the system of conductors in question. lines for axes
we
b
y
If
we take
these three
have evidently
shall
=
dS
0,
/
/
y fa dS
0,
&c.
Let us denote /
x
(j)
x
dS by Qx
,
*/
//
/.
I I
/
y $ v dS by
Qy
,
and
f / *$ / */
^ by
ft
;
A SYSTEM OF INSULATED CONDUCTORS
180
[l8o.
then the polarisation in any line whose direction cosines referred to these axes are a, p, y, due to unit force in that line is
180.]
Of
the energy of the polarisation of a system of conductors
placed in a field of uniform force.
Let the system be referred to
its
principal axes.
be the forces parallel to these axes respectively.
Let
Let X, Y,
X$ x
Z
be the
density at any point on any conductor which would be produced and Z. Then alone acting, Y$ y and Z(f> z the same for by
X
X(j> x
Y
the density when potential of the three forces is
+ Y(j> y -\-Z(f)
The
z
is
all
three forces act.
-(Xx+Yy + Zz). The work done
in constructing the system against the external
forces is then
taken over the surface of every conductor. The work done against the mutual forces of the system itself is
Yy + Zz) The whole work
is
therefore
taken over the surface of every conductor, and the work done against the mutual forces of the separated electricities is the
same expression with the
But the axes being
ffx^y dS Therefore the energy
positive sign.
principal axes,
=
0,
Pix
=
0,
&c.
is
72 or
181.] If the conductors be free to move in any manner, either by translation or rotation, they will endeavour to place them-
I
IN A FIELD OF UNIFORM FOECE.
8 2.]
181
such positions as that the above expression within brackets shall be maximum, given the resultant external force.
selves in
If the system consists of a single rigid conductor Q x Q and y If Qz will not be altered by any motion of the conductor. placed in a field of uniform force and free to move about an axis, ,
position of stable equilibrium will be that in which the axis of greatest polarisation coincides with the direction of the force. If the system consist of many conductors, and they be not so
its
distant that their mutual influence
may be neglected, they will tend either by rotation or translation to assume a position in which the axis of greatest polarisation shall coincide with still
the direction of the force; but Q x Q y , and Q a will in this case generally vary with change of position of the conductors. ,
We
182.]
have hitherto treated
(j)
x,
cf)
y
,
&c. as the densities of
the distribution produced on a conductor in a force.
If
we extend our
definition,
field
of uniform
and define // xfydS to be
the polarisation in direction x of an uncharged conductor placed in any electric field, $ being the density of the induced distribution,
The
we can prove the
electric
polarisation
following proposition of any spherical conductor due :
without
to
any equal in magnitude to the resultant force at the centre of the sphere due to that distribution multiplied by the cube of the radius, and is in the same distribution
of
electricity entirely
it is
direction with that resultant.
X
be the ^-component of force at the centre of the the external system. Then is the ^-comto due sphere centre due the to at the electrification. induced ponent of force
For
let
X
But the last-named component
if
or
is
be the density of the induced distribution.
Hence Similarly
JJx(rdS=r*X.
jjy
Y,
182 SYSTEM OF CONDUCTORS IN A FIELD OF FORCE.
Z
be the components of force parallel to this establishes the proposition. If v be the volume of the sphere, if
Y
and
y and
[l 82.
z,
and
CHAPTER XL SPECIFIC INDUCTIVE CAPACITY.
ARTICLE 183.] HITHERTO, in accordance with the plan laid down in Chapter IV, and with the view of giving the two-fluid theory as complete a development as possible in its most abstract form, all
space has been regarded as
made
up, so far as electrical pro-
two kinds conducting space, through which the fluids pass from point to point under the influence of any electromotive force however small, and non-conducting or insulating space, through which no force however large can perties are concerned, of
;
cause such passage of the fluids to take place. Conducting spaces are necessarily occupied stances,
spaces
prominent among which are
may
all
metals.
by
actual
sub-
Non-conducting
be occupied by actual substances, called non-con-
ductors, insulators, or dielectrics (the last term having reference to their transmission of electric action as distinguished from the passage of the electric fluids), such as dry air and other gases,
wood, shell
A
lac,
sulphur, &c., or these spaces may be vacuum. at one time regarded as a perfect
vacuum was
perfect
insulator.
184.] Faraday was the
periments performed by
to point out, as the result of exthat there is a considerable diversity him, first
of constitution in dielectric media, in reference to their electric different substances possess the properties that, in fact, while ;
common
property of refusing passage to the electric
are nevertheless
endowed with very
fluids,
they
different electrical properties
in certain other respects. If, for instance, there be, as in Faraconcentric two conducting spherical surfaces, day's experiments,
and the space between them be
filled
with any dielectric substance, e on itself
the potential of the inner sphere due to any charge
SPECIFIC INDUCTIVE CAPACITY.
184
[185.
hitherto investigated (which according to the simple theory
should be -
>
where r
is
/
the radius) will be found to depend on
the nature of the dielectric medium in the space between the And therefore the charge on the sphere necessary to spheres. a given potential, or the capacity of the sphere, is produce greater
for
some of such substances than
for
others.
The
It is dielectric, in Faraday's language, has inductive capacity. than for solid the and for air less dielectrics, any permanent gases
and rather less for vacuum than for air. 185.1 In order to explain this phenomenon, Faraday adopts the hypothesis that any dielectric medium consists of a great number of very small conducting bodies interspersed in, and separated by, a completely insulating medium impervious to the passage In his own words, If the space round a charged of electricity. '
'
'
*
'
filled with a mixture of an insulating dielectric, as of turpentine or air, and small globular conductors, as shot, the latter being at a little distance from each other, so as to be
globe were oil
insulated, then these
and action exactly be the condition and action of the
in their condition
'
resemble what I consider to
'
particles of the insulating dielectric itself.
'
charged, these
1
were discharged, they would all be polarised again upon the recharging of the globe/ The properties of such a medium closely resemble, as far as
'
little
If the globe were be polar ; if the globe return to their normal state, to
conductors would
their mechanical action
all
concerned, those of a magnetic mass, of Faraday's shot being in fact each by Poisson, when polarised equivalent to a little magnet, except that in dealing with magnetic masses the polarisation is usually under-
as conceived
is
'
'
stood to be in parallelepipeds instead of in spherical particles, and Poisson's investigations are therefore applicable. (See
Memoires de PInstUut for 1823 and 1824.) The mathematical theory has also been treated by Mossotti with especial reference to Faraday's theory, but by a different method from that here employed. 186.] In accordance with this hypothesis of Faraday's, we will consider the dielectric as consisting of a great number of very
1
SPECIFIC INDUCTIVE CAPACITY.
86.]
185
small conducting- bodies, not necessarily spherical, and separated first object is to find the by a perfectly insulating medium.
A
form assumed by Poisson's equation when the conductors become infinitely small.
In this medium take any parallelepiped whose edges, h, k, I, are parallel to the coordinate axes. Let these edges be very small compared with the general dimensions of the electric field, but yet infinitely great compared with the dimensions of any of the little conductors in question. The face Tel of the parallelopiped will intersect a great number of these little conductors. If the medium be subjected to any electromotive forces, there will be on each conductor
whose
superficial density
conductor
Let
X
/ /
>
ds
=
an induced distribution of
we
shall denote
by
electricity
Then
$.
for each
0.
be the average value over the face kl of the force and therefore normal to kl, due to the whole electric
parallel to #, field,
including the induced distributions on
whether intersected by kl or not.
We
all
will first
the conductors
assume that the
corresponding forces Y, Z are zero. In this case the average value per unit area of kl of the algebraic sum of the induced distribution on the intersected conductors which
lies to
the right, or positive, side of kl
is,
by
the principle of superposition, proportional to X. Let it be were the unit QX, Q being the value which it would have if
X
of force. If the forces Y and Z are not zero, then the quantity of the induced distribution on any individual conductor lying to the right of kl will generally depend on Y and Z as well as on X.
But
if
the conductors be in
all
manner of
orientation indiffer-
ently, the quantity of free electricity to the right of kl, due to and Z, will disappear on taking the average because for any
Y
;
conductor,
from
if
Y or Z
the total density of the induced distribution arising for any position of the conductor be calculated, then
on turning the conductor through two right angles about an axis parallel to #, the corresponding density in the new position will be equal to that in the former position, but of opposite sign.
SPECIFIC INDUCTIVE CAPACITY,
186
We
shall for the present confine ourselves to the case in which the conductors are orientated indifferently in all directions, and we shall define a medium in which this is the case to be an
isotropic
medium, and any other
to be a heterotropic
medium.
evident that in an isotropic medium the quantity of the induced electricity in the conductors intersected by the faces hk It
is
or h I of the parallel opiped which lies to the positive side of those faces respectively, due to unit force in direction y or z, is the
same is
as the corresponding quantity for the face kl.
That
is, it
Q.
The total electricity included within the parallelopiped will consist of (l) p 7ikl, the quantity of the given electrical distribution, which is supposed to exist independently of the condition 187.]
of the
medium within
h
whether
Jc I,
it
be vacuum or dielectric
;
(2) the sum of the induced superficial distribution on those parts of the conductors intersected by the faces of the parallelopiped which lie within its volume. The part of (2) arising from the
two
Jcl
faces will be
UQX respectively,
In
like
and
-klQX -
and
their
sum
Tiki
dx (QX)
therefore
is
manner the part of
(2) arising
from the faces hk,
Jil
of
the parallelopiped are
and
-Kkl-
respectively.
We
obtain therefore
parallelopiped,
Now the
let
for
which we
the whole electricity within the
will call E,
N be the normal force
parallelopiped
at any point on the surface of measured outwards. Then on the two kl
faces the average value of
-X
N is and
+X + h^ dx
1
SPECIFIC INDUCTIVE CAPACITY.
88.]
187
Therefore for these two faces
respectively.
Nds Similarly, for the other
dX hkl +~ dx
=
two
Nds
we
faces
'
=+
~ rl
shall
have
Y
hkl,
dy and Therefore for the whole surface of the parallelepiped
But
=
/YV
+ hkl(dx
Therefore
But from
-
^
dy
(1)
= hkl Hence we obtain
or, if
we write 1
and
7
if T
=
^,
be the mean potential, so that JL
cfo
This
+ 477(5
"
is
isotropic
v
dx
= -=dV '
>
I
= -rdV '
Zi
^
= -dV
>
'
dz dz dy^ dy the form assumed by Poisson's equation
medium
as
now under
consideration.
in such
K
an
evidently
depends on the form, number, and position of the conductors, It is called the dielectric is, on the nature of the substance.
that
constant.
188.] Again, if p
+p' be the whole
free electricity in the
element
SPECIFIC INDUCTIVE CAPACITY.
188
[l8 9
of volume dxdydz, including both that of the general and that of the induced distributions, evidently
.
electric field
Hence we have d
dV
,
dV
*
dV
d_
dz
"fa
and phkl is the sum of those portions of the induced distributions on the conductors intersected by the faces of the parallelepiped which lie within its volume. 189.] If over
given
surfaces there be superficial electricity of the
any
electric distribution, the equation
dx
^
dx
'
dy
^
'
dz
dy
x
dz
becomes, as in the cases previously investigated, (2)
dv or
,dV (
dv
where the
suffixes relate to the
media on either side of the surface
of separation.
Let
(1)
and
(2) denote
two media bounded by a plane
1 C.2
surface
X
that K, Q, and have the suffixes and 2 in these media respectively. Let Clt
AS, such
be two parallel planes on either side of Then, by the preceding, the superficial
AB.
induced electrification in the
space between
C1 and AB, and C2 and AB, per unit area of the planes, is + Qi-^i an(^ ~~ Qz^-z respectively, and the B Fig. 28.
between C^ and
C2
is
Q 1 X1
Q 2 X2
per unit area
oiAB.
two planes Ci and C2 be made to approach each other they become infinitely near, this gives a superficial electrifi-
If the till
total induced electrification in the space
SPECIFIC INDUCTIVE CAPACITY.
189
cation over AJ3, the surface of separation of the media (l) and from induction on the such that conductors, (2), arising
dv if
the normals be reckoned inwards from the surface in case of
each medium.
Also
if
the electricity per unit volume arising- from such />', we have seen that
induction be called
Therefore our equations
may
be written
=
0,
and a' are called by Maxwell the apparent electricities solid and superficial respectively, and Faraday's hypothesis of the dielectric medium supplies us therefore with a physical meaning //
for these quantities.
It follows from the equation (2) that at the surface of separation of any two isotropic media, in which the constant has the values K-l and 2 respectively, if there be no real electri-
K
K
fication
on that
surface, that
is,
if
o-
=
0,
the normal forces on
either side of the surface are to each other in a constant
ratio.,
namely
be the superficial electrifi^llia at any point 190.] Now let of the surface of any one of the small conductors in jbhe neighbourhood of any point P, (x, y, z) in the dielectric, and let the sum of the integrals "sffxQdS, or
1
1 to $ US
taken over the
4 surfaces of all these conductors within uni o volume, be denoted
by
o-
x,
assuming that the distribution of
electricity
on each
SPECIFIC INDUCTIVE CAPACITY.
190
[190.
conductor and the distribution of the conductors themselves
is
P
constant throughout that volume and the same as it is at thus a-x is a physical property of the dielectric at analogous to the pressure p referred to unit of surface at any points in a fluid ;
P
Let a be the average mass, and other similar quantities. distance between two planes parallel to y, z, touching any small conductor, i. e. the average breadth of a conductor parallel to #, n be the average number of such small conductors in It follows that the number of conductors in an elementary parallelepiped dxdydz is n dxdydz^ and the number
and
let
unit of volume.
intersecting the dydz face
must be
dx
or n a dydz.
a
Now
xl be the x coordinate of the left-hand plane parallel yz touching any conductor, the average amount of the electricity lying to the right of any other plane parallel to yz interif
to
secting the same conductor
must be // -
tydS, the integration .
being over the surface of the conductor, rr since
N)dS =
i.e.
it is
* - /rr / x
a
(/>
dS,
1 1
0.
amount of
Therefore the
electricity
on the small conductors
by the left-hand dydz face of the parallelepiped dxdydz and lying to the right of that face must be intersected
nadydz
But nilxfydS Therefore jrm
is
is
I
-
j
or
n/
I
axfrdS. dydz.
the quantity above designated by a x
.
the amount of electricity per unit surface
on the rin| hand of a plane at P parallel to yz situated on small conductors intersected by that plane it is the same quantity as is denoted by QX in Art. 186; similarly for a- y and 0-3. ;
The
the direction of
From
o-.
(See Chap. X, Art. 178).
Arts. 186 aiuj 188
it
follows that
//,
the density at any
SPECIFIC INDUCTIVE CAPACITY.
1 9 1.]
P
point
of induced
electricity
101
belonging to the conducting
surfaces, is
dx dz dy and that if o- x cr y) v z be discontinuous over any surface S through P whose normal has direction cosines I, m, n, then there will be ,
a superficial electrification arising from the induced electricity S, the density .of which
on the small conductors over the surface at
Pis I
where
8
and
crx
(a- x
-
a-' x
+ m(o-y- a'y) + n(
)
the values of
a'x are
cr
x
and
v'x
on opposite
sides of
at P.
If
V be
the potential at any point of the
field arising
from
the induced charges on the small conductors and from these must satisfy the equations alone, then from Art. 189
V
dV dv
And unique,
_
d
since the solution of these equations is determinate
and
we must have
r=
rrrp'dxdydz
+
rr
where the integration extends over the whole field, a result in which is at once obvious from the physical meanings of p
fact
and
(/.
191.]
"We know that the average force in direction x over plane parallel to yz due to an electrical distribution
finite
any whose algebraic sum is m placed on the positive side of the plane, and at a distance from it very small compared with the dimensions of the plane, is 2irm. If therefore any such plane be situated in the medium under consideration, the average force upon it arising from intersected conductors will be -2770-3 from those on the right-hand side,
and again 4770-,,. 2770-3. from those on the left-hand side, or on the whole and if we limit ourselves to the consideration of the little conductors situated between two planes very close to ;
and
parallel to the supposed plane
on opposite
sides of
it,
the
192
SPECIFIC INDUCTIVE CAPACITY.
4770-3. parallel to average force will be for non-intersected conductors is zero *.
x, since
the value of
m
medium we take
It follows therefore that if in the dielectric
and each perpendicular to v, the resultant of
any two planes very
close together,
,
,
,
* The proposition in the text is an important one and may be proved rigorously as follows Let there be an infinite number of polarised molecules between two infinite planes parallel to yz, and not intersected by the planes. Let the equations of the = a t and x = a2 . Let V^ and V2 be the values of the potential from planes be x the polarised molecules upon the respective planes; and let dy l dz l dy 2 dzz denote elements of their surfaces, and dv^ and dv2 elements of their normals measured outwards from the space between the planes. Applying Green's theorem to the space between the planes, using x and 7 for :
,
we have
functions,
rrr CCdv dy^dZi + CTdVj J -j- dyzdzy Illy. dv d "> JJ JJJ JJ .
ai
I I
ctt
"
I
I
>
rrr the last term
/ / /
JJJ
xv*V dxdydz
including
^
superficial density or discontinuous
and
/ / /
- 4tr
x v 2 V dxdydz
Also
I I I
(;r-
/ /
JJ
dv.
+ T~)d>3
for surfaces of
<".
&c.
,
Since the distribution within the planes
are separately zero. / I fl~ dytdzz
rr# ,dv
is
algebraically zero, //
^-
and
xtydxdydz
are
if
I
and dt/i ^~dyidz
/ -=
/
d + and
i
l
resp respectively,
be the density at any point
anes.
I
irt dy9 .dsg -
the
mean
/
/Fi^efoj
= 4w
force in direction of x,
/ / /
xdxdydz.
and C the area of either plane,
ll' where
H
is
the volume between the planes.
Therefore
r-a
These conclusions are equally true when the planes are of any magnitude, provided the distance between them is infinitely small compared with their linear dimensions, in which case
/
y
JJ
becomes the same quantity as that
n above denoted by a x and the proposition
is
proved.
1
SPECIFIC INDUCTIVE CAPACITY.
9 I.]
193
small conductors situated between these planes and not inter47r
the planes parallel to x, y, and z will be 47rcr 2 47:0-3., 4 irtT yt same as force would the the have been from the respectively, included conductors
if
the parallel planes including the point
had been perpendicular to
#, y, or z respectively. If instead of taking a region between two parallel planes whose distance is very small compared with their linear dimen-
we had
considered a space inclosed within any small sphere medium, then we might prove that the force at the centre of the sphere arising from all the small conductors entirely sions
in the
included within
it
has for
its
33--
--
components
A
-
and
cr a
3
(r x
,
cr
v
,
respectively.
If instead of a
medium
of small conductors with electricity on
their surfaces distributed according to the law of induced elec-
under the action of a constant force with components were considering a medium composed of small dis-
tricity
X,
Z we
J",
crete molecules, each separately containing
bution according to
the algebraic sum molecule is zero, and
if
we denoted by
any point in any molecule, and by
for unit of
in the
by
iJx
0-3.,
y
,
cr
z
o-
x)
>
vy
the electrical density at and o-z the sums ,
triple integrals
where
<
is
superficial,
we should
still
have
the components of a vector; inasmuch as by changing such that the directionf, 17,
= I.
(#, ^, z)
scfydv &c. being replaced
any other rectangular axes are l> m, n, we get cosines of
VOL.
electrical distri-
volume in the neighbourhood of any point P,
medium, the
to
an
any law, subject only to the condition that of the distribution in or upon every such
S
/ /
/ ((lx
+ my + nz) (f)dv
SPECIFIC INDUCTIVE CAPACITY.
194
and the results arrived at in the preceding
[
1
92
-
articles as to the
density, solid p or superficial , of the electricity at
any point, and the value of the potential F', would hold good of this medium. Comparing two electrified systems in all respects 192.] similar, except that in the
uniform value
j5T
l5
one the dielectric constant has the
and in the other the uniform value
K
2
,
we
from the form of Poisson's equation above obtained, that, if has at each point in either system the same value as at the
see
V
corresponding point in the other system, all the volume and superficial densities must as between the two systems vary directly as K, the dielectric constant. Conversely, if the densities be the same at all corresponding points, the potential at corresponding points will vary inversely as K.
The
electrified system, is
medium with
which the constant is unity, in therefore the same as if all the charges
dielectric constant K, for
any
a uniform dielectric
effect of substituting
air, in
on the conductors were reduced in the ratio 1 :K, or as if the repulsion between the masses e and e' at distance r were /
/
instead of
^
It follows that the charges necessary to
is the capacity of the system, must is called in similar systems vary directly as K. For this reason the specific inductive capacity or briefly the inductive capacity of
produce given potentials, that
K
>
the medium.
The phenomenon observed by Faraday, using two spheres separated by a homogeneous
conducting
medium,
is
a particular case
concentric isotropic
of the general result of this
article.
The conception and treatment of lines, tubes, and fluxes developed in Arts. 96-103 of Chap. V are equally applicable to an isotropic dielectric with any value of K, either uniform or variable, and might, indeed, have been applied to If we establish the equations above obtained in such a medium. above of the viz. proved, integrate each term equation 193.] of force
dz
SPECIFIC INDUCTIVE CAPACITY.
I93-]
over a space inclosed within any closed surface
and
S,
195
we
get
m, n be the direction cosines of the normal to any element dS of the surface, this becomes if
I,
/
or
and If,
KFn dS'= 4 Tim;
Fn is the normal force
measured outwards at each point of the algebraic sum of the included electricities. therefore, as in Art. 102, any tube of force be limited by the
where S,
1
m
is
S and $', and if F and F' be the normal forces S and S' respectively, and if K and K' be the in-
transverse surfaces at points on
ductive capacities at those points, then the equation becomes
^K'F'dS'or as it
may
be written
/7Vd,S"- ff'f& = 4w
jm-
an equation expressing the same physical property as that of Art. 102, inasmuch as
the addition to the electricity included in the limited tube of force arising from the polarisation of the small conductors in the is
medium. In a medium with a continuously varying specific inductive capacity and finite volume densities, the force F obviously varies continuously both in direction and magnitude. If however there be an abrupt transition from one dielectric medium to another at any surface S, then, whether there be an dielectric
S or not, there is, as we have seen, a charge over the from polarisation of the small conductors, called arising O 3
actual charge on
S
SPECIFIC INDUCTIVE CAPACITY.
196
[194.
generally the apparent electrification, although both from theory and experiment it is proved to have an existence as real as what is called
by
this charge
distinction the actual charge
;
if a'
be the density of
we have seen that
where Fl and F2 are the forces normal to 8 at the point in the two media respectively, each supposed to act from medium L If therefore there be no actual electritowards medium 2
K
K
.
on S we have
fication
9
-
or
= 0,
as above shewn.
And
if
i
and
i'
be the inclination of the lines of force to the
normal to S before and
after the transit over $,
tan
i
=
(1
+
~f,
-
.) '
tan i'
as above proved.
All the properties of tubes of force, elementary or otherwise, hold good in proved in Chap. V, for a dielectric of uniform
K
medium with varying K, if we substitute for F (the any point) the quantity KF. This quantity KF is some-
the case of a force at
times called the induction.
Should there be an actual charge a over equations wonld give
8,
then the ordinary
being determined as before. 194.] As an illustration of the application of the preceding results, let us consider the state of the electric field when two (/
media of different but uniform inductive capacities are separated by an infinite plane surface, and an electric charge is situated at a given point in one of them. Suppose the plane of the paper to be perpendicular to the plane of separation; let YEY' (see Fig. 29) be the line of inter-
1
SPECIFIC INDUCTIVE CAPACITY.
94.]
197
section of these planes, and let the charge m be situated at the in the medium whose inductive point
m
is
capacity
being Let
2
kl3 that of the other medium
.
mEm' be drawn
bounding plane, and
m'E = mE
=
perpendicular to the m' be so taken that
let
suppose.
,
Let the distances of any point P from and m' be called r and / respectively. If V and be the potentials on the left
m
,
V
and right of the plane YEY' respectively, then V and must satisfy the following conditions, ^ and 2 being uniform in each
V
/
medium
V on the plane and each vanishes at
(1)
V=
(3)
VF+
Now
Fi
:
r
=/
-3
=
over the plane and the differential coefficients
of the functions
portional to
- and
along the normal to the plane are pro-
and -^ respectively.
It follows therefore that the conditions (l) satisfied
infinity,
and
(4)
may
be
by assuming
V
A ~ + r
B
and
r
and properly determining A, B,
_,
V'=
C r
C,
+
D r
and D.
Since p is zero to the left of YEY' except in the neighbourhood of m, the condition (2) requires that A should be equal to -j-
>
and since p
is
zero everywhere to the right of YEY', the
SPECIFIC INDUCTIVE CAPACITY.
198
D should be zero,
condition (3) requires that
[195.
and therefore
)
r
m T-
(1) gives
(4) gives
whence we get
2m
_
-^ 2m
/
and the problem is completely determined. If o-' be the superficial electrification of polarisation, or so called apparent electrification, over the plane, then
-
m
f^i-
^i-^xo *-!
1 /.
7 CT
^) If k^
=
1
and
>^
2
=
oo
,
27T7-
3
we have
agreeing, as they should do, with the results obtained for an infinite conducting plane in presence of a charged point, in Art. 105.
195.] As an instance of a similar treatment let us consider the case of a sphere composed of a dielectric medium of specific
inductive capacity ky brought into a field of uniform force
F
in
air.
Before the introduction of the sphere the force throughout field was in a given fixed direction, which we will take for
the
F
the direction of the axis of x.
SPECIFIC INDUCTIVE CAPACITY.
I95-]
199
If the origin be measured from any point, as for instance the with which the centre of the dielectric sphere is made to point coincide, then, before the introduction of the latter, the potential of the field was Fx+C.
V
Let a be the radius of the sphere, and let V and be the and inside of the sphere, then the conditions to
potentials outside
be satisfied are
:
(1)
V= V
(2)
V V= V
(3)
k-j
V
and
at the sphere's surface,
Fx + C
becomes
at
infinity ; 2
+
dv l
2
V=
everywhere
=0
at the surface.
dv
Now we know from forms Aas
t
or
Art. 107 that a potential
0,
To?
either of the
gives a normal force at any point on the
-y
sphere's surface proportional to
V2 V
;
provided
a?,
that in the
and
satisfies
case of the
the condition
Bx
form
being
chosen, the point is not infinitely near to the centre. If therefore we make of the form
V
and make
V
of the form
satisfied identically,
and
(3)
Bx+C, we
(3)gives
.Bx = _*_
is
A
have condition
(2)
and B.
= Bx+C = 7 = Ax+C at the
_
ZFx
therefore the potential
the sphere
shall
shall be able to satisfy conditions (1)
by properly determining
(1) gives 7'
And
and
+
_Ax-, 2
V
w
or
surface, or
B=
A.
3^T
= ,!=__
throughout the region external to
given by the equation
and the potential
V within the sphere 3Fx
is
given by the equation
SPECIFIC INDUCTIVE CAPACITY.
200
The density of so-called apparent is given by the equation
,
that
is.
[195
a.
electrification at the surface
= -3F k-lx =
4ir
k+2a
n
If k become infinitely great, becomes
Tjl >
a
4TT
as already
determined for a conducting sphere in a uniform field. obtained above are obviously those outThe potentials Fand
V
side
and inside of the surface due
to the superficial electrification
BF k-l k+2 "
x '
ITT
together with the potential of the this
potentials of
kl Fx k+2 That
is
field
Fx+C.
If
we
subtract
V
V
and respectively, it appears that the within and without the surface respectively are
from
latter
~a
kl F a?x^
and
s
k+ 2
respectively.
1
to say, a superficial electrification
px
over a sphere's
surface produces potentials
4 7f
.
-pa*,
and
4 a*x p TT
,
within and without the sphere respectively, and therefore gives a uniform field of force within the sphere. 1950.]
We proceed next to
medium not
consider the more general case of a
necessarily isotropic.
In that case the quantity of electricity of the induced distributions on the little conductors intersected by the kl face of the parallelepiped above mentioned, which lies to the right of that face, depends generally on the forces T and Z> as well as X.
By the principle of superposition it must consist of three portions and Let it be denoted by y proportional to X, respectively.
Y
In
like
Z
manner the quantity of
electricity
distributions on the conductors intersected
of
the
induced
by the hi and
hk
Of THf
SPECIFIC INDUCTIVE CAPACITY.
196.]
faces of the parallelepiped, which lies on the positive side of these faces respectively, may be denoted by
where
=2
xdS
j j
=2
d8
=2
the electric density at any point of a small conductor, the double integration extending over the surface of each conductor and the summation 2 extending over the conductors in
$ being
unit volume, as described above, Art. 190. We shall generally omit 2 for the sake of brevity in cases where there is not likely to be
any doubt
as to the meaning.
196.] Again, the induced electrification $ at any point on any conductor consists, by the principle of superposition, of three portions proportional to X, J, and
Z respectively.
We will
denote by X$ x the density of the electrification due to the force X, Y$ y that due to and Z$ z that due to Z, so that
Y
t
the whole density at any point on the surface of any conductor be
We
^ = ^ x+ QxyY+ Qx&
have then
the integrals being over It follows that
Similarly
we
all
conductors in unit volume.
shall obtain,
Q >
l,
x
v
Y+Qy
dS+
,
ry ^
dS +
zy
.
dS,
SPECIFIC INDUCTIVE CAPACITY.
202
[197.
and therefore
&c.
The
= &c.
properties of the coefficients
&c. were investigated
ffx9 d8,
=
Q xx Q xv
in
,
Chap.
,
&c. or
X;
and
as
there
Q yx &c. proved, Q xv If I, m, n be the direction-cosines to the normal to any plane drawn in the medium, the quantity of electricity on the little ,
conductors intersected by unit area of that plane which on the positive side of it is l
Z will
medium be
isotropic, as
above defined, the forces
lies
Y and
not affect the quantity of electricity to the right of the
In
plane kl.
ffx
(f>
y
fact, in
dS
And
=
0,
an isotropic medium
Uy
x ifc dS
z
= 0,
dS
=y
<\>
v
dS
and
ifz
=
z
x
dS
=
0.
$ z dS.
In that case
and
JT
= + 1
197.] If the
medium be not
4-7T
isotropic, the integrals
/ /
x $ y dS,
&c. are not generally zero for all directions of the coordinate axes. But it was shewn in Art. 179 that for any system of conductors in a field of constant force there exist three directions at right angles to each other, such that if these be taken for axes, each
of the integrals \\x (j> y dS, &c.
is zero.
SPECIFIC INDUCTIVE CAPACITY.
197.]
203
We may therefore describe a small sphere about any point in our medium, and find three perpendicular directions such that, if these be taken for axes, each of these integrals vanishes, if taken And we may
throughout the sphere.
call these three directions
the principal axes of electric polarisation at the point in question. With these directions for axes we have
=
(T
=
<*
But
xj
will not be generally true that
it
(T
y
,
and
respectively,
o-
g
we
being in this case proportional to X, Y, and
Z
will write
X'=
and
r= Z = f
And we may
'
write
Kv =
These ratios
K K K x,
y,
z
have a determinate value at every
point in a heterotropic medium, but may vary from point to Also the directions of the principal axes may vary from point.
point to point. and If x y
K K
K
and the directions of the & be constant, the axes also medium, whether isotropic or constant/ principal An If be otherwise, heterogeneous. not, is said to homogeneous. ,
isotropic
>
medium
is
therefore
homogeneous
if
K be constant.
SPECIFIC INDUCTIVE CAPACITY.
204
K
K
the same reasoning by which z vary continuously, x) K y) an isotropic medium we obtained the equation d /r^dV. d , T7 dV. d T,dV.
If in
[198.
r
,
+ --(Kv v dx (K--) dx ' dy dy
) '
leads, in case of a heterotropic
+ 4irp = (K) dz
+ dz
v
'
medium
if
the coordinate axes be
the principal axes, to
d cfcc
dV.
.
x (K v
(fcc
) '
+
d
,
d
_
(K v y
dy
)' 4-
K
-jdz
.
^ dV^
4 w/> s (K v ) + '
= 0.
K
constants for Also if j5Tla; Z1J/5 JTlg and 2x 2y , K2z be the two heterotropic media separated by a plane whose direction cosines are I, m, n, we have at the surface of separation ,
,
corresponding to equation (2) of Art. 189. Let X' be the average force which would exist on the plane kl if all the intersected conductors were removed. Then., it follows from Art. 191,
equations
that
X'
is
connected with
X
by the
X = = X-\- TtQx X if the axes be the principal axes, = KX X,
so that
X'-K
~X~ KX
or
>
K
x is the ratio between the average force which would exist on the plane Icl if all the intersected conductors were removed and the average force which does exist over that plane in the
medium. 198.] As shown in Art. 180, the energy of the unit of volume is
medium per
-\ and the
little conductors, if each free to rotate on any axis, will so use their freedom as to make the expression within brackets the
1
SPECIFIC INDUCTIVE CAPACITY.
99.]
205
greatest possible, by placing themselves in suitable positions ; that is, they will endeavour so to place themselves as that the axis of greatest polarisation shall coincide with the resultant force ; so that, for instance, if
the conductors will so place themselves as that the principal x shall coincide with the direction of the force.
axis
If they be perfectly free to move, this object will be effected any direction of the resultant force ; and as in that case there
for
no polarisation in any direction at right angles to the
will be force,
the expressions
are zero. isotropic
But
Such a medium medium.
will
then have
all
the properties of an
unless the conductors be perfectly free to move, or are medium will in general be heterotropic.
spheres, the
199.] It appears from the preceding that the numerical value of the dielectric constant in any isotropic medium must depend
K
upon the form and density of distribution of the small conductors within the medium. Suppose now that these are spherical, and that A is the fraction any volume within the medium which is occupied by the whole of the small conductors within that volume. of
Suppose also that the average force within the medium in the neighbourhood of any point is X, parallel to the axis of x. Since the force within each conductor
is zero, it
follows that
the average force in non-conducting space in the neighbourhood of the point in question
Now
must be
1
X
.
A
the electrical distribution on the surface of each sphere force arising from it within the sphere,
must be such that the
together with that from all the other electricity in the field, shall be zero throughout that sphere. If the spherical distribution were very rare it is clear that the force arising
from
all
the electrical distributions except that in
SPECIFIC INDUCTIVE CAPACITY.
206
[2OO.
any one sphere must be sensibly constant throughout that In other words, the sphere is situated in a field of sphere.
jr
constant force
-
J.
A.
'
parallel to the axis of x.
Therefore the polarisation of any particular sphere must be
-4-77
>
where v
is
A
1
Hence the
the volume of the sphere; see Art. 182.
polarisation per unit of volume,
3A above by QX,
is
X--
1
which we denoted
and therefore q\ 1
or, as
A.
is
supposed very small,
K
1
A
+ 3A.
This amounts in
fact to regarding each sphere as polarised independently of the If A be not very small, so that we have to consider the rest.
mutual influences of the spheres, the reasoning
is
precarious and
cannot be insisted upon.
We may also
construct a composite medium, portions of which shall consist of a dielectric whose constant is l9 and other
200.]
K
portions consist of a dielectric whose constant
a
medium be uniform throughout any
distribution inter se of the
two
is
K
2
space, that
dielectrics be the
.
is,
same
If such if
the
for unit
of volume in any part of the space in question, the problem presents itself for consideration, what is the average force in
such a composite medium due to the induced distributions within it ; or, as we may otherwise express it, what must be the value
K
for dielectric of K, in order that a uniform medium with the same effect as the have constant may composite medium.
solution of such a problem depends on the manner in which the two dielectrics are distributed inter se. If, for instance, the
The
K
medium 2 is in separate masses bounded by closed surfaces dispersed through the medium ly the solution of the problem will depend on the shape as well as the number and magnitude
K
of the separate masses.
"We might endeavour to determine the density of an induced distribution on those surfaces which, if they were filled with the dielectric lt would cause the normal forces on opposite sides of
K
SPECIFIC INDUCTIVE CAPACITY.
2 CO.]
207
rr
the surfaces to bear to each other the ratio -=?
by any other method, we could
find the value of
composite medium, the value of
K
at once
is
If
/ /
by
x
known
that, or
ds for the
to be
rr
JJ The problem
substantially the same as that of the determination of the dielectric constant in a single medium. If, for is
K
be contained in spheres, and they be 2 from each other that their mutual influence may be neglected, and the whole system be regarded as placed in a field of uniform force X, it will be found that the density of the
instance, the dielectric so distant
induced distribution upon them which causes the normal forces within and without the spheres to have the required ratio is
3X cos
proportional to
0,
X
being the external
force,
and
0,
as
angle between the radius of any point on a sphere
before, the
and the direction of #. Let the density be a
= n x 3X cos 0, 47T
where n
is
n X within any sphere.
force
This distribution gives a Consequently the normal force
a ratio to be determined.
within any sphere
is
(l-n)XcoaO. The normal
We
force outside of a sphere is
have then by the condition respecting the forces
KI (1 and
from which
or,
K
X being very
small,
+ 2 n)
= A*
2
(1
n),
CHAPTER
XII.
THE ELECTRIC CURRENT. ARTICLE 201.] HITHERTO we have been engaged in the development of the so-called two-fluid theory of electricity in its application to Electrostatics, or the conditions, on that theory, of the permanence of any electric distribution, one essential condition being that the potential shall have the same value at every point in a conductor. far in
The
results arrived at are so
agreement with experiment as to justify the acceptance
of this theory as a formal explanation of electrical phenomena. now proceed to consider how far the theory can be adapted
We
the explanation of observed phenomena in another class of cases, those namely in which different regions of the same conto
ducting substance are maintained by any means at unequal potentials.
Suppose, for example, that two balls of any given metal and at the same temperature, originally at different potentials, are held in insulating supports, and connected together by a wire of the same metal
;
then
it
is
found that after an interval of
time inappreciably small, the potentials are reduced to an equality at all points of the conductor thus formed of the balls and wire,
and that the
charge on the ball of higher potential has been diminished, and that on the ball of lower potential has been inWith the conception and language of the two-fluid creased. total
theory there has been in this short interval a flow of positive electricity in the one direction along the wire, or of negative electricity in the
opposite direction, or both such flows have
taken place simultaneously. If a magnetic needle be suspended near to the wire, a slight transitory deflection of this needle may be observed during the process of equalisation of potentials,
and
it
might be
possible
THE ELECTRIC CURRENT.
203-] with a
sufficient
209
length of wire and apparatus
of sufficient
delicacy to detect a slight rise of temperature in the wire.
202.] Methods exist whereby the inequality of potentials in different parts of a conductor may be restored as fast as it is destroyed, and in such cases certain properties are manifested in the conductor and its neighbourhood so long as this inequality is
maintained.
For instance, if the conductor be very small in two of its dimensions in comparison with the third, in ordinary language a wire, the deflection of the needle is no longer transitory but persistent, so long as the inequality of potentials is maintained, the
amount of such
deflection
depending upon the amount of
the inequality, and the dimensions and constitution ef the wire heat also continues to be generated in the wire at a rate depend;
ing upon the same circumstances. Also if the wire be severed at any point, and the severed ends connected with a composite conducting liquid, thus forming a heterogeneous conductor of wire and liquid, chemical decomposition of the liquid will ensue at a rate dependent on the difference of potentials, and the nature of the wire and liquid.
According to the two-fluid theory, there must be under the given circumstances a permanent flow of one or both electricities
between the unequal potential regions, of a transitory flow spoken of above, and the wire
like nature to the is,
in the language
Of of that theory, spoken of as the seat of an electric current. course the existence of such a current is as purely hypothetical transference of some as that of the electric fluids themselves.
A
kind there must
be, for it is indicated
by the
respective gain
and loss of electrification in the two connected conductors, but whether that transference be a material transfer as implied two-fluid theory, or a formal transfer like a wave, or the transmission of force as in the case of a tension or thrust,
by the
The current, as it is are not in a position to determine. a as must be phenomenon by itself, called regarded designated,
we
into existence under certain conditions,
investigated
by
203.] The VOL. i.
and subject to laws
to be
independent observation.
question indeed
might P
arise,
how
far
are
we
THE ELECTRIC CURRENT.
210
[204.
warranted in regarding current phenomena as indicating the absence of electrical equilibrium ? When parts of a conductor are maintained at permanently unequal but constant potentials, a certain state of the field ensues,
which
is
also
permanent, and
it
might be
said that
we have
in such a case a system in equilibrium although not To satisfying the conditions required by the two-fluid theory.
can only be replied, that in the case of electrostatical equilibrium we have a system permanent of itself whereas in a constant current the permanence always necessitates an expendi-
this
it
;
The former case ture of energy from some external source. resembles the mechanical equilibrium of a heavy body on a horiThe permanence of the latter case resembles that zontal plane. of a heavy body dragged uniformly up an inclined plane, and requiring at each point of
its
course the expenditure of external
work.
Laws of the Steady Current
in a Single
Metal at Uniform
Temperature.
204.] (l) The intensity of the current is the same at every point. We have mentioned certain physical manifestations accom-
panying the current, viz. thermal, chemical, and magnetic. These are capable of measurement and it is reasonable to re;
gard these measurable
effects as exhibited in the
neighbourhood
of different portions of the current as giving a measure of the It is found experiintensity of the current in those portions. mentally that in the case of a steady current these effects are
the same throughout.
pended
assuming
stances to be the
its
Wherever the magnetic needle
is
sus-
distance from the wire and other circum-
same
the same deflection results.
If the
wire be of equal section in every part, then equal portions are heated at the same rate, and in whatever portion of the wire the liquid conductor above described action also takes place at the same rate.
This law
is
introduced, chemical
is evidently consistent with the two-fluid theory, according to which we regard the current as a flow of either fluid across any transverse section of the conducting wire.
THE ELECTRIC CURRENT.
205-]
211
Ohm's Law.
(2)
This law, which is universally accepted, asserts that If a uniform current be maintained in a homogeneous wire whose surface intensity
of
is completely enveloped by insulating matter, the the current in the wire is directly proportional to the
electromotive force
and
the difference ofpotentials at its extremities), inversely proportional to the resistance of the wire; the (i.
e.
mathematical expression of the law being 7 current intensity
',
E is the
The quantity here
T?
=
electromotive force,
where I
-
s
JK
and
R
the
is
the resistance.
the resistance depends upon the and the wire and upon the material transverse of section length of which it is composed. For wires of the same substance it is
called
proportional to the length directly and the transverse section and Ohm's law asserts that if through a wire the
W
inversely,
electromotive force
wire
W
E produces
a current
I,
the electromotive force E' produces a current
the fractions
E -=-
and
E' -~r
will always bear the
another so long as the same wires ^Tand
R
and through another
be the resistance,
-^J\i
is
W
same
may
then
ratio to
one
are employed.
If
called the conductivity,
be denoted by K, then Ohm's law
/',
and
if this
be expressed in the form
that no trans205.] If the insulation of the wire is perfect, so ference of electricity can take place across its surface, the direction of transference at each point must, in the permanent state, be but this direction parallel to the axis of the wire at that point, also perpendicular to the equipotential surface through The wire is in fact a tube of force, and if it be of that point 1 .
must be
uniform section the resistance through each element of length ds is
proportional to ds, and therefore
we have
i <x
,
or
j^
s
1 This coincidence of direction of the electromotive force at any point and of the current through that point does not hold good in the case of anisotropic substances. At present and hereafter, in the absence of special notice to the contrary, it must be understood that we are confining ourselves to the consideration of isotropic
substances.
P 2
THE ELECTRIC CURRENT.
212
[206.
Hence, by Art. 102, there can he no free within the substance of the wire. any point This condition is satisfied according to the most generally
constant at each point. electricity at
accepted theory by regarding the electric current as consisting of equal quantities of positive and negative electricity flowing in opposite directions. In this case the potential F, at any point distant s from a fixed point in the wire's axis, must be given by the equation
V = Rs+C,
and i.e.
R
is the constant resistance per unit where length ; the axis be a straight line parallel tc x we have Rx + C, the potential is that of a field of uniform force.
F=
if
If the wire be not of uniform section, then the resistance
along a portion of length ds along the axis is, by Ohm's law, proportional to ds directly, and to dS the transverse section inversely ; hence
we have ice
-jdS,
dS is
or
out, proving (by Art. 102) that there within the substance of the wire.
is still
constant through-
no
fr^e electricity
Ohm's law may be generalised as homogeneous isotropic conductors with all dimensions finite. For, from what has been said above, it
206.]
The statement
of
follows for all forms of
their
follows that the current flows from one elementary region to another of such conductors along elementary tubes of force. If
we regard such
tubes as wires in which the current obeys
Ohm's
law, this leads us to the equation for the intensity of the current over the elementary area dS of the equipotential surface through
any point
dV i oc
70
-r- dS.
ds
dV or
-=-
_
as
.
is
constant.
ds
An
equation which, as before stated, proves that there is no free electricity within the substance of any conductor through which If the conductivity be variable a permanent current is passing. and be denoted by K, this becomes
may here be interesting to prove the following proas an illustration of the resistance of a conductor, position, 207.] It
THE ELECTRIC CURRENT.
208.]
213
namely If one electrode of a conductor be in communication with the earth, and the other with a conducting sphere charged originally with any amount of electricity, then the resistance of the conductor is the reciprocal of the velocity with which the radius of the sphere
must diminish, in order that the potential of the
may remain
constant, notwithstanding the loss of electhrough the conductor *. Let the initial radius of the sphere be #, and the mass of its initial charge be then the original potential V will ;
sphere
tricity
M
M
i,
be
a
M
dM
and da be the simultaneous small decrements of and a in the small time dt, then, since by hypothesis V is constant, we must have If
~
M _dM ~ da
a
But
if
R
be the Resistance of the conductor,
we have
.
K
Multiplying these equations together we get
"
\_
dt_
Bda 1 da_ ~-' On
208.] (a)
the Value
A series
of the Resistance in particular Cases.
of wires of the
same material but
different trans-
verse sections joined together in series end to end. &c.
A3
A2
Al
At
Fig. 30.
Let the
wires,
n in number, be
A^A^ A 2 A 3 AZ A^ ,
H R3
the resistances in these wires be RL, 2 tials at the points A lt A2 A B9 &c. be ,
1
The
proposition
is
,
,
7lf 72 73 ,
&c.
Let
Let the poten-
&c. ,
&o.
And
let
taken from Mascart and Joubert's Electricity and
THE ELECTRIC CURRENT.
214
[208.
the intensity of the current, which must be the same in each wire, be i.
Then, by the continuity of the current and Ohm's law, we have
_ Fv l~ F _ F2~ F '
'
2
'
R,
R,
If therefore
R be the
(/5)
The
3
_
fc
FF n~ '
r
FF
_ _ v ~ ~R,+ R
n+i
n+1
I
Rn
z
.
resistance in this case,
resistance in the case of a multiple arc conductor.
The conductor is called a multiple arc when it is formed, as in the figure, of a number of separate wires branching off from A,
Fig. 31.
the extremity of one wire a A, and converging to the extremity of another wire Bb at B. Let F Fb be the potentials at A ,
and B respectively. In this case the electromotive force in each separate transit If R^ R2 &c. be the from A to B is the same, viz. Fa Fb .
,
resistances of the wires, %, them, and i the current in ,
if j5
2
i2
,
&c. the currents flowing through
A a, we
= R = 2
&c.
have
= in Rn = Fa- F
6
R
The current flowing ,
.
equal to
JL RI in
_L
J_
R
Rn
2
any particular
wire, as for instance
i
v
iE ^H \
If
,
be the resistance sought, and therefore
JL
is
i
we denote the
conductivities
by
K
: ...
K
2 ...
^
n,
we have
It follows from (a) that the resistance in a wire of given
THE ELECTRIC CURRENT.
210.]
215
uniform section varies directly as the length of the wire, and it follows from ((3) that the resistance in a conductor consisting of any number (n) of similar and equal wires placed side by side is
- th of the resistance of any one of the wires taken singly, % hence the resistance of a wire of given substance and given length varies inversely as the area of the transverse section, or generally the resistance in a wire of given material varies as the length directly and the transverse section inversely, as
already stated. 209.] If we have a homogeneous wire of which the area of a transverse section at distance s from one end is /(), the resistance
per unit of length at the same point rs
of the portion s of the wire
is
/
JQ
is
.
/W
,
and the resistance
$s
/W
.
Hence,
F
if
F
be the*
a at distance potential at the end in question, the potential i flows from that end, is found from
s,
when a current
Currents of much greater complexity may occur in practice and are of great importance. We shall later on investigate a more general case of a system of wires traversed by electric currents.
of wires, 210.] When the conductor is not a wire, or collection seen that have but a continuous conducting substance, we
Ohm's law may be expressed by the equation
**?? ds
the intensity of current which traverses an equithe neighbourhood of any point potential elementary area dS in is an element of the line of force at which the potential is F", ds
where
*
is
K
is a constant at all points in the subthrough the point, and stance depending on the nature of its material. When any given regions of such a conductor are kept at uniform given unequal potentials, a permanent current state
THE ELECTRIC CURRENT.
216
[211.
soon established the given equipotential regions are in such a case generally termed electrodes, and sometimes sources or sinks of electricity, according to the direction of the current flow from
is
;
or towards them.
When sink,
these electrodes are two in number, one source and one as in the case of a wire or wires, determine a value
we may,
of the ratio of electromotive force to current intensity which will remain constant so long as the substance and position of
the electrodes
constant, and this ratio
is
is
spoken of as the
resistance of the system ; the electromotive force is the difference of the constant potentials of the source and sink, and the
current intensity is measured by the rate of transference from source to sink per unit of time. 211.] As a particular example let us take an infinitely extended and very thin conducting plate, bounded by parallel planes and pierced by two cylinders P and Q which are maintained at
given constant potentials. If the mean plane of the plate be that of x, y> and V be the potential at any point, the conditions that there shall be no electricity within the plate, and that the equipotential surfaces are all normal to the plate, lead to the equations
free
V F=0,
^=0. dz
2
Hence the problem may be treated as one in two dimensions and the electrodes may be regarded as circles with radii
only,
equal to those of the cylinders let these radii be a and l > and let the constant potentials be 9 and Vq respectively. ;
V
The equation
in F, or
r
dx*
may
be
satisfied
F=C +4
~ df
by assuming
r
1
logr,
+4
2
logr2 +
&c., or
C+2A logr,
where the quantities r l9 r2 &c. are the distances of the point x, y from any assumed fixed points, and these points must be so ,
taken that
V
is
equal to
Vp
and
ferences of the circular electrodes.
Vq
respectively at the circum-
THE ELECTRIC CURRENT.
211.]
Let Op and O q be two points within the that each potential r^
and
the image of the other in
is
V at
any point be taken equal
spectively, then
Vp
the
all
C and A
>
where
r2
of the point from O and O rep q required conditions will be fulfilled,
be taken to satisfy the conditions
= CA log f
rz
Vq = C A inasmuch
C A log -
the distances
r2 are
provided
to
P
and Q such circle, and let the
circles
own
its
217
as
is
log
r2
at the circumference of P,
at the circumference of Q,
constant over each of these circumferences.
**2
Since
V
is
constant whenever
is so,
r2
it
follows that the
equipotential curves are circles each one of which is conjugate and Q. The orthogonal trajectories of to the centres of
P
such
circles, or
the lines of current flow, are circular arcs each
passing through these centres, and,
if
da
and
circumferences of these circles respectively, current in unit time in terms of
Vp
V
q
be found at the do
we can
find the
in the form of
R
V
whole
^=
V
is then called the resistance of the the quantity system, reciprocal being the conductivity.
-
;
its
In the particular case of the radii a and 6 being equal, and very small compared with ./, the distance between the centres, we find from the above equations each
A
Let
i
= \ V~ V
"
be the current in unit time over the arc ds of the
circumference of the
P
electrode, then, since r2
stant and the direction v of the line of flow
ofP,
is
.
i
T
,
sensibly con-
along the radius
AK ,dV -K^ -ds=K-^dV T~ds - --ds, a dr dv T
i
is
218
SYSTEMS OF LTNEAE CONDUCTORS.
[212.
K
where
is the conductivity of a unit length of a prism of the conductor of unit breadth. Thereforev the total current in unit
time over the It
is
plate,
P
and
is 2-nai, or 2irKA. proportional to the thickness 8 of the write Kb, the current per unit time will be
circumference
K
clear that
we
if for it
is
K is now the conductivity through
where
whose edge
Writing
is
the unit of length.
for
A
and the resistance of the system
is
V~
2 log3
i-
,
we get the
a
log
On Systems of Linear
A
V
the value already found -
current per unit time equal to
212.]
a cube of the substance
Conductors.
two of whose dimensions are very small
conductor,
compared with the third, as for instance a wire,
is
called a linear
conductor.
We
have had occasion to consider certain properties of linear Firstly, we have seen that if such a conductor be
conductors.
divided into several parts through which a current flows conC, &c., the resistance of the whole is the secutively, as
A,
sum
of the separate resistances of the several parts. Hence, in case of a homogeneous conductor at uniform temperature, if the
known we can determine the when a current is flowing.
potentials at the ends are at any intermediate point
For instance, extremities are
let
Va
let the resistance of
rpb .
Then
if i
APB
and
7b
.
potential
be a wire the potentials of whose Let P be an intermediate point, and
the portion
be the current,
AP
be rap and that of ,
PB
be
SYSTEMS OF LINEAR CONDUCTORS.
2 1 3.]
219
Hence
which determines Similarly, if i
Fp
.
we
be given, but the potentials are not given,
can determine the differences of potential Fa Fp and Va Vb Again, in case of two or more wires connected in multiple arc, .
we have shown
that
if
Vm Vb
be the potentials of the extremities
K
the currents in the several wires are respectively Vb ] l (Va where the conductivities of the &c. are (V a &c., z v K^ F&),
K
J
K
And we
wires.
and
sum It
can therefore determine
all
b are given, or the difference of potentials of the currents is given.
assumed that the wires are
is
all
Va
the currents if
T
Fa
F
b,
if
the
of the same metal, and at
uniform temperature. 213.]
The points of junction of the wires are
called the electrodes.
In the above simple case we have only two electrodes. But we may conceive a system of wires meeting in more than one point.
For instance, to take a case a little more complicated, APB, AQB, and the
let there
be two wires
^B
two intermediate points P and Q connected by a third wire. If the potentials at given,
P and Let
A
and
we may determine
B
are
those at
Q, as follows.
K K Kpq be the conductivities pb ,
ap ,
PB, PQ. must be
Then, since the
zero,
sum
of the three wires
AP,
of the currents flowing from
P
we have (r -7,)
= o.
(F,-
= 0,
(I
Similarly,
*.. (F.- F.J + Z,,
(F- F.) + *M
F.)
K
from which, the conductivities being known, the two unknown and thence the curV and be F can determined; potentials, p q ,
rents are
known.
If instead of the potentials, the current at
A
and leaving
it
at B, be given,
we have
entering the system three linear equations
C,
SYSTEMS OF LINEAR CONDUCTORS.
220
to determine the differences of potential
Va
f^i,
[214.
^o~ ^p
Va -Vq namely, C = Kap (Va - Vp) + Kaq (Va - Vq\ ,
The points and a current 214.]
The
P
and
Q
will generally
case in
which
be at different potentials,
PQ QP. P and Q happen or
will pass along
to be at the
same
of special importance. In that case no current passes potential in PQ, and the potentials at every point in either wire are the is
same as
if
P and
there were no metallic connexion between
Q.
Tkali "
VA K^
(V'
(
A
current will pass in one or other direction along and Q are at the same potential, that is, unless
unless
PQ,
P
This principle resistance.
is
made use of in instruments
Suppose, for instance,
AX is
for
measuring
a wire whose resistance
BX
be a conrax is required. Let ductor whose resistance rxb is known. Place
AX
conductor
and
XB
so as to form one
AXB. Let AEB
be a uni-
form wire, E a point in it. If E and be joined by a wire, a current will pass along it in one or
X
JB
other direction, unless the potential at is the same as at E.
X
We
increase or diminish the distance of
E
from
EX
A
until a
needle suspended near shows no deflection when an electric current is made to pass from A to B. Then we know that the potential at
X is the same as that at E; AE
and therefore
SYSTEMS OF LINEAR CONDUCTORS.
2 1 5.]
which determines
known
rax
This
.
is
221
the principle of the instrument
as Wheatstone's Bridge.
215.] In a more general case, there may be n points, or elecby wires of known conductivities.
trodes, connected each to each
Vl
Let
.
.
Vn
.
be the
potentials
at
the
several
electrodes,
the currents which enter the system from without at p these electrodes respectively, taken as negative when a positive (?
c2 ,
...
cn
Then the current
current leaves the system.
and we have
for the electrodes
P
and
in
AB is
Q
= K p (V,- Va + K, f (Vp - Fi) + &c.) = *(?,- r.)+r. (v,- r)+&4 ,
op
and
so
on
Now
)
'
for each electrode.
since
no
electricity
can be generated or destroyed within
the system, the sum of the currents entering the system at the electrodes must be zero. That is, e1
Therefore only
1
ft
+ c2 +...+on =
of the
c's
all
0.
are independent.
we
are only concerned with the differences of the are n i independent quantities of the form there potentials, Also, since
A
_
we have n
_
independent linear equations of the form between the independent quantities. subsisting 1 of the quantities be given, the equations If therefore any n
In
all
1
2n2
suffice to
determine the others.
For example,
if
the entering
1 of the electrodes, we can detercurrents c be given at any n mine all the differences of potential. And if all the differences of potential are given we can determine the currents.
If
we
differentiate the equation
A
for
any
electrode, as P,
we
obtain Similarly, differentiating the equation for
Since
Kap = Kpa
,
it
A we
obtain
P
due to the follows that the potential at A is equal to the potential at A
introduction of unit current at
due to the introduction of unit current at P, and so on.
222 GENEKATION OF HEAT BY ELECTRIC CURRENTS. On
in
the Generation
AB
VA
and VB
1
6.
of Heat ly Electric Currents.
216.] Suppose a uniform current of intensity of resistance R, a linear conductor
potentials
\2
/ to be
existing
with
terminal
.
a transference, per unit time, of electricity / from the extremity A to the extremity of B. Now if elt 2 &c. be the charges upon a system of conductors AH A 2) &c., and if Tl} Vz , &c. be the corresponding potentials and
There
is
,
W the
electric
energy of the system, we have proved that '
de
now under
In the case
consideration, the charge at the extremity A of the conductor, where the potential is TA) is diminished by Idt in the time dt, and that at the extremity J9,
where the potential is FB) is correspondingly increased by the same quantity. Hence, since VA is greater than VB there is by ,
the process a diminution of the electric energy of the system in
time dt equal to
(VA
But by Ohm's
law,
-VB)Idt.
we have
R Therefore the diminution of electric energy, owing to the existence of the current, in the time dt
= RPdt.
K This
is
is
the work done ly the electrical forces in the
field in
time dt in the passage of the current 7 through the conductor,
and this work done, or
electric
heat evolved in the conductor
energy
AB in
lost,,
must reappear
in
the same time.
If therefore / represent the Joule heat- equivalent, the heat evolved per unit time will be
RI*
~J" Joule was the
to prove by direct experiment that the rate of evolution of heat in any wire through which a current passes first
2l8.]
GENEKATION OF HEAT BY ELECTRIC CURRENTS. 223
proportional to the square of the intensity of the current,
is
and we now see that this result follows directly from Ohm's law and the principle of the conservation of energy. 217.] If the current, C, having been generated in a system, be allowed to decay by the resistance R, the value of the current at time t after the commencement is C*~ Rt Hence the total .
when
quantity of heat generated
the current has ceased
is
*+**& = 1C*.
&C* */o
For
this reason
\C Z
is
sometimes called the energy of the
current.
It is supposed here that the current during this process is uninfluenced by any other current, or by any magnetic field, as
we
shall see later that electric currents in the
same
field
exert
mutual action on each other.
On
the Generation of
Heat
in a System of Linear Conductors.
218.] In the simple case of a number of wires in multiple are arc, we have seen that R^C^ = -S2 2 = &c., where Clt 2 &c. the currents, and R lt R^ &c. the resistances in the respective ,
wires.
If the total current
Cl + C2 -f &c.,
distribution of the current
among
or 2 C be given, this is the the several wires which makes y
the heat generated per unit of time a minimum. For *2,R Z C is the heat generated, and the condition that this should be mini-
given 2 C is that R C = R2 C2 = &c. The same property can be proved (Maxwell's Electricity and Magnetism, 283) for the more general system, provided there be no internal electromotive forces.
mum
For
let
Cay Cb)
&c. be the given currents entering a system of Let Cps be the
linear conductors at the electrodes A, H, &c.
current in any wire PS determined according to the process above described, so that or
cps = (vp -r,)Kpi Vp -V, = Rpt Cpt
Ohm's law by
,
.
Let us next suppose the same
total current constrained to flow
through the system according to any other
mode
of distribution,
224
ELECTROMOTIVE FOBCE OF CONTACT.
without however altering the or to
any
Cp8 + Xps
of the currents flowing from current in PS be
Let,* for instance, the
electrode.
instead of
sum
[219.
Cps
.
Then the heat generated Ohm's law is 2.RC 2
in the distribution according to
.
And
the heat generated in the constrained distribution
But
for each electrode A> JB,
is
or
currents
is
&c. the
sum
of the entering
unaltered.
Hence, for any electrode as A,
2Xa = 0.
or
2^RCX=Q.
Hence
Therefore the heat generated per unit of time in the con2 2 in is , and exceeds that generated 2 the original system by the essentially positive quantity 2
2KC +2RX
strained system
EX
.
Electromotive Force of Contact.
219.] Up to this point we have introduced the restriction that the conductors with which we are concerned shall be of the same
substanee throughout.
The reason
of this restriction, which in
strictness is equally required in electrostatic investigations, will
now
be considered.
Volta believed that when two different metals were placed in contact, the potential of one of them was always higher than that of the other, and this without any disturbance of electric In fact, that instead of the condition of electric equilibrium.
=
Constant throughout equilibrium being V the condition should really be, ducting space,
ing space
V
is
composed of substances of
C19 V =. C2 V =.
all
continuous con-
when such conductdifferent
materials,
&c., in the regions occupied by these substances respectively; the values of the constants Clt C2 (?3 &c. being dependent upon the nature of the substances, and the ,
(?3
,
,
electric distribution in the field
,
subject only to this restriction, that in every case of electrostatic equilibrium of a compound ;
ELECTROMOTIVE FORCE OF CONTACT.
221.]
225
Cr Cg of the constant potentials of any two given substances should always be the same at the same conductor the difference
temperature. 220.] For instance, if a zinc wire and a copper wire were held by insulating- supports, and brought into contact at one end of each,
the potential of each wire would be the same throughout, but that of the zinc would exceed that of the copper by a quantity
always the same for the same temperature.
If platinum were
substituted for copper a similar result would be observed, but the difference of potentials (the temperature being the same as before)
would be
If platinum, and copper were similarly connected, less. the platinum would stand at the higher potential, and the constancy of temperature being still maintained, it would be found that the excess of potential of zinc over copper in the first case,
supposed above, was equal to the
sum
of the excesses of the
potentials of zinc over platinum and platinum over copper in the two last cases. This difference of potentials is generally called the electromotive contact forces of the two metals, and is for
metals
A
and
B denoted
by A/B.
considered as positive if the metal of higher contact potential is placed before the line and negative if the reverse, so It
is
A/B + B/A = 0, and if there were three metals J, B> and C whose electromotive contact forces at any temperature were A/B for A and B and B/C for B and , then the electromotive contact force for A and C at the same temperature would be A/B -f B/C or in other words, for the same temperature we have that
;
the equations
A/B + B/C+C/A=0.
and
A
B
at any temperature stand at a higher potential than B, it is said to be electropositive with to be electronegative with regard to A. regard to j5, and with his followers regarded all metals as having Volta 221.]
If the metal
in contact with
B
certain specific affinities for the positive fluid, so that in cases of contact the electropositive metal of the pair becomes charged
A
similar effect is positively with reference to the other metal. on this view supposed to attend the contact of all conducting
VOL.
I.
Q
CUEEENTS THEOUGH CONDUCTORS.
226
[222.
bodies, whether metallic 01* non-metallic, solid or liquid, in the absence
In the case of composite liquid conductors of chemical action. chemical decomposition ensues on contact, and the electromotive is on such decomposition diminished, or reduced to the difference of potentials being, on this view, contact ; the absence of chemical action. dependent upon
contact force zero
According to the views entertained by other physicists, the dependent upon the medium in which the touching bodies are situated, and is in all cases the difference of potential at contact is
The
remit of chemical action in that medium.
latter hypothesis
by experiment*, and the as cannot be subject being yet thoroughly decided. regarded Meanwhile we may, without waiting for a solution of the difficulty, develop the laws of this electromotive force of contact, so
is
to a certain extent at least borne out
far as
they have been experimentally determined. Ohm's law as originally enunciated contemplates a wire of
222.]
homogeneous substance throughout. The laws of current intensity and of evolution of heat in the case of wires in series require modification when these wires are not of the same materials :
v a vb
V~
~m~
A
B
~F
2
Fig- 34-
For example, let there be two wires of metals A and B touching at m. Let the potentials of A at the free end and at m be Vv and Va9 and let those of B at the corresponding points be
Let Ra and R b be the resistances of the A and B wires Then by Ohm's respectively, and let i be the current intensity. law ?2 and
V
b.
Ra if
R
Ra + R b
b
F! be greater than
T
29
and
if
R
R be total resistance as
* See a Paper by Exner, Phil. Mag. vol. x. p. 280, and works there cited. third view, suggested by Professor Oliver J. Lodge, is that each metal in the absence of contact with another metal is at lower potential than the surrounding air by an amount depending on the heat developed in its oxidation, that on contact the potentials of the two metals become equal, the more oxidisable metal receiving a positive and the other a negative charge.
A
PELTIER EFFECT.
224-] previously defined,
the term
B/A
positive or negative according as negative with regard to A.
number
If there had been any &c., the equation
B
227
being, as above explained, electropositive or electro-
is
of wires of metals
A lt A2 A 3 ,
,
would have been
R So that Ohm's law might
still
be enunciated for such an arrangeYl 7J were
ment, provided the external electromotive force
increased by the electromotive contact forces at the respective junctions, regard being paid to the signs of these forces, and the resistance being the sum of the resistances in the respective wires.
223.] In the multiple arc arrangement with initial and final wires of metals A and _Z?, and connecting wires of metals m-^ m 2 ,
&c., if i be the current intensity in
wire
and
A
or B,
and
i
r
mr with resistance Er) V^ and 7J the potentials 7r and Vf of m r at the junctions, we have V V r ,
,
that in the of
A and
B,
,
r
V V
Rr
Rr
Rr >
and therefore
So that
same expression
in this case also the
results as in the
already investigated, provided the external electromotive force be increased by B/A.
homogeneous multiple
arc
in the case of 224.] The expression for the energy dissipated wires in series also requires to be modified when the wires are
not of the same metal throughout. If as in the last article there be two wires of metals
and the notation of that
The
A
and B,
be retained, we have energy per unit time as the current
article
total loss of electric
A Q2
passes from the free extremity of
to that of
B=
(J^
V^)i.
228
PELTIER EFFECT.
[225.
Therefore the whole heat generated must be
J But by the equation above obtained Therefore the heat generated
is
be positive the heat generated in the compound wire of resistance R by the passage of the current of that
is to say, if
B/A
Hi 2 intensity
i is less
than
=-
J
,
or
what
it
would have been had the
wire been homogeneous, by the quantity
than
by j/
'-
T
/
if
A/E
be positive
:
'
' ,
that
is
and
is
greater
to say,
when
a current in passing through a circuit of heterogeneous metal wires traverses a junction from an electronegative to an electropositive metal there is absorption of heat at the junction, and the contrary, there is evolution of heat in the passage from electropositive to an electronegative metal.
on an
225.] This absorption and evolution of heat at metal junctions first observed by Peltier, and the phenomenon is called after his
was
name it is physically analogous to the absorption and evolution of heat accompanying chemical dissociation and combination respectively, the electricity at the junction being raised to a ;
higher, or sinking to a lower potential in the respective cases, just as the chemical potential of the dissociated or combined elements is raised or depressed. The actual amount of heating or cooling as experimentally observed is always less than the theory requires, and in some cases is of the opposite sign ; indicating, apparently, that the whole electromotive contact force of Volta is not to be sought in the mere metallic contact, but in the action of the surrounding medium.
CHAPTER
XIII.
OF VOLTAIC AND THERMOELECTRIC CURRENTS. ARTICLE 226.] IF any number of wires of different metals are joined together in series, and are kept at the 2 1/3, l
3fl9
M
,
M
M
M!
M
z
M
3
l
Fig. 35-
M
same temperature throughout, the wire of metal l beginning and ending the series, it follows from the laws of contact action above stated that each wire is at the same potential throughout its length, and that the beginning and wires are also ending l at the same potentials, inasmuch as the sum of the electromotive
M
contact forces
M^M^ -f M2 /M3 + M /M 3
l
is
zero
be formed by joining the free ends of the If however
we
M
;
l
substitute for the
hence
if
a circuit
wires no current
wire a composite and thus conductor the L, circuit, the electroliquid complete motive contact forces L/MZ and L/M% are modified, the liquid L will ensue.
M^
being at the same time decomposed. According to the extreme views of the Volta contact theory, the last-mentioned electromotive forces disappear with the de-
M
M
at their 2 z and composition, the liquid L and the metals points of immersion in that liquid are reduced to the same potential, the
electromotive
contact forces Jf2/M"3
metallic junctions are no longer compensated
and
L/MZt
,
by the
&c. of the forces
and a current ensues through the wires and
L/M2
liquid.
Suppose, for instance, the liquid be dilute sulphuric acid and the metals be plates of zinc and copper partially immersed and having their unimmersed ends attached to platinum wires, so long as these platinum wires are not united to each other, the zinc, the copper, and the liquid stand, according to this theory, at the
same potential (V suppose), but the platinum wires attached to and the zinc and copper plates are at the potentials
VZ/P
VOLTAIC CURRENTS.
230
V+P/C
[227.
If now the platinum wires be united, can no longer be maintained, inasmuch as
respectively.
electric equilibrium
the two portions of the same platinum wire are now at potenHence a tials differing from each other by Z/P + P/C or Z/C. take must the of flow place through platinum wire electricity
from the copper to the zinc plate, raising the potential of the zinc and depressing that of the copper. The inequality of the potentials thus produced in these im-
mersed plates is again destroyed by the action of the liquid, which is at the same time decomposed, oxide of zinc being formed at the zinc plate, which is dissolved as soon as formed,
and hydrogen being given off at the copper plate, and thus a permanent current ensues in the closed circuit of copper, platinum, zinc, liquid, copper, and in the direction indicated by the Such an arrangement is called a Voltaic order of these words. current, the vessel containing the liquid and plates is called a Voltaic cell, the decomposable liquid is called an electrolyte^ and its decomposition on the passage of the current is called electroThe intermediate platinum wire is in no respect essential lysis. to the process, which would have equally taken place if the copper and zinc plates had been in immediate external contact with each other. 227.] According to the theory of the Voltaic circuit, above explained in outline, the potential rises discontinuously at the metallic junction or junctions outside the cell, and falls continu-
ously throughout the rest of the circuit ; the whole electromotive force of the current is sought for in the contact force at the junctions, the function of the chemical action in the cell being
limited to the continued equalisation of potentials within the cell as fast as the equality is destroyed by the electric flow.
According
to the chemical theory of the circuit,
which
is
now
more generally
accepted, a discontinuous change of potential takes place at the junctions between the metals and the liquid, those being the points at which, as we shall see presently,
energy for the maintenance of the current absorbed.
only
it is
The
known
is
in fact evolved or
true theory of the cell is not finally settled, that the chemical decomposition is an essential
VOLTAIC CURRENTS.
229.]
231
part of the phenomenon. It is possible however to develop certain fundamental laws of the action, which are essentially the same
whatever be the metals constituting the plates, and whatever be cell, provided it be capable of electrolysis.
the liquid in the
228.] The plates by which the current enters and leaves the cell are called electrodes, that by which it enters is called the anode,
and that by which it leaves is called the cathode, the two elements into which the electrolyte is decomposed are called the ions, the element appearing at the anode is called the anion, and that appearing at the cathode is called the cation.
Let the metal electrodes be called
P
and
N respectively, and
the two constituents, or ions, into which the liquid is resolved be called TT and v respectively. On the passage of the current the ion
TT
will appear at the electrode
electrode P.
The
(1)
and
Then
it is
ratio of the
N
t
and the ion
masses in which the two constituents
v appear at the electrodes is that of their
The absolute mass of each ion
(2)
time
v at the
found that TT
combining weights.
so deposited per unit of
proportional to the strength of the current, or in other each unit of positive electricity transmitted a certain for words, is
mass of each ion This
is
deposited at the corresponding electrode. called the electrochemical equivalent of that ion. is
So long as the electrolyte is the same, the ions into which decomposed are the same, whatever the metals constituting and the same ions appear at the anode and the electrodes cathode respectively. One or both of the ions may be com(3)
it is
;
pounds, and the same constituent which in one electrolyte becomes an anion, may in another electrolyte become a cation. (4)
The source whence the energy,
requisite for the maintenance
the arrangement of the elements of derived, the electrolyte and the immersed plates in a combination of lower chemical potential energy than that which existed anterior of the current,
is
is
to the current.
The
action of the typical cell described above of zinc and copper plates in diluted sulphuric acid may be supposed to be as follows. The chemical arrangement before the circuit was
229.]
completed was
Zn H^SOl
. . .
H SO^ 2
.
Cu
;
VOLTAIC CTJKKENTS.
232
and during the existence of the current
ZnSOv The and
zinc
the
first
zinc
...
[230.
it is
ff^Cu.
combines with the oxygen of the water (ff2 0), oxide is then replaced by the zinc sulphate
Zn S0, which being soluble leaves the zinc plate free for further The potential chemical energy of Zn S04 is less than action. that of 7/2 0, or, as more practically expressed, the heat evolved and Zn 0, S03 is greater than that by the combinations Zn required for the decomposition of the current energy. 230.]
A
feeble current
H 0, the difference furnishing 2
might have been obtained with water
cell, the chemical arrangements before and after the completion of the circuit being
only in
the
and
ZnO.^H^Cu
respectively. But in this case, since the oxide
Zn is insoluble in water, the zinc plate would soon, by its oxidation, become unfit for action, and the current would cease. may however use this
We
ideal case as
In
an example.
this case the ions
TI
and v are
and
H
2
respectively.
Taking
unity as the combining number for hydrogen, that of oxygen Therefore one gramme of zinc is 8, and that of zinc is 32-53.
-
takes in combination with oxygen the place of 32-53
grammes
g
of hydrogen, each combining with o &'5 o
,
or -246
gramme
of
oxygen. The heat evolved by the combination of one gramme The heat which would of zinc with the oxygen is 1310 units *. be evolved on the combination of -7^-^ O '53
gramme
of hydrogen
with the oxygen, and which is therefore absorbed on their disTherefore for every gramme of zinc sociation, is 1060 units. * The object in this and the two following articles being illustration only, the absolute numerical values are of less importance. The system of units and the numerical values are those employed in Hospitaller's Formulaire pratique de VElectricien, English Edition, p. 214.
VOLTAIC CURRENTS.
232.] oxidised the
excess
of
heat evolved
233
over
(1310 1060) units; that is, 250 units. Again, for every unit of current 00034
In other words, -00034
oxidised.
that
absorbed
gramme
of zinc
is
is
the electrochemical equivalent of zinc. Therefore for every unit of current the excess of heat evolved over that absorbed is -00034x250. And this is
equivalent to an
amount
of mechanical
X
where /
Now
work
34 x 250 '
100000
Joule's factor.
is
if
is
F be the electromotive
the amount of heat evolved of heat evolved
force of the
And
Fi.
is
by unit current
is
That
F.
cell, i
the current,
therefore the
amount
is,
34x250 "
'
100000 231.] It
is usual, as
above
said,
to
dilute sulphuric acid, the formula for this case we may suppose that the 2
H
the
H
2
Zn.
4
H
.
must be added
to the
One gramme
ZnO,
is
250 units above mentioned.
of zinc combines with -246 of a
oxygen H of heat.
.
formed, and the hydrogen combines with the fiOSi forming The heat evolved by this last-mentioned combination
place oxide of zinc, Then the Zn set free.
first is
employ instead of water OS03 In which is 2 is decomposed, and in
gramme
of
being decomposed with the evolution of 250 units 2 And 1-246 grammes of oxide of zinc combine with S03
with the evolution of 360 units.
Adding together 360 and 250,
we obtain 610 units as the total heat evolved. are 232.] In the cell known as Darnell's cell the electrodes of one zinc and copper, but there are two liquid electrolytes, them saturated solution of sulphate of copper in contact with the copper, and the other dilute sulphuric acid in contact with the zinc, the mixing of the liquids being prevented by a porous diaphragm which does not interfere with the electrolytic conduction, that
is
to say,
the liberated ions pass through the
diaphragm but the liquids do not. The following posed to be the action of such a cell.
may
be sup-
234
VOLTAIC CURRENTS.
[233.
electrolysis of the II2 S0 in contact with the zinc gives a chemical action identical with that of the last case, but does not as in that case remain free. It passes the hydrogen 2
The
rise to
H
through the diaphragm and displaces an equivalent of copper in the sulphate of copper Cu <S04 giving as a result S04 and 2
H
,
,
depositing the copper on the copper plate. In estimating the electromotive force of this battery the dissociation and combination of the water counteract each other, 2
H
and the resulting force is the difference between the heat of combination of zinc with S0 and that of copper with the same element.
The heat of combination fin $04 we have already found to be units, being the sum of Zn.O (1310 units) and ZO.SO^ (360 units), and the heat of combination CuSO is 881 units. The difference, i. e. the thermal measure of the chemical action, is 789 units. The product of this 789 by TT nnnr the electro1670
chemical equivalent of zinc, gives the thermal measure of the chemical action for each unit of electricity transmitted, and this result again, multiplied
by
Joule's factor, gives the electromotive
force of a Daniell's cell in the ordinary mechanical units. 233.] The electromotive force of a cell in which unit in centimetre-gramme-second electricity transmitted is
It
is
The
is
done
for
work
each unit of
taken for the unit of electromotive
force.
A
Daniell's cell gives in practice about 1-079 unit resistance is an Ohm, and may be defined to be
called a Volt.
Volts.
measure
The 48-5 metres of copper wire of one millimetre thickness. unit current is called an Ampere, and is the current generated by an electromotive force of one Volt in a conductor whose resistance is
one
Ohm.
234.]
The electromotive
force of a cell
may
be expressed in
general terms as follows.
Let the
effect
into
grammes
Then
e
and
'
See Fleeming Jenkin's Electricity. of the unit current be to decompose a constituent of one ion
TT,
and
e'
grammes
of the other ion
are the electrochemical equivalents of the
v.
two
IT and v. Let 6 be the quantity of heat absorbed in the combination of unit mass of TT with the corresponding mass of v, and let 0' be
substances of
VOLTAIC CURRENTS.
236.]
235
the heat absorbed in the same combination for unit mass of
Then
e
= 0'
e' is
v.
the heat evolved in the circuit for every unit
of current.
And
in mechanical units J#e
If the chemical action be
cell.
the DanielPs
it
cell,
still
is
the electromotive force of the
more complex,
as in the case of
remains true that the heat evolved
is
is to be found as the proportional to e or e', and algebraic sum of the heat evolved and absorbed by all the chemical changes
from which the ions
result.
235.] By increasing the dimensions of the cell we do not increase the electromotive force of the circuit, but we diminish the resistance within the cell,
and we therefore
increase the intensity of
the current, especially in cases where the external portion of the circuit is of a small resistance, and where therefore the resistance of the
cell
becomes appreciable
;
and the same
result follows
when
several cells act together, the zinc plates being severally connected, and likewise the copper plates, for this arrangement is
in its electrical effects the
same as
if all
the zinc and
all
the
copper plates were severally combined into one zinc and one copper plate with areas respectively equal to the aggregate areas
and copper plates in the separate cells. If however the zinc of one cell be united with the copper of the next, and so on in order, the cells are said to be in series, the of the zinc
electromotive force of the separate
is
cells,
sum
of the separate electromotive forces and the arrangement is called a voltaic or
the
galvanic battery. If the discontinuous rise of potential in case of a circuit formed of a single cell be E, then in case of a circuit formed of
two or more there
are
cells in series it will
cells.
According
be repeated as
many
times as
to the Volta theory, this
rise of
potential takes place at the junction between the copper of the first cell and the zinc of the second, and so on. According to the chemical theory the change takes place in
between the metals and the liquid according to either view the electromotive force of n cells in series is n times that
each
cell
of a single cell. 236. ] If we have a
number of
cells
of different kinds connected
ELECTROLYSIS.
236
[237.
in series, the electromotive force of the series will be the algebraic sum of the electromotive forces of the separate cells.
We
might for instance place a single Daniell's more two powerful batteries, connecting the zinc
cell
between
plate of the
single cell with the terminal zinc plate of one battery, and the copper plate of the single cell with the terminal copper plate of another battery. Then in calculating the electromotive force of
the system, we must take that of the single cell as negative. In that case the current is forced through the single against
its
own
cell
electromotive force.
A cell in which no chemical actions can take place on the the current, evolving more heat than is absorbed, cannot of passage maintain a current. But it may be possible by connecting its poles 237.]
with another battery to force a current through it and this current may have the effect of decomposing the liquid of the first cell, work being done in it by the external battery against the chemical ;
forces of the cell itself.
Such a
cell is called
an
electrolytic cell.
238.] Cases exist in which the ions formed in an electrolytic cell do not escape, but enter into new combinations within the cell.
Such new combinations, since work has been done against the chemical forces in forming them, are of higher chemical potential than the original combinations which they replace.
They may be capable of decomposition and
restoration to their
original condition under the influence of a reverse electric current, in which case heat will be evolved, and the cell in its new state will be a Voltaic cell capable of maintaining a current. Such a cell is called a secondary cell, or an accumulator, because the
work done called
in producing the first chemical changes, or as it is charging the cell, is stored up in it, and may be made
available, as required, to
maintain an electric current.
Such is in its essential features the theory of the Plante battery and other allied forms, in which, when charged, one plate is of lead and the other consists of peroxide of lead, and the liquid used is generally dilute sulphuric acid. The cell when so charged maintains a current from lead to peroxide through the liquid, and from peroxide to lead outside, the chemical change being the conversion of the peroxide into protoxide of lead. Then by forcing
ELECTROLYSIS.
240.]
237
a current through the cell in the reverse direction the protoxide again converted into peroxide.
is
239.] Clausius has suggested a theory of electrolysis, supposing that the molecules of all bodies are in a state of constant agitation ;
that in solid bodies each molecule never passes beyond a certain distance from its mean position but that in fluids a molecule, after moving a certain distance from its original position, is just ;
as likely to
move
further from
it
as to
move back
the molecules of a fluid apparently at rest
changing their
positions,
of the fluid to another.
Hence again. are continually
and passing irregularly from one part In a compound fluid he supposes that
not only the compound molecules move about in this way, but that in the collisions that occur between the compound molecules, the molecules, or rather submolecules of which they are composed, are often separated and change partners, so that the same individual submolecule is at one time associated with one sub-
molecule of the opposite kind, and at another time with another.
This process Clausius supposes to go on in the liquid at all times, but when an electromotive force acts on the liquid the motions of the submolecules, which before were indifferently in are now influenced by the electromotive force,
all directions,
so
that the
positively charged submolecules have a greater tendency towards the cathode than towards the anode, and the negatively charged
submolecules have a greater tendency to move in the opposite Hence the submolecules of the cation will during direction. their intervals of freedom struggle towards the cathode, but will be continually checked in their course by pairing for a time
with submolecules of the anion, which are also struggling through the crowd but in the opposite direction. of electrolysis be or be 240.] Whether this view of the process not accepted as corresponding to a physical reality, it gives us a clear picture of the process, and is in accordance with the prinBy means of certain assumptions more or cipal known facts.
we may extend the hypothesis to the explanation of the process of electrical conduction, at any rate through a liquid, as to do this it is only necessary to suppose that each submolecule less plausible
when acting
as
an ion
is
charged with a definite amount of elec-
238
ELECTROLYSIS.
tricity, in
accordance with the statement
amount of
electricity transferred
[241.
made above that the
is the same same number of liberated ions, the charge of the cation being positive and that of the anion negative, by which conception the conduction current becomes assimilated to a convection current,
during electrolysis
for the
perhaps more correctly, to the transfer of motion along a row
or,
of equal and perfectly elastic balls in contact. Many difficulties the of this way hypothesis, as for instance the present themselves in fact that certain ions are anions in some electrolytes and cations in others.
We do not
stop to consider these difficulties in detail,
because the whole hypothesis, while useful in furnishing a mental picture of these processes, is not essential to the enunciation
and mathematical development of the laws by which they are regulated.
241.] In practice, Voltaic
cells,
especially single fluid cells, are
liable to certain defects, the chief of these
being irregularity of electromotive force arising from the accumulation of the ions at the electrodes, thereby causing what is termed electrolytic polarisation,
That such an or an electromotive force opposed to the current. accumulation of the ions would engender this opposing force is
any rate on the hypothesis of Clausius, because, having their electric charges at their respective electrodes, with parted there is no longer any action tending to keep them in this obvious, at
and they necessarily tend to recombine. This opor negative electromotive force is not so obvious in its posing in effects a Voltaic cell, because in such a cell there is at all position,
times a preponderating positive force, but exhibited in an electrolytic or resisting cell.
it
may
be clearly
If the electrodes of
such a cell be platinum plates and the contained liquid be water, then so long as the current is maintained oxygen is given off at If the current be the anode and hydrogen at the cathode.
suspended and the platinum plates externally connected by a wire, a current will pass through this wire in the reverse direction, that is to say from the anode to the cathode, and the liberated gases in the
The
cell will
recombine.
special defect arising
of current which
it
from polarisation
produces.
the irregularity If the current be suspended for is
THERMOELECTRIC CURRENTS.
243-]
any experimental purpose and then again renewed, with greater intensity than
Of
is
239 it
will start
ultimately maintained.
Thermoelectric Circuits.
If a circuit be formed of wires of two or more metals same temperature, the contact differences of potential are
242.] at the
consistent with each wire being at uniform potential throughout its length, and therefore produce no current.
But the
contact difference of potential
is
a function of the
temperature at the point of contact. If therefore the junctions be at unequal temperatures, it is not generally possible that each wire should have constant potential throughout its length. We therefore expect that a current will ensue. 243.] It has been shown by Magnus that in an unequally heated complete circuit of a single metal no current is produced
by the inequality of temperature.
On there
the other hand, Sir W. Thomson has shown that generally an electromotive force from the hot to the cold parts of
is
the same metal, or from cold to hot, according to the metal and the temperature, but that in a complete circuit the total electrois zero. As in order to prevent a current from flowing from copper to zinc in contact, it is necessary that the potential of the zinc should exceed that of the copper by the
motive force
quantity Z/C ; so in order to prevent a current from flowing from an element of the zinc at temperature t + dt to an adjoining element at temperature t, it is necessary that the potential of the
second element should exceed that of the
first
by a
certain
the potential be constant, there is an electroquantity adt, motive force vdt. This quantity a is for any given metal a function of the temperature, and may be positive or negative, or, if
but has generally different values originally called by metal in question.
Thomson the
for different metals.
specific
It
was
heat of electricity for the
V
According to this law, if a and Vb be the at the ends of a wire unequally heated, the electropotentials /*<* motive force in it is
Va -Vb +
Jb
Since
o-
is
for
any given metal a function of the temperature
THERMOELECTRIC FORCES.
240
alone, it is evident that for
any closed
however the temperature vary,
=
a-dt
0,
[244.
circuit of
one metal,
or there
no electro-
is
which agrees with the law of Magnus. This difference of potential, due to difference of temperature, is frequently called the Thomson effect,' and a- the coefficient of the motive
force,
'
Thomson
effect.
has shown experimentally that throughout
Professor Tait
ordinary temperatures, and probably at all temperatures, o- is It is positive for proportional to the absolute temperature.
some metals, negative
for others,
and
is
nearly zero for lead.
It follows from the above statements that in a circuit of
two
metals with unequally heated junctions we have to consider two causes, each of which may produce a current, viz. the unequal contact differences of potential -at the junctions, and the electro-
motive force due to variations of temperature in the same metal. 244.] It is found that in general an electric current flows
accompanied with equalisation of the unequal If JR be unless these be artificially maintained. temperatures the resistance of the circuit, i the current, the electromotive force
round the
is
Ri.
electric
circuit,
Such a
circuit is called a thermoelectric circuit, or thermo-
The electromotive
couple.
force is
found to obey the
following experimental laws. I. If the temperatures of the junctions be t and t and if p qt A p/.q be the electromotive force when A and B are the two
B
metals,
and
B p/ C when B and C are q
the two metals, then
This was proved by Becquerel. II. If a thermoelectric couple
and B, and p and (f be
be formed with given metals
A
electromotive force with junction temperatures A p/q>B, and with junction temperatures q' and q be
if its
A q'/q B,
then the electromotive force of the couple when the junction temperatures are p and q will be
that
A*/t B
is,
This also
is
=
due to Becquerel.
THERMOELECTRIC FORCES.
245-]
The
III.
A
to
J3,
direction of the current, that
or from
B
is
to A, at the hot junction
241
whether
be from
it
depends on the mean
temperature of the junctions. When the mean temperature of the junctions for a given pair of metals is below a certain temperature T, dependent upon these metals, the current sets in one direction through the hot junction, and when the mean temperature is above T the current sets in the opposite direction, or the electromotive force This was discovered by Seebeck.
The temperature
T
reversed.
is
called the neutral temperature for the In an iron and copper couple this pair of metals employed. is
neutral temperature is, according to Sir W. Thomson, about 280 C. When the mean temperature of the junctions is below
the current sets from copper to iron through the hot junction, and when it is above this the current sets from iron to this,
copper through that junction. IV. For any constant temperature of the cold junction, the electromotive force is the same when that of the hot junction is
T+x,
as
when
it
is
T
x,
and
maximum when
a
is
it is
T.
This was established by Gaugain, and results from Tait's exIt may be expressed thus The electromotive force periments. :
of the couple between temperatures
The following phenomena 245.]
is
t
and
t
is
proportional to
a mathematical explanation of these
:
If the difference of temperature between the two junctions be very small, as dt, the electromotive force of the couple must be proportional to
by
(t> ab dt,
where
it,
and
for the metals
is for <j> ab
mean temperature
A
and
B may be
denoted
the given metals a function of the
of the junctions, and is taken as positive when A to at the hot junction. It is called
B
the current sets from
the thermoelectric power of the two metals at temperature t. It follows from II. that if the temperatures of the junctions be tQ and tlt where t^ 1 is finite, the electromotive force is
$ ab dt which }
Again, VOL.
I.
if
for the
given metals
we take any
is
a function of
tQ
and
rf
x
.
particular metal for a standard, and
R
THERMOELECTRIC FORCES.
242 denote
it
by the
suffix
o, it
[246.
follows from I. that the electromotive
force for the couple in which the metals are temperatures of the junctions t and 1} is
-
A
and
Z?,
and the
r Jt 't
If the reference to the standard be understood, we may call the thermoelectric power of the metal A. And in that case <j) a
the thermoelectric power of the couple formed of the metals
A
and
with junctions at temperatures ri *^
It
Q
and
tf
x
is
^0
usual to take lead as the standard metal.
is
The
t
functions
the same metal
$ a and
may
may
be positive or negative, and for
be positive at some temperatures and nega-
tive at others.
It
deduced from the experiments of Professor Tait that
is
each metal perature;
-j~ that
for
has a constant value independent of the temis,
=
at
/3,
where
t
is
the absolute temper-
same metal. Hence if t and x be the lower and upper temperatures of a circuit of metals A and A,
ature,
and
a, ft
are constants for the tf
Also
and
is
if
T be
the neutral temperature at which
>=
0,
proportional to
as stated in IV.
246.] Adopting a method originally suggested by Thomson, represent the thermoelectric powers of different metals
we may
at different
temperatures by a diagram.
Let
the
abscissa
THERMOELECTRIC FORCES.
246.]
243
represent absolute temperature, and for any given metal let the ordinate represent its thermoelectric power, that is, the thermo-
power of a couple composed of that metal and lead, with the temperatures of the junctions infinitely near that denoted by the abscissa. electric
It follows then from the constancy of
is
,
-
that the locus of
d>
at
a right line inclined to the axis of x at the angle tan" 1 -^-
5
and that
for any given abscissa, as that corresponding to 50 C., the difference between the ordinates of any two metals represents the thermoelectric power of a circuit of the two metals at that
temperature. In the annexed diagram atures below 50 C. lead is ,
,
ZERO CENT
,
jj
positive to iron and negative to copper ; from 50 to 284 C.
we
see that for temper-
50C
positive to iron and negative to lead ; from 284 to 330 iron is positive to copper
copper
is
and negative to lead; 330 lead is positive to and negative to iron. rally, if for any two
NM be
above copper
GeneFig. 36.
metals
the difference of the ordinates at temperature
t,
and
E
M'N'
at temperature t', and if be the neutral point, the thermoelectric power of the couples
with the junctions at graphically area
t
represented
MEN-M'EN',
and
t'
is
by the whether
M'N'
be at temperature below or above E.
Fig. 37-
So long as the lower temperature represented by altered, the difference between MEN and M'EN' has
MN
is
un-
its
greatest value when the higher temperature is at E, the neutral point. It becomes zero when the mean temperature of the junctions
is
the neutral temperature. Further,
if
M'N' and
M"N" be taken E 2
at equal distances from
244
ENERGY
DISSIPATION OF
E on either side results agree
of
it,
[247.
MEN-M'EN'= MEN-M"EN".
These
with IV.
247.] Next, let us consider a circuit of three metals
and CA, the junction A being at temperature t and C at temperature 3 2
t ly
AB, BC,
B at temperature
.
,
We may
imagine three lead wires
AD^B,
BD
2
C,
and
CDZ A
connecting the junctions, and forming three distinct circuits.
The electromotive force of the circuit ABC is the sum of the electromotive forces of the three circuits
AB^A,
BCD2 B, CAD3 C, Fi
together with that of the circuit composed of the three
8
lead wires
AD
l
BJ) 2 CD^ A.
But, by the law of Magnus, the electromotive force of the latter circuit
is
zero.
Hence the electromotive r^
rt3
a
h
In
any
like
force of the circuit
dt+
b
Jtz
dt
+
ABC is
/*! c
dt.
Jts
manner we can express the electromotive
circuit of different metals
force
due to
with unequally heated junctions.
We
may suppose further a circuit composed of alternate 248.] wires of two metals only, A and B, and each alternate junction at the lower temperature tlt and every other junction at the higher temperature t2 If there be n pairs, the total electromotive force of such a .
circuit
is,
by the
last article,
The pairs are said to be joined in series. By this means the electromotive force of a thermoelectric couple can be multiplied at pleasure. Such an arrangement is called a thermoelectric pile.
Of
the energy of the current in a Thermoelectric Circuit.
249.] current.
Energy, as alove shown, is necessary to maintain the In the case of thermoelectric circuits, now considered,
IN THERMOELECTRIC CURRENTS.
250.]
245
no energy is supplied from without, nor are there, so far as we know, any chemical actions between the metals, or between them and the surrounding medium, from which the requisite energy can be obtained.
We
infer that the
energy required for maintenance of the the conversion of part of the heat of the supplied by metals into another form of energy, namely, that of the electric
current
current.
work.
is
This might conceivably be employed to do external if not, it will be reconverted into heat by the
But
resistance of the circuit.
As in the working of a heat engine, the entropy of the system must be diminished by the process, that is, there must be equalisation of temperature. It is found that at the neutral temperature for any two metals a current passing the junction has no heating or cooling effect.
The
Peltier effect changes sign at that point.
But'
if a couple be formed with the hot junction at the neutral temperature, the cold junction is nevertheless heated, although the heat cannot be derived from the cooling of the hot
junction. It is evident, therefore, that the current itself
must have a
heating or cooling effect. For instance, in an iron and copper circuit, with the hot junctions at the neutral temperature, either a current in iron from hot to cold must cool the iron, or a current in copper from cold to hot must cool the copper, or both these effects take
place.
And
it
may
be inferred that the heat so
gained or lost is compensated by a change in the potential of It was this consideration that led Sir W. Thomson the current. to the discovery of the electromotive force in unequally heated portions of the same metal.
250.]
writers (Mascart and Magnetisme ; Briot, Theorie
The method adopted by some
Joubert, Legons sur VElectricite et le la Ckaleur] is as follows.
Mecanique de
It is assumed that the
heat generated, as unit current passes from potential Va to potential 7b , is always 7a Fb whether the fall of potential be gradual as ,
in a single metal, or abrupt as at the junction of two metals. the case, the electromotive force of a couple formed That
being
GENERAL SYSTEM OF LINEAR CONDUCTORS.
246
of metals
A
B whose hot and must be
and
respectively,
cold junctions are at
H,-H + Jf\cra -
[251.
^ and
t
Q
b )dt,
to
where
H
l
H
and
are the Volta contact differences of potential at
the junctions.
When
1
tl
becomes
infinitely small, this
dH that
(pab
IS,
Further,
if
-jT-
+
"a
becomes
"b
the current be infinitely small,
we may regard such
a circuit as a reversible Carnot cycle. Then, if 6 Q be the heat absorbed at temperature t, taken as negative when heat is evolved,
/T
dt
=
for the entire cycle.
When ^
And
^o
becomes
infinitely small, this
becomes
the contact difference between two metals
is
zero at their
neutral temperature. are now in a position to treat a more general case 251.] of a system of linear conductors than that considered in Art. 215, in which the wires were supposed to be all of the same metal
We
and
at the same temperature. In that case, the potential of all the wires which meet in any electrode as is the same at the
P
common
extremity P, and may be designated, in the case of each When the wires are not of wire, by the common symbol Vp the same metal, we may suppose that instead of being in .
immediate contact with each other at the electrode P, each
is
GENERAL SYSTEM OF LINEAR CONDUCTORS.
251.]
in contact with a small wire or disk of
247
some standard metal at
Vp
were the potential of this connecting metal at P, then the potential of any other wire as PA of metal (), + x(^P )> where x( a ^p) suppose, at the extremity P would be If
that point.
^
represents the contact electromotive force from the metal (a) to the standard metal at temperature tp similarly if Fq were the -,
potential of the connecting metal at the electrode Q, where the temperature is tq the potential at the extremity Q of the wire ,
PQ
would be Pg + x(^)'
^ n estimating the currents therefore
we may regard the potentials of the all wires at any electrode as equal
in terms of the potentials
common
of
extremities
provided we
increase the electromotive force in
PA
any wire as
by the quantity
xK)-xK)The Thomson
effect
treated of in Art. 243 will produce a
similar increase of electromotive force of the form
ftp /
vdt,
Jtq
which may be expressed in the form If therefore
Epq
be the electromotive force arising from a PQ, the expression for
any, in the course of the wire the current in that wire will be battery, if
and similarly
Of
for each of the
course the wire
metals, or
may
PQ
consist of
remaining wires. itself be composed of dissimilar
may
two wires communicating with the
which cases the requisite liquids of an interposed battery, in corrections are obvious.
CHAPTEE
XIV.
POLARISATION OF THE DIELECTRIC. ARTICLE 252]. IN the preceding chapters we have endeavoured to explain elect rostatical phenomena by the and Green as the result of direct attraction
method
of Poisson
and repulsion at a distance, according to the law of the inverse square between the positive and negative electricities, or electric fluids. As explained at the outset, in Chaps. IV and V, we do not assert the actual
We assert merely that the electrobetween conductors are as they would be if the existed, and conductors and dielectrics had the properties
existence of these fluids. statical relations
two fluids
attributed to
them
in those Chapters.
Faraday and Maxwell made an important step in advance. They assume all non-conducting space to be pervaded by a medium, and refer the force observed to exist at any point in the electric
field,
not to the direct action of distant bodies, but medium itself at the point considered.
to the state of the
Faraday was led by his experimental researches to believe in the existence of certain stresses in the dielectric medium in presence
of electrified
bodies.
Maxwell shews that
if
the
medium
consist of molecules with equal and opposite of charges electricity on their opposite sides, or, as we expressed it in Chapters and XI, polarised, these stresses would in fact dielectric
X
See Maxwell's
Electricity, Second Edition, Chap. V. There would be at every point in the medium a tension along the lines of force, combined with a pressure at right-angles to
exist.
them, and by such tensions and pressures all the observed phenomena may be accounted for without assuming the direct action of distant bodies on one another.
It
is
true, as
Maxwell
says, that some action must be supposed between neighbouring molecules, and that we are no more able to account for that
than for action between distant bodies.
And
if
only electro-
H XJNTVr
X^C4LIF03N^^ POLARISATION OF THE DIELECTRIC.
253.]
249
phenomena were concerned, it would be perhaps of little importance whether we attributed them to direct action of distant bodies or to a medium, so long at least as the electric fluids and statical
medium were equally hypothetical, and had no other duties to perform than to account for the phenomena in question. The advantage of Maxwell's hypothesis is that it connects the the
electricity and magnetism with those of light and radiant heat, both being referred to the vibrations of the same medium. There is, in fact, in the phenomena of light, inde-
phenomena of
pendent evidence of the existence of Maxwell's medium, whereas there is no independent evidence of the existence of the two
The medium
fluids.
therefore has better title to be regarded two fluids have.
as a vera causa than the
No
treatment of the subject can, in the present state of know-
ledge, be more complete than Maxwell's own in Chapters II and of his work, and it is necessary to study those chapters in order
V
properly to understand his views. The whole subject of statical electricity has also been treated very fully from Maxwell's point of view in the article
'
'
Electricity in the Encyclopaedia BritanNinth It may, however, Edition, by Professor Chrystal. nica, be of some advantage to obtain the same results from a slightly different starting-point.
253.] In Chap. XI we had occasion to treat of a particular case of a polarised medium, a medium, namely, in which are interspersed little conductors polarised under the influence of given forces. If the induced distribution on the surface of any conductor be
denoted by $, the quantity
xtydS, taken over
all
the con-
ductors in unit of volume, was defined to be the polarisation in x per unit of volume.
direction
We will now adopt a rather more sation.
Let us conceive
general definition of polari-
a region containing
an
infinite
number
of molecules, conductors or not, each containing within it, or on its surface, a quantity of positive, and an equal quantity of nega-
Let P be a point in that region, and about P be taken a unit of volume, containing a very great number of the molecules in question. Let us further suppose
tive, electricity.
let there
POLARISATION OF THE DIELECTRIC.
250
[253.
that throughout this unit of volume the distribution of the molecules in space, as well as the distribution of electricity in individual molecules,
may
be regarded as constant, and the same Let $ dx dy dz be the
as in the immediate neighbourhood of P.
quantity of electricity of the molecular distributions within the
Then \\\$dxdydz throughout
element of volume dxdydz. the unit of volume
zero; and
is
we
will define
/
/ /
x$ dxdydz
taken throughout the unit of volume to be the polarisation in direction x at P.
Let
I] Ixfydxdydz
meanings
=
ax
for the axis of y
\
and
and
let
vy
,
a s have corresponding
z.
P
If a plane of unit area be drawn through parallel to the And the the of it will certain of molecules. intersect plane yz>
XI
(Art. 190) shews, that the quantity of electricity belonging to these intersected molecules which lies on the positive side of that unit of area is a-x Similarly if the
reasoning of Chap.
.
plane were parallel to az, or xy, the quantity of electricity of the intersected molecules on the positive side of the unit of area would be (T y or tr a in the respective cases. If the direction-cosines of the normal to the plane were m, n, the quantity of electricity of the intersected molecules lying on the positive side of the unit of area would be lo- x mo-y + n(r ,
+
ls
.
For, by definition, the polarisation in the direction denoted by I,
m, n
is or
that
/
/
/
(lx
+ my + nz)
dx dy dz,
is,
= that
=
I
x
dxdydz + m III a-
is,
Hence
a-xt
a-
y}
and
o-
z
y<j>
dxdydz + n
z$ dxdydz
',
=
are components of a vector.
If the distribution be continuous, so that o-x , a- and v z do not vy change abruptly at the point considered, the same reasoning as employed in Chap. XI shews that the amount of the distri-
POLARISATION OF THE DIELECTRIC.
2 5 3-]
bution within the elementary parallelepiped dxdydz
where 9
~~
Should the values of
^
dax
{
dx
251 is
oand v z change abruptly at the y point in question, there will be over the unit of area a supero-x
,
,
or quasi-superficial distribution a-x a'x where
,
we have seen
Also, as
in Art. 190, the potential T, of such a
any point x, y^ z, is determined by the rr
polarised distribution at
equation
where r
is
the distance of the point
x, y, z
from the
superficial
element dS, or the solid element clx'dy'dz', as the case may be. Hence it appears that such a system of polarised molecules as
we and
are supposing gives rise to localised distributions with solid superficial densities of determinate values throughout given
regions and having the same potential at every point of the field as
would
result
Conversely,
from such if
localised distributions.
we had an
electric field
with given localised
charges, we might substitute for it a system of polarised molecules in an infinite variety of ways, the physical properties of which, so far as we are concerned with them, would be in all respects identical with those of the given localised charges. For if p and o- were the densities, solid or superficial, at
any
point in the supposed system of given charges, and if the polarisation and arrangement of the molecules were such that (o^, o- y ,
and
0-3
being as above defined), d(T x
d
d(rz
_
\
at each point of the field, then we should have the same density at each point as is given by the localised charges, and the
V
at each point would also be the same as in the case potential of the localised charges, being determined by the equation
F== /T
JJ
r
+ /YT JJJ
POLARISATION OF THE DIELECTRIC.
252 As
the values of
ar
x9
[ 2 53-
a z are subjected to only one equation of may clearly be chosen in an infinite
o^,
condition (A) these quantities variety of ways.
Among
all
the possible values of
a-x
,
y
and
,
tr,
we
'
IT
shall con-
sider only *
~
JL
4w
"
21
~~
(Ty
dx
JL
'
47T
^1
s
~
JL 4^r
dy*
'
*
dz
These relations will satisfy (A) identically, since by Chap. Ill
-',-.,
for all points
And
where
AV dV dV -7-
d#
>
-=-
^
>
-7-
ds
vary continuously.
also
ZK-o-'sHw^-o-'^ + ^s-cr's) 1 .dV dV .dV dV
=r
dV
dV'
(T
over surfaces of discontinuous values of these coefficients. It appears then that such a system of polarised molecules not only produces at all points in space the same potential as the system of volume and superficial distributions for which it
was substituted, but
selves to reappear.
also
It can be
the same in the two cases.
the distributions them-
causes
shewn For the
also that the
polarised
energy
medium
is
is
in
a state of constraint, because the separated electricities are not allowed to coalesce and neutralise each other. Work has been
done upon
it
in producing- this state.
the constrained state of the dielectric
In ordinary experiments is
produced by the introis the work done in
duction of charged bodies, and the work
* With the distribution of polarisation assumed in the text, if a small cylindrical region be described in the medium whose generating lines are parallel to the force at the point and infinitely smaller than the linear dimensions of the bounding planes, the force at any point within the cylinder is that arising entirely from the polarisation of the molecules completely included within the cylinder, and the total force from all the rest of the molecules is zero. If the polarisation were magnetic, this result would be expressed by saying that the law of magnetisation is such that the magnetic induction at every point is zero.
POLARISATION OF THE DIELECTRIC.
253-]
253
charging' them, but according to this theory the energy resides, not in the charged bodies, but in the dielectric. 1
The energy in unit volume see Chap. V, that is,
+h The energy of the
W
is
R
2 ,
h
entire system estimated in the
throughout the whole of
But
of the polarised system
same way
is
dielectric space.
this is also the expression for the
energy of the originally given system according to the ordinary theory, as shewn in Chap.
The two systems
X.
are therefore for all purposes equivalent. conceive that the molecules of all dielectrics are
We may
capable of assuming such polarisation as required for this hypoIf, as we have hitherto supposed, vacuum be a perfect
thesis.
dielectric, it
becomes necessary
for the hypothesis to conceive it
as permeated by a non-material ether, the molecules of which are capable of such electric polarisation. And if the existence of such an ether be assumed, it may be that in case of other dielectrics,
the electric polarisation resides in the ether rather
than in the molecules of the substance.
We may further
suppose that the essential property of con-
ductors, as distinguished from dielectric media or insulators, is that their molecules are incapable of sustaining electric polarisations, or that the substances of
conductors are impermeable by
the supposed ether, and therefore that no electric force and no free electricity can exist within them.
We
might thus construct a theory of
electrostatics founded
dielectric, just as the ordinary theory founded on the property of conductors. In the ordinary theory the electromotive force at any point is
on the polarisation of the is
the space differential of a function 7, which is constant throughat out any conductor, and satisfies the condition V 2 F+4irp=
Assuming points where there is free electricity of density p. that no case of electrostatic equilibrium has yet been discovered, which can be proved to be inconsistent with the ordinary theory, all
STRESSES IN POLARISED DIELECTRIC.
254
[254.
follows that the supposed dielectric polarisation must, when there is equilibrium, be the space differential of a function F, it
which
is
constant over and within every closed surface bounding 2 4 itp at all points in the satisfies V
=
T+
the dielectric, and dielectric.
The Stresses in the
Dielectric.
^
of an electri254.] If any closed surface 8 separate one portion system from the other portion 2 as, for instance, if the whole
U of E2 outside
fied
,
be inside, and the whole of S, then this hypoan explanation of the phenomena without assuming any direct action between E2 and E. For if the polarisation of
j^i
thesis suggests
be given in magnitude and direction at each point in
given at each point on S. Then we know that if charge of every conductor within S be given, and electrification within
S be
S, -=-
is
dv the form and
'
if all fixed
given, V has single and determinate
all points within S. It follows that all electrical
value at
phenomena within
H
S,
which in the
ordinary theory are due to the action of 2 are on the polarisation hypothesis deducible from the given polarisation, that is the ,
dV
given value of -=- at each point on >
8.
We
might then always substitute for the external system E2 a certain polarisation on 8, without affecting the equilibrium of An example of this substitution has already been given lm
U
(Art. 58) for the case
where S
then a distribution over
S whose
force as the external system at
If
S be not
follows
Let
is
equipotential
an equipotential T> density
is
--
exerts the
For
same
47T
any point within
we
surface.
S.
obtain a corresponding result as
* :
Yl be
the potential of the external,
T2
that of the internal
system, and V V^ + T2 the whole potential. The whole force in direction x exerted by the external on the internal system is
throughout the space within * This investigation
is
8.
taken from Maxwell's Treatise, Second Edition, Chap. V.
STRESSES IN POLARISED DIELECTRIC.
254-]
255
But
V F =V 2
and within
S
Hence
whole force
The
tlie
object
2
F.
2
is
is
to express this in the form of a surface integral
over S. If
we can
find three functions X,
dx
dx
J",
Z, such that
da
dy
then evidently, by Green's theorem, -
over the surface S.
V This
is
the required surface integral.
Let us assume
dVdV dVdV dx dy
Then
**
satisfy the condition.
The
quantities p xx p yy &c. are the six components of the on the surface S due to the polarisation of the dielectric. If S be an equipotential surface, we have ,
,
stress
dV dx
p. Rl
dV
= _ Rm dV = -
,
dy
dz
Rn,
STRESSES IN POLARISED DIELECTRIC.
256 where
E
is
-
The ^-component of stress
R
is
2 54-
the normal force, and therefore
a
that
[
2
is
then
Similarly the y- and ^-components are
1.
O7T
That
is,
force
R acting on
the stress
is
normal to 8 and t
is
equal to that of the -n
If
S be
stress in
the surface electrified to a density J
any element of
it
,dV
/
dx
Now
an equipotential surface, thus, in this case,
at right angles to
8 IT {lpxx H-
+m
dV
dV =
+n
dz
dy
we
find the
...............
(1)
mpxy + npxa }
dVdV o?c
dV Multiplying (l) by 2 Sir {tyxx
3-
+2w
dy
and subtracting from
+ mp.xy + np
x!}
}
(2),
dVdV T--Jdz dx
we
..
(2)
obtain
=-lR*.
Hence the components of tension perunit
area are
If therefore these stresses exist at every point of the surface
/S,
no matter how they arise, they produce on the interior system EI exactly the same effect as, according to the theory of action at
SUPERFICIAL CHARGE ON A CONDUCTOR.
255-]
257
a distance, would be produced by the attraction and repulsions due to the external system Ez .
255.] According to the theory of dielectric polarisation as explained in Art. 253, the so-called charge on a conductor is to
be regarded as the terminal polarisation of the dielectric as belonging in fact not to the molecules of the conductor, but to the adjacent molecules of the dielectric. (Maxwell's Electricity^ ;
Art. 111.)
So long
as
equilibrium,
we it
are dealing with a system at rest and in statical indifferent for all purposes of calculation
is
whether we regard the charge as belonging to the
dielectric or
to the conductor.
It solid
however possible to induce in any conductor or other body the state which in the ordinary theory is called a
is
charge of electricity; and it is possible to move the body in this state from place to place through air without destroying its charge. It should seem therefore that although the electric force at
of the
any point in
medium
air
be due to the polarised state and not to direct action of the
may
at the point,
charged body, and although the polarised particles be always those of the dielectric, yet the ultimate cause of the may be in the body and not in the dielectric.
phenomena
And
this
appears to be Faraday's view, where he says (1298), 'Induction appears to consist in a certain polarised state of the particles into
which they are thrown by
the
'
electrified
body sustaining the
action?
Certain experiments have been appealed to as shewing that the electrification, whatever it be, is in the dielectric and not in the conductor.
If,
for instance, a plate
of glass be placed
between and touching two oppositely charged metallic plates, and these be then removed, it will be found that they exhibit If they be replaced and scarcely any trace of electrification. connected by a wire, a current passes of the same or nearly the same strength as if no removal had taken place. See Jamin, Cours de Physique, Leon 36.
A
similar result
was obtained by Franklin with a Leyden
the metallic coatings of which were moveable. VOL.
i.
s
jar,
MAXWELL'S DISPLACEMENT THEOEY.
258 256.]
[256.
to this point we have not dispensed with the twoand the law of the inverse square in electric action, is only by the use of that theory that we have proved
Up
fluid theory
because
it
the properties of our medium. All that we have done introduce a somewhat different conception of an electric and the distributions of which it is composed. If any advance
to be made, it
is
Faraday and Maxwell
as follows
must be
is
to
field,
in the steps of
:
We
observe that in the polarised medium the relation between the force at any point and the polarisation at the point is given 4770-,,., &c. by the equations JT
=
These equations are of the same form as those which express the relation between the force existing at any point in an elastic body in equilibrium and the molecular displacement at the point.
In treating of elastic bodies we regard these relations as ultimate facts based on experiment. We might then regard the corresponding equations for the dielectric as ultimate facts, without resorting to the two fluid theory for their explanation. We might regard the dielectric as an elastic medium capable of
being thrown into a state of strain, and presenting when in that state the phenomena which we call electric force and electric distribution.
257.] According to the theory in this form, no action is exerted by the electricity in any part of the dielectric on that in any other part, unless the two are contiguous. might thus dispense with the notion of action at a distance, on
We
which the ordinary theory of the
ordinary theory
actions with which
is
is
founded.
Another
characteristic
the instantaneous nature
of the
For, according to that theory, if any change take place in electrical distributions in any one part of space, the corresponding change takes place at the same instant in every other part however distant. The substitution of the
it
deals.
medium
for the direct action
between distant
bodies, suggests that these corresponding electrical changes may not take place at the same instant, but that electrical influence
may be
propagated from molecule to molecule through
the
ELECTRIC DISPLACEMENT.
258.]
259
certain velocity. And herein lies the strength For, as Maxwell discovered, if electrical effects
medium with a of the theory.
propagated with finite velocity through an insulating medium, such velocity is the same as that of light, or so nearly the same as to leave no room for doubt that the two classes of are
phenomena
are physically connected.
Again, an
medium, if thrown by any forces into a on removal of those forces immediately recover its original condition. There is a time of relaxation. Certain phenomena, such as the residual charge of a Leyden jar elastic
state of strain, does not
Maxwell's Electricity, Chap. X), lend countenance to the supposition that a dielectric medium influenced by electric forces does not immediately, on the removal of those forces, recover its (see
original condition.
258.]
We
proceed to consider the meaning of the term
displacement as used
by Maxwell,
for
electric
which purpose we must
revert to the conception of the two-fluid theory. If through any point in the medium of polarised particles a plane be drawn perpendicular to the v direction of the re-
R
at that point, the density
per unit area of that plane, of the electricity on the particles intersected by that plane and on the positive side of it according to It's direction is, sultant force
as
we have
seen,
or,
determined by the equation
In Maxwell's view, any
field of
electromotive force in the dielec-
accompanied by a strained state of the particles of the dielectric or of the pervading ether, a displacement or transfer
tric is
T)
of positive electricity equal to - - per unit area of surface of the particles taking place
on the positive
side,
from each particle to the adjacent particle
along with an equal displacement of negative
electricity to the adjacent particle
+
or
-
electricities
on the negative
side.
The
do not coalesce or neutralise each other
within each particle, but a polarised state is set up throughout the field, each particle being in a strained state owing to the
ELECTRIC DISPLACEMENT.
260
separations of the electricities within
it,
[259.
the result being graphic-
ally represented thus,
Fig- 39-
the shaded sides of the particles A,
J5, C,
&c. indicating positive,
and the unshaded sides negative, electrification. According to this view the displacement is the process hy which the polarised
It
is
We
been brought about.
state of the particles has
rally denote the polarisation
by
cr,
easily seen that in a dielectric
shall gene-
and the displacement by/.
medium/ =
cr.
259.] If the field were one of uniform force parallel to a line from left to right across the plane of the paper, the total displacement or transfer of electricity across all
planes perpendicular to that line would be the same and equal to -
per unit of area.
If the field were such as corre-
G>
sponds to what
is
called
an
elec-
point 0, i.e. a charge within a very small volume about 0, the particles would be polarised as in
trified
Fig. 40.
the figure, the displacement (sup-
posing no other charge in the field) taking place concentrically from within outwards, and the quantities o-x , a y and a- z being ,
so determined that d
-j dx
+
da-,.
dcr*z
dy
dz
^+
at all points without the small region,
da v dx
dy
within that region where p
charge at 0.
is
"
= and that
dcr,
dz
determined to give the requisite
261
ELECTRIC DISPLACEMENT.
260.]
The law of resultant polarisation exterior to may in this ease be determined, and the consequent law of force, if it be assumed that the resultant polarisation a is symmetrical about the point with which sensibly coincides. be we must have if this assumed For
=
a-
$
(r)
and in the
direction of r the distance of each point from 0. Therefore
=
(r)
-
d(T x
.
Therefore since
-^ax
or
,
+
y
d(T y
-=-^-
260.] If in any polarised
and
If-
find the integral
R
the angle between that the result
know
$
+
^ (r)
do-,,
-=*-
az
ay
or
=
=
field
,
=
(f>
(r)
-
= 0,
;jr
we -
describe a closed surface
over that surface where
S
9
is
and the normal to 8 at each point, we is the total quantity of the 9 where
is
M
M
S; that is to say, the whole quantity of the electricity lying without the surface S on all the molecules intersected by S would be -M, and the whole quanelectricity situated within
S to the adjacent external in other words the total quantity of particles, within closed surface whatever is unalterable. any electricity The electricity behaves in all respects like an incompressible tity of the electricity displaced across
would be
9
+M
9
all space, and the introduction of any quantity closed into any region is accompanied by an efflux of a corresponding quantity from that region. have so far supposed the whole region to be dielectric or
fluid
pervading
We
non-conducting, but the introduction of conducting substances does not affect the result. The special property of conductors is that their molecules are incapable of polarisation, or that the substances of conductors are impermeable by the other whose molecules may be thus polarised.
ELECTRIC DISPLACEMENT.
262
[261.
Suppose now that our closed surface S was intersected by a conductor C, and let us replace S by another closed surface made up
Q
of the
external
portion,
to
(?,
of
S
and
another surface S'QS'
41.
very nearly coinciding it, the dotted line
with that of the conductor and external to indicating the continuation of S within C.
The
integral
/ /
-
-
taken over this new closed surface
be equal to all the original mass of the included since the charge on the conductor must be zero on electricity, the whole ; and since there is no polarisation within C we have as before the total quantity of electricity transferred across the
will, as before,
S
original by displacement equal to the mass within it, the only difference being that instead of such transference being throughout molecular, as it is in the dielectric, it is a transference in
mass across the conductor. This transference by displacement differs from that by coninasmuch as when the force ceases the state of strain
duction,
and the displacement cease
also,
and
all
things return to their
original condition, there being no permanent transfer. 261.] Recurring again to the simple illustration of the field of uniform parallel force, suppose the force, remaining uniform, to C, vary from time to time, then the state of the molecules A, &c. in Fig. 1 also varies, the shading becoming darker as the ,
and lighter as it diminishes. If the displacement at any instant were /, it is clear that this variation of the force and consequently off would produce
force increases,
a transference of electricity across any plane perpendicular to the force in
all
-~respects analogous to a current of intensity dt
along the lines of
force.
For example, suppose the field to be that of the dielectric between the plane armatures of a condenser, and suppose these armatures
ELECTRIC DISPLACEMENT.
262.]
263
by a wire. Then a discharge would take place from left to right the effect of this discharge the wire through would be to diminish the displacement within the dielectric to be connected
;
from
left to right, or to produce a counteracting displacement from right to left, inasmuch as the positive charge of the lefthand and the negative charge of the right-hand armature
would diminish at the rate u per unit time, through the wire. Therefore
we should have -~ Cvt
-f
u equal
if
u were the current
to zero, or a closed
current would flow through the whole apparatus of dielectric, armatures, and wire. And this is what is supposed to take place in every case of transference
by conduction.
262.] Hitherto, throughout this chapter, we have treated our dielectric as being what may be called a pure dielectric with In the case of impure dispecific inductive capacity unity. electrics like those treated of in
Chap. XI, we
may
either, as in
what has preceded, retain the conceptions of the two fluids with distant action, or adopt Maxwell's more simple conception of a displacement connected with the force by a law regarded as an ultimate fact (Art. 256).
On
the former hypothesis we may, as is done in Chap. XI, assume the intermixture of small conductors. In the figure annexed let the plane of the paper be
supposed parallel to the axis of as, and let the line AB be the intersection with that plane of a plane drawn
through any point P in the medium perpendicular to that axis, and let the dotted line be the intersection with the same plane of the paper of a surface as nearly as possible coincident with the aforesaid plane perpendicular to a?, but so drawn as not to intersect any small
^ ri S- 4 2 -
conductors, this surface will not differ sensibly from the plane. ax be the density per unit area of this surface of the elec-
If
upon the polarised dielectric molecules intersected by and lying to the right or positive side of the surface, and or and o-a be corresponding quantities for planes through P y
tricity it
if
parallel to xz
and xy respectively,
it
follows from
what has
ELECTRIC DISPLACEMENT.
264
[262.
been already proved that the density /> of actual charge in the medium at P is determined by the equation
But
K
if
XI
Chap.
be the
d(T x
d(T y
d(r z
dx
dy
dz
inductive
specific
_
capacity,
we know from
that
whence we may, as in the preceding case, choose as our solution ox a y) and a-,,, and their resultant o-, the equations
for
vx
,
= K dV -yThe
j
dx
4?7
= -K dV -7-
y
4-77
polarisation
"
j
z
K dV = TT~ '
477
ay
and
K
=
a-
.
dz
as before, measures the displacement at
o-,
.
477
any
follows from Art. 193, Chap. XI, that the total dispoint, over any closed surface is equal to the total quantity placement of electricity within the surface, as in the case of pure dielectric
and
it
media.
According to Maxwell's point of view, we should ignore the analysis of the action on the inverse square hypothesis altogether, and regard the equation ff
where /
is
=_^. B 477*
f=.R
or
477*
the displacement, as an ultimate fact expressing the
between force and displacement in any isotropic medium, with the requisite modifications for heterotropic media, in which
relation
=
X
*= ~^- Y
-\r
V
-45;*
~V
'
"*
=
~^- Z Z
r? '
K K K
when the
axes are principal axes; and since are z x y the not not in resultant will generally equal, displacement this case be necessarily coincident with the resultant force R. ,
,
f
T7-
The equation
f
=.
.72, if
+
K
be capable of assuming
all
1 and co, expresses the relation between force and displacement in all bodies, the lower limit (l) corresponding to air or rather to vacuum, and the higher limit (-f oo) to
values between
conductors.
265
ELECTEIC DISPLACEMENT.
263.]
263.] In ideally perfect insulators, there
as
is,
we have
said,
no permanent transfer arising from displacement the force ceasing, the polarisation and displacement also cease, and any passage of ;
on the cessation of the an rebound or retransfer of electricity across the by equal same plane backwards. Such perfect insulators do not exist in electricity across a plane is succeeded
force
Kecurring, for instance, to our uniform force illusthis force or the corresponding polarisation of the
nature. tration,
when
A, B, C, &c. particles reaches a certain intensity, the particles become incapable of retaining their state of strain, and the +
and
electricities in
perpendicular to left to right,
R
each particle intermix.
Across any plane
transfer of positive electricity from of negative from right to left ; in fact a tem-
and
there
is
porary current, and each particle returns to its unstrained state. If however R were maintained constant, there must be a renewed
displacement to the same extent as before, so that we have what equivalent to a permanent transfer of electricity, a current
is
from left to right, the intensity of the current u being the rate at which the electricity is transferred in each particle across any transverse section, and which is connected with the force 1\ TJ
by the equation u If the
force
R
=
if r
be the resistance within each particle.
remained constant the polarisation a- or the f must be renewed as
corresponding and equal displacement fast as
it
is
destroyed, so as always to satisfy the equation
-n o-
=
;
in other words, there
must be a continually recurring
displacement or transfer from particle to particle equal per unit of time to the quantity u *.
Again, suppose that the diminution of polarisation or transfer current u was absolutely impossible, i.e. that the insulation was perfect but that the force varied, producing therefore a variable * The actual historical displacement or transfer at any time must be distinguished from the instantaneous displacement or polarisation; this latter is the transfer or displacement which the state' of the field requires in accordance with the above theory, and is what would take place if there were no conduction the former, in case of conduction, is the sum of the continually renewed instantaneous displacements, required for the polarisation of the field which have taken place up to the instant considered. ;
VOL.
I.
T
266
ELECTRIC DISPLACEMENT.
[264
The result is equivalent to polarisation cr or displacement f. a transfer of positive electricity from left to right at the rate per unit of time of -~at If both the conduction current u and the variable force, and
consequently variable
f coexisted,
the resultant effect would be
equivalent to the current of intensity u
We have already
264.]
+ -~ dt
considered the case of a condenser with
plane armatures connected by a wire, and have seen that the discharge current u in the wire is accompanied by an equal and opposite current -j- in the dielectric, but in point of fact the u/t process which, in this case, takes place almost instantaneously is in effect, though much more slowly, always going on throughout
For no substance
the dielectric.
is
absolutely and completely
non-conducting, the molecular constraint is continually giving way, there is a continual passage of electricity from the positive to the negative
armature causing a diminution of force and u be the conduction
If polarisation throughout the medium. the force, and current, r the resistance,
X
any
instant,
u
where
we
=X T
have, as
,
and
f the
shewn
/=
displacement at
in Art. 261,
K >
R.
4-7T
Therefore
= /.
X=X
and
and
e
Kr ,
X
f being the initial values of /and X. These equations express the law of decay of the efficiency of condensers. 265.]
According to Maxwell's doctrine, as we have already
said, all electric currents flow in closed circuits.
267
ELECTRIC DISPLACEMENT.
265.]
Let us recur
to the case of the
charged particle of Art. 259, rest within the medium,
which we have hitherto regarded as at and suppose the charge to be unity.
If it move from one position to another we have in effect a current of electricity from to (7. But from another point of view the effect is the same as if the particle in the first position
were annihilated, and another similar particle placed
second position; that unit
in the
as if
is,
a particle with
positive
charge were placed in the second position, and a particle with unit negative charge superadded to the positively charged particle in the first position. If denote the
first,
0' the
P
second position, any point in the displacement at space,
P
due to the placing of a negative particle or annihilation of a positive particle at
PO.
The displacement
a displacement
is
P
at
-
in direction
due to the positive particle at 0'
is
a displacement 47T If
O
f
be infinitely near to 0, and 0(7
equation to the resultant as follows
Let
ZPOO' =
PO = and
PQ^ = e
Let OQ = 3 a cos 0. P0'= r-a cos 0, and
Then
0.
r,
r
3acos0
r
P(>
if
= r-3a cos 0, PO
r2
r is
parallel to
O'Q.
Its
therefore
dy -~ ax Therefore
=
= ^ dx
SacosOsinO a
can find the
2acos0
r-acosd
Therefore the resultant
= a, we
:
3 a cos 2
.>
and
y x
= tan a0.
2
equation
is
ELECTRIC DISPLACEMENT.
268 The
solution of
where
c is
This
is
is
a variable parameter. the equation of a system of closed curves having 00"
common
for a
which
tangent.
any quantity of positive electricity moving particle is equivalent for the of positive electricity), we have a flow or purpose to a flow It thus appears that if flows from to (/ (for our
current of electricity at every point in space, in direction forming closed curves with the line 0(7. From which it would seem, as we have already said, that there is no real change of position of all the positive electricity in space. Or, in other words, either kind of electricity behaves like an incompressible fluid, and the quantity of it within any finite space cannot be increased or diminished. If,
and
on the moving particle be unity, to 0', that is a distance a in unit of time,
for instance, the charge
it
move from
the current in 00'
P be
a point in the plane bisecting 0(7 at right-angles, and r be measured from 0, the displacement current from right to left through a ring of the plane is a.
between the distances
r
If also
and r + dr from
is
The whole displacement current from right plane
is
and
is
to left through, the
therefore a.
therefore equal to the current from left to right in 0(7.
Hence, according to Maxwell's view, all electric currents in nature flow in closed circuits. This theory will be found to lead to important consequences when mutual action of electric currents.
we come
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