Geophysical Field Theory and Method, Part A, Volume 49: Gravitational, Electric, and Magnetic Fields (International Geophysics)

a'P

az

(1.131)

3. Spherical system

It is clear that Eqs. 0.118) and (1.129) have the same meaning because

they both characterize the density of terminal points in a very small volume with the center at point p. 4. The simplest expression for the divergence is, of course, in a Cartesian system. However, it may be a source of confusion since it is a great temptation to consider div M as a sum of derivatives of field components Mx ' My, and Mz ' with respect to corresponding coordinates, and inter-

1,6 Flux, Divergence, Gauss' Theorem

47

pret div M as a measure of a change of the field, but it is not true.' For instance, if in the vicinity of some point vector lines do not have terminal points, div M equals zero regardless of how the field M changes near this point. To emphasize the real meaning of divergence we will consider a segment of the elementary vector tube (Fig. 1.12c). The surface surrounding this volume consists of the vector surface of the tube Stand two cross sections dS I and dS z . Taking into account that the field is tangential to the vector surface and

We have for the flux through the closed surface

or

Therefore, 1

a

divM = cr(p) ae{M(p)cr(p)}

(1.133)

since b.V = de a ( p)

Here de is the length of the tube segment. In accordance with Eq, (1.133) divM can be treated as the rate of a change of the flux (but not the field) in the direction of the field normalized by the cross-section area of an elementary vector tube. 5. In deriving Eq. 0.128) we assumed that all derivatives of the field with respect to coordinates exist in the vicinity of the observation point. In other words, unlike Eq. (1.118), this expression for the divergence is valid at the usual points only. Now we will consider the special case when one of the field components is a discontinuous function and, correspondingly, Eq. (1.128) cannot be applied. Similar behavior of a field is often observed at interfaces between media with different physical properties. To determine the distribution of terminal points of vector lines on such a surface we will consider its element dS and surround it by a cylindrical surface Sc' as is shown in Fig. 1.12d. The total flux through this surface can be presented as a sum of

48

I

Fields and Their Generators

three flux corresponding to the lateral surface S t , as well as dSlpI) and dS/pz)·

Inasmuch as dS z = dS n, dS I = -dS D, where n is the unit vector directed from the back to the front side, we can rewrite Eq. (1.134) as (1.135) are the normal components of the field at the Here M2)(Pz) and M~I)(PI) front and back sides of the surface, respectively. With a decrease of the cylinder height, the integral value at the right-hand side of Eq. (1.135) tends to zero, provided that the field has finite values, but points PI and Pz approach point P on the surface. Correspondingly, instead of Eq. 0.135) we obtain

The next step is obvious. To characterize the distribution of terminal points on the surface S, it is natural to divide both parts of the last equation by dS. Then we have ¢M·dS dS

- - - = M(Z) - M(1) n

n

or ¢M'dS

dS

1

«;

=--

adS

Here qrn is the number of terminal points located within an elementary area dS. Correspondingly, one will introduce the surface divergence ¢M ·dS

DivM=lim

=

2.a ~

~S

~S

= M(Z) - M(1) n

n

as

~S

~

0

(1.136)

In accordance with Eq. (1.136), a difference of normal components of the field from both sides of the surface defines the density of terminal

49

1.6 Flux, Divergence, Gauss' Theorem

points. In particular, in the vicinity of those points of the surface where the normal component M; is a continuous function, terminal points are absent. Let us remember that a discontinuity can manifest itself as a change of either a magnitude or a sign of the vector field, or both of them. Thus, we have introduced three equations characterizing a distribution of vector lines. 1

¢M . dS = -;;qrn divM Div M

=

¢sM'dS lim - - - -

(1.137)

LiV

= M(2) - M(l)

n

n

on

S

Now, making use of the concept of terminal points of vector lines, let us derive one of the most important relations of field theory. Suppose that an arbitrary volume V, surrounded by the surface S, is mentally divided into many elementary volumes LiV. Then we will calculate the total number of terminal points inside the volume. First of all, by definition it is equal to the surface integral.

On the other hand, the number of terminal points inside any elementary volume dV is divMdV

and therefore the total number of these points inside the volume V is

f.v divM dV Thus, both the surface and volume integrals characterize the same number of terminal points of vector lines, and we obtain Gauss' theorem. ¢.M' dS

s

=

f.v divM dV

(1.138)

This equality plays a fundamental role in the theory and interpretation of geophysical methods. It results from the fact that Eq, (1.138) establishes a connection between the field on the surface S and its values inside the volume V. For this reason Gauss' formula is often used to derive equations that allow us to find a field somewhere inside the volume, as its values on

50

I

Fields and Their Generators

Fig. 1.13 (a) Gauss' theorem applied to a two-dimensional model; (b) the two-dimensional analogy of divergence; (c) voltage; and, (d) voltage and edge lines.

the surface S are known either due to calculations or measurements. It is also proper to notice that it is because of the application of vector lines we have been able to derive Gauss' formula in such an extremely simple way. To conclude this section we will obtain an expression for the divergence in the two-dimensional case as well as Gauss' formula, assuming that the field does not vary along some straight line 2' (Fig. I.13a) With this purpose in mind we will consider an elementary volume Il V, surrounded by the surface S, which consists of two elementary plane surfaces perpendicular to the line 2' and located at a distance Ilh from each other and the lateral surface Sf. Inasmuch as the field is tangential to two first surfaces, the flux through them equals zero and, correspondingly, for the total flux we have

~M'dS= S

f M·dS

(1.139)

Sf

Then, taking into account the fact that the field does not change along line 2', but

dS = 11 de' Sh ;

dV=dSllh

1.6 Flux, Divergence, Gauss' Theorem

51

we obtain for the two-dimensional divergence divsM =

Is t M· dS

¢tM' v de = .:. . :. . _

~V

~S

¢tMv dt

as

~S

~

0

~S

(I.140)

Here v is the unit vector perpendicular to the closed line t, and M; is the field component in this direction. Applying the same approach as in the three-dimensional case, div" M can be expressed in terms of derivatives as (1.141) It is clear that this equation directly follows from Eq. (1.129) if we let h 3 = 1 and JM 3 / Jx 3 = 0, and it is valid at the usual points only. In those cases when there is a line where the normal component of the field is a discontinuous function, one can characterize a distribution of terminal points along such a line in the following way:

div" M

=

A:.tM· v dt lim 'j'

= M(Z) v

~t

M:

M(l) v

(1.142)

M:l)

Z Here are components of the field perpendicular to the line ) and from both its sides (Fig. I. 13b). Thus, we have derived three equations describing a distribution of terminal points in the two-dimensional case, as is summarized below.

~M'v

'f/

1 dt= -qrn Q'

¢t M' v dt div" M = lim .:....:---~S

div" M =

M(Z) v

(1.143)

'

~S~O

M(l) v

Now we will consider some plane surface S, bounded by a closed contour t. Then, mentally dividing this surface into many elements dS and applying Eq. (1.140) for every such elementary surface ~Si we have div" M ~Si

=

¢,M . v dt t j

(1.144)

52

I

Fields and Their Generators

Here t; is the contour surrounding the surface AS;. Then, performing summation of this equality for every elementary surface and taking into account that integration along any internal contour t; is performed twice with opposite directions of v, we obtain Gauss' formula for the twodimensional case.

fs divSMdS=~M

t

.vdt=~Mvdt

t

(1.145)

In conclusion let us note that often it is convenient to present div M as divM= V·M 1.7 Voltage, Circulation, Curl, Stokes' Theorem In the previous section we introduced the flux of the field through a closed surface, as well as divergence, to characterize the distribution of terminal points of vector lines. Here we will study in detail a second type of tools used in field theory, circulation and curl. First we will consider only those fields that can be described with the help of normal surfaces. These results will then be generalized for any vector field. Consider normal surfaces sm and assume that (a) They are drawn with a density proportional to the field M. (b) The field M and the direction of the edge line L'" of a normal surface (Fig. 1.14a) form the right system (Fig. t.n.n To begin we will introduce a concept of the voltage as the integral (1.146)

along some path t with terminal points a and b. Let us make two comments.

1. Usually this notion is applied to electric and electromagnetic fields, but we will use it for all vector fields. 2. As will be shown later, the voltage generally depends on the path of integration as well as the position of the terminal points a and b. However, there is one special type of field that plays a very important role in the theory of some geophysical methods for which the voltage is path independent. Now let us attempt to evaluate the voltage by making use of normal surfaces. First, consider an element of the vector line, dl'm. By definition it

1.7 Voltage, Circulatlon, Curl, Stokes' Theorem

53

Fig. 1.14 (a) Voltage and edge lines; (b) illustration of curl; (c) illustration of curl; and,

(d) surface analogy of curl.

is perpendicular to the normal surface sm, and correspondingly its direction coincides with normal n. The voltage along this segment is dV=Mdr

(1.147)

since the angle between M and dl'm equals zero. It is clear that the number of normal surfaces dN z intersecting this element dt'" is (1.148)

Thus, the voltage and number of normal surfaces are related by 1 dV= -dNz

(1.149)

f3

or dN M . dl'm = M dr = _z

f3

54

I Fields and Their Generators

Now determine the voltage along an arbitrary oriented element dl' (Fig. 1.13c). Inasmuch as the projection of dl' on the vector line passing through point p is intersected by the same number of normal surfaces as the element d/, we have

dN z = 13M dt'" = 13M dteos(M, dl' )

= 13M· dl'= 13 dV

(1.150)

Again the voltage characterizes the number of normal surfaces that pass through the element d/. As Eq. (1.150) shows, the orientation of dl' has a strong influence on the voltage. For instance, if dl'is a segment of the vector line dl'ID, then there is a maximum number of passages of normal surfaces, and correspondingly the voltage also reaches its maximal value. If the element dl' is tangential to the normal surface, it is not intersected by normal surfaces and therefore the voltage vanishes. Finally, if elements dl' and dr form an angle exceeding 90 the number of passages of normal surfaces through dl' becomes negative, as does the voltage. In particular, if the elements dl' and dl'm have opposite directions, the voltage is negative, but it has a maximal magnitude. Thus, a change of the orientation of the element dl' at vicinity point p leads to a change of the voltage. Now making use of Eq. (1.150) and applying the principle of superposition, we obtain for the voltage V along an arbitrary contour t 0

,

V=

f

b

1 M· dl'= -Nz {3

at

(1.151)

Here N z is the amount of normal surfaces of the field intersecting the path t , and either their positive or negative contribution is defined by a mutual orientation of normal surfaces and the element d/. As follows from Eq. (US!), a change of the position of the path t does not change the voltage if the number of passages of normal surfaces through these paths remains the same. Having established a relation between the voltage and the number of the normal surfaces, let us introduce the concept of the circulation as the voltage along an arbitrary closed path.

V=¢.M'dl'

r

By analogy with the flux through a closed surface, we will raise the following question. Does the circulation characterize some essential features of the field behavior? The answer is yes and it can be obtained with

1.7 Voltage, Circulation, Curl, Stokes' Theorem

55

amazing simplicity. In fact, suppose that one normal surface sm passes through a closed contour t, and its edge line Lm is located outside the surface S, surrounded by the contour t (Fig. Ll Sd), In this case the normal surface intersects the contour t twice, so that at one time its normal n forms an angle with the contour segment dl'; which is less than 90°, while the second time the angle between them exceeds 90°. Correspondingly, the number of intersections by this normal surface equals zero, and therefore in accordance with Eq, (1.151) the circulation is also equal to zero. Generalizing this result, one can say that if all normal surfaces do not have edge lines L" inside the contour t, the circulation ¢(M' dl' equals zero. Next we will assume that the edge line of the normal surface intersects the surface S bounded by contour t. This means that this path is intersected by the normal surface only once; that is, the number of passages of this surface through the contour t is equal to either + 1 or -1 (Fig. 1.14a). As was pointed out above, the sign of the passage is defined by the mutual orientation of the edge line Lm and the path dt: Thus, in this case the circulation is

~M' 'Y

dl'=

1

+-{3

Now integrating we can conclude that in general, as there is an arbitrary number of normal surfaces, the circulation is

~M'd/=-Nz (

1

f3

(1.152)

Here N 2 is the total number of edge lines L m intersecting a surface S, bounded by the contour of integration t. Let us make several comments that describe this equation. 1. N z is an algebraic sum of positive and negative terms. Those edge lines that form a right-handed system with the vector d/ give a positive contribution, while the others define negative terms of this sum. 2. Any surface S bounded by the path t is intersected by the same number of edge lines L m, since they are closed. 3. A change of a position of the contour t does not affect the circulation if this path does not intersect the new edge lines or a sum of new intersections equal to zero. Similar behavior is observed in the case of the flux through a closed surface. Performing a deformation of the surface surrounding the volume, the flux will not change if the surface does not

56

I

Field~

and Their Generators

intersect terminal points or the sum of the number of terminal points equals zero. 4. Equation (I.152) establishes a very important fact, namely, that the voltage of this field along a closed path (circulation) characterizes the number of edge lines of normal surfaces that intersect any area bounded by this path. It is clear that the circulation presents itself as an analogy to the flux through a closed surface, since the latter describes the amount of terminal points within the volume surrounded by the surface. Moreover, as will be shown in the next section, edge lines and terminal points play a role of geometric models of two types of field generators: vortices and sources. This is the main reason why both functions

¢,M'dl' t

and

can be considered the most fundamental concepts in the theory of fields. In accordance with Eq. 0.152) the circulation along a contour t is defined by the total amount of edge lines L m passing through any surface bounded by this contour, but its does not provide any information about their distribution. Because of this, let us introduce a new tool that allows us to characterize both a density and a direction of these lines. With this purpose, unlike the case of terminal points, it is natural to use a vector. First, consider a small contour I' surrounding the plane surface dS (Fig. I.14b). As always, the directions of dl' and dS obey the right-hand rule. It is also assumed that within dS edge lines are distributed uniformly and all of them have the same direction. As is seen from Fig. I.14b, the number of edge lines intersecting the surface dS depends on its orientation. For instance, if the path t is located at the plane of edge lines, they do not intersect dS, and correspondingly the circulation equals zero. Let us choose a contour t around the point p, which is located in the plane perpendicular to the edge lines. Then it is obvious that a maximal number of these lines intersect the area dS, and the direction of the normal n (dS = dS n) is tangential to the edge lines. Inasmuch as the number of edge lines N 2 is

their density is defined as

N3 ¢M'd/ dS = f3 dS

(U53)

1.7 Voltage, Circnlation, Curl, Stokes' Theorem

57

To characterize also the direction of the edge lines we multiply both parts of this equation by unit vector L'8, which is directed along these lines, and then obtain N _2 LID dS 0

= a. ~

jM'dl' LID dS 0

(1.154)

The vector given by Eq. 0.154) is called curl M, curlM =

dl'

~M'

dS

LID

(1.155)

0

or curlM

= lim

~M'

dl' liS L'8

as

liS -) 0

and it describes the density as well as the direction of edge lines in the vicinity of the source point p. For example, if curl M is negative, the edge lines have a direction opposite to that of the normal n. 5. In deriving Eq. (1.155) we have to choose a small closed contour, for instance a circle, around point p to determine the circulation. However, it does not, of course, mean that the geometry of the field is such that its vector lines should also be closed around this point. 6. As in the case of divergence, it is appropriate to replace the cumbersome procedure of integration on the right-hand side of Eq, (1.155) by differentiation. With this purpose, let us introduce the vector B, whose component in any direction is characterized by unit vector p, defined as 1

B = -~M'dl' p dS p 'Jf

(1.156)

Here dS p is the elementary surface with its center at the point p located in the plane perpendicular to p, t is the contour surrounding this surface, and its direction and that of vector B form a right-handed system. In particular, along coordinate lines we have (1.157)

Here dS k is the element of the coordinate surface perpendicular to the line t k , and t [dSk] is the contour bounding this area. Multiplying each component Bk by the corresponding unit vector and adding all three

58

I

Fields and Their Generators

products, we obtain for the vector B

It is clear that along this vector its component has the maximal value, which is equal to the magnitude of B. Therefore, by rotating the surface dS p around point p the right-hand side of Eq. 0.156) B p reaches the maximal value, equal to IBI, as the direction of p coincides with that of B. At the same time the circulation ¢M . dl'in this equation is defined by the amount of edge lines U" piercing the area dS p ' and it is a maximum when p is tangential to these lines. In other words, in such a case the surface dS p is located at the plane perpendicular to the edge lines in the vicinity of the point p. Correspondingly, the vector B can be written as

B=

¢M'dl' dS

L'g

and comparison with Eq. (U55) shows that it coincides with curl M, curlM =B

(1.158)

but its components are defined from Eq. (1.157). Such presentation of curl M is essential since it gives us an expression for its components along coordinate lines. Moreover, Eqs. 0.157), (U58) are introduced without the use of normal surfaces and edge lines, and correspondingly these equations of curl M are applied for any kind of fields. In fact, curl M is usually defined by Eq, (U57). 7. Now we are ready to describe this vector using derivatives. First, consider a contour t, surrounding the elementary surface dS I with its normal directed along the coordinate line t I and consisting of two pairs of segments parallel to the coordinate lines t z and t 3' respectively (Fig. I.I4c). Taking into account the fact that the field M does not change along every segment, we have for the circulation

¢M . dl'= M( PI) . dl'( PI) + M( pz) . dl'( pz)

+ M( P3) . dl'( P3) + M( P4) . dl'( P4) or

¢M' dl'=Mz(Pl) dtz(pt) +M3(pz) dt3(pz) - M z( P3) dtz( P3) - M 3( P4) dt3( P4)

(1.159)

1.7 Voltage, Circulation, Curl, Stokes' Theorem

59

Inasmuch as distances between opposite sides of the contour are small, the corresponding differences of terms in Eq, 0.159) can be expressed through first derivatives. Then we have

or

a(h M )

A;M'dl'=

3

'Y

ax

3

dx dx _ Z

Z

a(h M )

z z dx dx

ax

3

Z

3

3

In accordance with Eq. (U57) the projection of curl M on the coordinate line t j is (1.160) since

In the same manner for the two other components we have

(1.161) curl 3 M

=

1 [ahzM z -hjh Z ax]

-

ahjM j] --ax z

Thus we have described curl M with the help of six derivatives in the vicinity of the point p, located at the center of the elementary volume formed by coordinate lines. Also, we present curl M in a more compact form as

curlM

1 =

hji[ a

h]h Zh3 ax] hjM]

hzi z

h 3i3

a

a

ax z

aX3

hzMz

h 3M3

(1.162)

As in the case of gradient and divergence, it is often useful to apply the nabla operator, and then we have curlM

= VX M

60

I

Fields and Their Generators

It is obvious that Eqs. (1.160)-(1.162) can be used only at usual points where derivatives of the field exist. To illustrate these results, let us present curl M in the three simplest systems of coordinates.

1. Cartesian system curl M

JM z Jy

JM y Jz

= -- - --

x

JM z JM x curl M= - - - - Y Jz Jx

(1.163)

JM y JM x curl M = - - - - z Jx Jy 2. Cylindrical system curl M=

~[JMz r

r

Jr.p

_ JrMcp] Jz

JM, JM z] curl M= [ - - - - cp Jz Jr curl M

1 [JrMcp JM r ] - - - --

= -

z

(1.164)

r

Jr

Jr.p

3. Spherical system curl , M

1 R sin

[J JM-e] sin (}Mcp - e Jr.p

= --.- -

J()

R 1 - [JM curl, M = - - - - sin (JRMcp] )-R sin () Jr.p JR

curl M = ~[JRMe cp R JR

_ JM R

(1.165)

]

J(}

8. As was mentioned above, the vector curl M is usually arbitrarily oriented with respect to the field M, and correspondingly normal surfaces cannot be, in general, used to study a field behavior. However, there is one type of field when at every point the angle between these vectors equals

1.7 Voltage, Circulation, Curl, Stokes' Theorem

61

90°, It occurs if the field M can be presented as

M=',O(p)gradU(p)

(1.166)

Here ',O(p) is a scalar function that can be differentiated. To prove that the vector M, defined by Eq, (1.166), and curl Mare perpendicular to each other, it is sufficient to show that M· curlM =

a

With this purpose, we will make use of Eq. 1.20:

a'(bXc)

-b'(axc)

=

-a'(cXb)

=

(1.167)

as well as two relations curl grad U =

a

(1.168)

and curl( ea) =

',0 curl

a + grad ',0 X a

(1.169)

Letting a = grad U we have curl M

=

curl ( ',0 grad U)

=

',0 curl

grad U

+ (grad ',0 X grad U)

Then taking into account Eqs. (1.166) and 0.168) we obtain curl M = grad

',0 X

1 grad U = - grad ',0 X M ',0

whence M

M'curlM= ',0

grad ',0 '(grad',OXM) = - - - '(MXM) =0 ',0

Thus, the curl M is perpendicular to the field M if Eq. 0.166) is valid. Such fields are called quasi-potential fields, and correspondingly there exists a family of normal surfaces perpendicular to the vector lines. In general, this is not true and because of this, curl M is defined by Eq. (1.156). At the same time it is proper to note that any field can always be presented as a sum of two fields.

Here M 1 is a quasi-potential field, while M 2 is described by the curl of some other vector.

62

I

Fields and Their Generators

Taking into account the fact that our main goal is to illustrate as much as possible the main concepts, such as curl, we will continue to use normal surfaces and edge lines with the understanding that the area of their application is limited. 9. It is obvious that Eqs, (1.156) and 0.162) have the same meaning. In particular, for a quasi-potential field they characterize the density and the direction of edge lines in the vicinity of the point p. At the same time Eq. 0.162) cannot be used near points where any component of the field is a discontinuous function, since the corresponding derivative does not exist in this case. As was pointed out earlier, a singularity in the behavior of a field is often observed at interfaces of media with different parameters. To characterize a distribution of edge lines located at this type of surface S in a simpler manner than Eq. (1.156), we will introduce a system of coordinates where lines t 1 and t 2 are tangential to the surface S; but t 3 coincides with normal n, as directed as always from the back to the front side (Fig. 1.14d). First, consider an elementary contour, located on the coordinate surface Sl around the point p. It is clear that the circulation along this contour is

Here M(2) and M(1) are values of the field from the front and back sides of the surface, respectively. Inasmuch as

but segments of the contour along the normal n tend to zero, we obtain (1.170)

In accordance with Eq. (1.151) this relation characterizes the number of edge lines intersecting an element of the coordinate line t 2. Correspond-

1.7 Voltage, Circulation, Curl, Stokes' Theorem

63

ingly, the density of their distribution is proportional to

¢M'dl'

--- = -

de2

(M(2) - M(1») 2

(1.171)

2

In a similar manner the density of edge lines passing through the element dtl of coordinate line t 1 is proportional to

¢M'dl' dt

--- =

M(2) _ M(l) 1

I

(1.172)

I

Correspondingly, the vector that describes a density of edge lines and their direction on the surface S can be presented as Curl M = (- MO) + M(1»)i + (M(2) - M(1»)i 2 2 1 I 1 2

(1.173)

and it is called the surface analogy of curl M. The latter can also be written in the form (1.174) Here M(2) and M(1) are the fields at either side of the surface. This result follows from the definition of the cross product a X b, if a is the unit vector, il

i2

n

aXb= 0

0

1

bx by bz In accordance with Eq. (1.174) the distribution of edge lines at the surface S is characterized by the difference of the tangential components of the field at either side of the surface. In particular, near those points where this component is a continuous function, edge lines are absent. Now let us generalize this result for an arbitrary field. With this purpose in mind, let us rewrite Eq. (1.156) as 1

B = curl M = --A:.M . dl' n n dS'tj Here n is the unit vector perpendicular to dS (dS Inasmuch as B;

=

B' n

or

curl, M

=

=

(1.175)

dS n),

n : curlM

64

I

Fields and. Their Generators

we have dS n . curl M = dS . curl M = WM . dl'

(1.176)

The left-hand side of Eq. 0.176) represents the flux of the field B = curl M through the elementary surface dS; that is, it characterizes the amount of vector lines of the field B = curl M passing through dS. At the same time, in accordance with this equation, the circulation along the contour t also describes the number of vector lines of the field B. In other words, the circulation along the contour t surrounding the elementary surface dS is equal to the flux of the field curl M through this surface. It is obvious that Eq. 0.176) is valid regardless of the orientation of the surface dS with respect to curl M. Applying this result to the surface analogy of curl M, one can say that in general Eq. 0.174) characterizes the density and direction of vector lines of the field curl M on a surface S. Thus, We have derived three equations that characterize either the distribution of edge lines of normal surfaces of the quasi-potential field M or, in a general case, vector lines of the field B = curlM. These equations are ~M'dl'=-N. 'f1

curl M =

1 {32

¢M'dl' dS to

(1.177)

Here to is the unit vector tangential to the vector lines B. Let us remember the following. In the first expression, t is an arbitrary contour and the integral defines the number of vector lines of the field B = curl M intersecting any surface bounded by the contour. At the same time, this circulation does not provide any information about the distribution of lines within such surfaces. In particular, the circulation can be zero even in the presence of vector lines of the field B. This happens when there are equal amounts of positive and negative passages of these lines through the surface. In the second expression t is a small contour surrounding the element dS, located in the plane perpendicular to the unit vector to' Correspondingly, curl M characterizes the density and direction of its vector lines.

1.7 Voltage, Circulation, Curl, Stokes' Theorem

65

Finally, the third expression describes a distribution of these lines on a surface S. As in the previous section it is natural to raise the question: Why do we pay so much attention to the distribution of vector lines of the field curl M? It turns out that the answer is very simple, and in the next section we will show that these lines allow us to visualize the distribution of the second type of field generators called vortices. 11. Now we will prove the second theorem that plays a very important role in the theory of fields. First we will start from the quasi-potential field and consider the dot product. curlM'dS Here dS is an elementary surface, arbitrarily oriented with respect to curlM. In accordance with Eq. 0.155) this product equals the number of edge lines passing through the element dS. Then, applying the principle of superposition, the total number of edge lines intersecting an arbitrary surface S is proportional to

fs curlM· dS

(1.178)

On the other hand, this number of edge lines is proportional to the circulation of the field along the contour surrounding this surface. (1.179)

¢.M'de' t

In both cases the coefficient of proportionality f3 is the same. Therefore, we obtain

f curlM' dS =~M' S

dl'

(1.180)

I

Here it is essential that the direction of the normal to the surface Sand that of the displacement along the contour de' are related to each other by the right-hand rule. Thus, we have proved Stokes' theorem for quasipotential fields, which establishes the relation between values of the field at points of the surface and the contour surrounding it. Next we will show that Stokes' theorem remains valid for an arbitrary field M. With this purpose in mind let us imagine that any surface S is represented as a system of elementary surfaces dS i , surrounded by the contour t i (Fig. 1.14d). In accordance with Eq. (1.176) we have dS; . curl M = ¢.M . dl' Ii

(1.181)

66

I

Fields and Their Generators

Then, performing a summation of this equality, written for every elementary area dS], and taking into account that integration along every internal contour is done twice but in opposite directions, we obtain again Eq. (1.181),

fs curl M.: dS =~M't

dl'

(1.182)

which is valid for any vector field M. This is Stokes' theorem. Undoubtedly both Gauss' and Stokes' theorem represent a foundation of the theory of fields applied in geophysics, and many important relations will be derived making use of these theorems. In this connection it is proper to notice that both theorems are valid at those points where field M does not have singularities. In particular, if such points are located at a surface and the field has finite values, the functions div M and curl M are replaced by their surface analogies, respectively.

1.8 Two Types of Fields and Their Generators: Field Equations

In the last two sections we introduced div M and curl M as operators that allow one to characterize the distribution of terminal points of vector lines and edge lines of normal surfaces of the quasi-potential field M, respectively. Now it will be demonstrated that these operators playa fundamental role in the theory of vector fields. With this purpose in mind, let us first consider the quasi-potential field M. M = 'P( P) grad U( p)

(1.183)

Here 'P(p) and U( p) are continuous scalar functions. Then, in accordance with results derived earlier we can write divM= Q

DivM=Q s (1.184)

1M' dl'= t

w*

curlM = W

CurlM = Ws

Here Q *, Q, Q s and W *' W, Ws are scalar and vector functions that describe the distribution of terminal points and edge lines. We will treat Eqs. 0.184) from two completely different points of view. First, suppose that the field M is known everywhere, and we would like to find places

1.8 Two Types of Fields and Their Generators: Field Equations

67

where, for some reason, vector lines and normal surfaces have terminal points and edge lines, respectively. At any point p, where the first derivative of the field exists, it can be done by calculating divM and curlM, Q(p) = divM

and

W(p)

=

curlM

(1.185)

while at interfaces, as one of the field components is discontinuous, we have and

(I.186)

Of course, instead of Eqs. (I.185), 0.186) one can use the integral form and

(I.187)

Indeed, by choosing a sufficiently small volume surrounded by a closed surface, and also a properly oriented small contour, one can again determine a distribution of terminal points and edge lines. Thus, in accordance with Eqs. (I.185)-(I.187) we can consider div M and curl M as well as the flux and the circulation as indicators of special places where terminal points and edge lines are located, and they characterize their distribution. This is the first approach of interpretation of Eqs. (I.184), which in essence gave the motivation to introduce divergence and curl. However, we will now treat these equations in a completely different way. Suppose that terminal points of vector lines and edge lines of normal surfaces are given, that is, functions and are known. Then, at the usual points where derivatives of M exist, we will treat relations curlM =W

and

divM= Q

(I.188)

as equations with respect to the unknown field M. At points where the first derivatives do not exist we have instead of Eqs. 0.188) their surface analogies. CurlM = Ws

and

DivM = Qs

(1.189)

Also, in accordance with Eqs. 0.184) the third form of the equations with

68

I

Fields and Their Generators

respect to unknown field M is

and

(1.190)

Let us note that the left and right-hand sides of Eqs. (1.188), 0.189) describe the behavior of the field M and functions Q and W in the vicinity of the same point, while Eqs. 0.190) relate to each other all these functions M, Q *, and W *, at different points. It is also essential that these equations do not contain derivatives and, correspondingly, they can be applied everywhere. In considering the first approach, we have seen that determination of the density of terminal points as well as that of edge lines and their orientation requires a differentiation of the field M, and as soon as the field is known, it is a straightforward problem that can be performed without any difficulties. A completely different situation holds when we would like to find the field M, assuming that functions Wand Q are known. Let us start from Eqs. (1.188) and raise three questions concerning these equations, namely,

1. How do we solve these equations? 2. What is the meaning of the right-hand side of Eqs. (1.188), 0.189)? 3. Do these systems define the field M uniquely? The first question presents itself mainly as a subject of mathematical physics, and it is not studied in this monograph. However, in some of these chapters several examples illustrating a solution to these equations have been given. Because of this we will restrict ourselves to their brief description only. In accordance with Eqs, (1.188) we have four differential equations with partial derivatives of the first order. In fact, in a curvilinear orthogonal system of coordinates this system is written as

(I.191)

1.8 Two Types of Fields and Their Generators: Field Equations

69

and

Suppose that there is also a surface of singularity where the field Mean be a discontinuous function. Then from Eqs. (1.189) we have

where all notations are given in the previous two sections. Again we have equations, but linear ones, relating unknown values of the tangential and normal components of the field from both sides of the surface S with given functions Ws(p) and Q/p). Let us examine the second question. First of all, we assume that everywhere in a space terminal points and edge lines are absent; that is, curlM = 0

divM = 0

(1.193)

=0

DivM= 0

(I.194)

and CurlM

Both pairs of equations are rather complicated, but it turns out that it is surprisingly easy to find the field M satisfying these systems. In fact, inasmuch as curlM = 0

and

CurlM = 0

(I.195)

open normal surfaces having edge lines are absent. Also, the field does not have closed vector lines. If there were even one closed vector line, the circulation ¢fM . d/ along this line could not be equal to zero since both the field M and the displacement dl' have the same direction. Then by definition, curl M and Curl M would differ from zero, too. From equations, div M

=

0

and

Div M

=

0

(I.196)

it directly follows that open vector lines, having by definition terminal points, are absent. Also we have to conclude that closed normal surfaces do not exist. If there were such surfaces, the flux through them would differ from zero since both vectors M and the normal n to the surface have the same direction. This result means that somewhere inside the volume surrounded by such a normal surface, there are terminal points of vector lines; that, of course, contradicts Eqs. 0.196).

70

I

Fields and Their Generators

Now we present results of this analysis in the form of a table. Equations curlM = 0 CurlM = 0

divM=O DivM=O

no closed vector lines

no open vector lines

no open normal surfaces

no closed normal surfaces

It is obvious that the same conclusion can be derived from the integral form of Eqs. (1.193), (1.194).

~M'

dl'= 0

and

f

~M' s

dS =0

(1.197)

Inasmuch as one can imagine only two types of vector lines and normal surfaces, namely open and closed ones, the table vividly shows that the field M, satisfying the systems 0.193), (1.194), must be everywhere equal to zero. In other words, to have a field M that is different from zero, there should be places in a space where either terminal points of vector lines or edge lines of normal surfaces or both of them are located; that is, without their presence the field cannot exist. Taking into account that only field generators such as charges, currents, stresses, etc., can create the field, we have to conclude that these generators of the field are located at those places where terminal points of vector lines and edge lines of normal surfaces are present. In essence, we can say that these points and lines have been introduced to visualize a distribution of field generators. Correspondingly, an interpretation of the system of Eqs, (1.193), 0.194) becomes obvious. Inasmuch as the field M satisfying this system does not have generators, it is natural that it equals zero.

M=O

(1.198)

Let us make one more step and classify field generators into two types, called sources and vortices.

I Field generators I

/

I sources I

~ I vortices I

Sources are a type of field generators whose distribution can be described by terminal points of vector lines. Classical examples of sources are gravitational masses and electric charges.

1.8 Two Types of Fields and Their Generators: Field Equations

71

Vortices are the second type of generators, and in the case of a quasi-potential field they can be described with the help of edge lines of normal surfaces located inside of some toroidal tubes. Unlike the first type of generators, vortices are characterized by both a magnitude and a direction. Electric current and a change of the magnetic field with time are well-known examples of vortices. It is natural to expect that different types of generators create fields of very different behavior, and for this reason it is proper to consider them separately. Depending on the type of generators, we will call fields either source or vortex fields, respectively.

I Field type I Vortex field

Source field

General case due to both the source and vortex generators Now it is time to return to Eqs. (1.188), (1.190). Taking into account .their importance, let us write them as one system again. ( a) (b)

~M' t

dl'= W*

curlM=W CuriM

-w

~M' s

dS =Q* divM = Q

(I.199)

DivM = Qs

On the right-hand side the functions W *, W, Ws and Q *, Q, Q s describe generators of the field, and on the left-hand side we have the field M along with its derivatives caused by these generators. What could be a more natural relation than that between the cause and its effect? And this fact is the main reason why Eqs. (1.199) are called systems of field equations in the integral (a) and differential (b) forms. At the beginning of this section we assumed that the field M is a quasi-potential field; that is,

M(p) =cp(p)gradU(p)

(1.200)

and correspondingly the system of Eqs. (1.199) was derived making use of terminal points of vector lines as well as edge lines of normal surfaces.

72

1 Fields and Their Generators

Before we demonstrate that this system remains valid for any type of field, let us investigate a distribution of generators of the quasi-potential field in terms of the function, 'P(p) and vector P, where P = gradU

(1.201)

Taking into account that curl M = curl 'PP = 'P curl P - P X grad 'P

= -P X grad 'P

(1.202)

since curl P

=

curl grad U == 0

and divM

div 'PP = 'P divP + p. grad 'P

=

within a constant scale factor, the generators of the field M at usual points are

w=

-

Q

'P div P

=

P X grad 'P

+ P . grad 'P

(1.203)

Analysis of Eqs. (1.202), (1.203) allows us to make several comments, such as 1. The field P, expressed through the gradient of function U, is the classical example of a source field and is often called the potential field. 2. Vortices of the field M are located only in the vicinity of points where vector P and grad 'P are not parallel to each other. 3. Sources of the quasi-potential field consist, in general, of two parts. One of them appears near those points where sources of the potential field P exist if 'P(p) =F 0, while the second part of sources arises in places where vectors P and grad 'P are not perpendicular to each other. Of course, these two types of sources can appear at the same as well as at different points. In particular, they can cancel each other if 'P div P

=

-

P . grad 'P

so that at some points the total density of sources vanishes. Now we will consider surface distributions of sources and vortices of the quasi-potential field. Inasmuch as

1.8 Two Types of Fields and Their Generators: Field Equations

73

we have CurlM

=" X H( 'Pz -

'PIHPz + PI) + ('PZ + 'PI)(PZ - PI)}

where 'PZ' P z , and 'PI' PI are values of the scalar and vector functions at the front and back sides of the surface, respectively. Taking into account Eq. (1.201) we can show that the tangential component of the potential field P is a continuous function; that is,

Therefore, the surface vortices are Ws

=" X t( 'Pz -

'Pl)(PZ+ PI)

= ('Pz -

'PI)" X

pay

or Ws =

- pay

X Grad 'P

(1.204)

Here

is the average value of the potential field P at the surface. By analogy we have for the surface sources Div M = M(Z) n

M(1) n

=

m

"1""'2

p(2) n

m

..,..1

p(1) n

or Q s = p av • Grad 'P + 'Pav DivP

(1.205)

Here av

'P

=

'PI + 'Pz 2

It is obvious that Eqs. (1.204), (1.205) are very similar to the corresponding Eqs. (1.203), and as in the case of volume distribution of field generators, it

is appropriate to make the following comments:

1. Surface vortices arise at the vicinity of points where a discontinuity of function 'P(p) takes place and also where the tangential component of the potential field P is not equal to zero. 2. In general, there are two types of surface sources. One type arises exactly in places where sources of the potential field are located if 'P I 'Pz· At the same time the second type of source appears near points of discontinuity of function 'P(p), provided that the normal component of the field P does not vanish.

*-

74

I

Fields and Their Generators

Let us illustrate the quasi-potential field by one example. As follows from Ohm's law the current density j(p) and the electric field E(p) are related in the following way: j(p)

y(p)E(p)

=

where y( p) is the conductivity of the medium and, in general, changes from point to point. Inasmuch as the electric field of constant charges is a potential field, we have j(p) =

~y(p)gradU

That is, the current density field is a quasi-potential field caused by sources as well as vortices. As was pointed out above in deriving Eqs. (I.199), both vector lines and normal surfaces have been used, and correspondingly its application was restricted to the quasi-potential fields only. Now we will generalize this result and show that it is natural to treat Eqs. (1.199) as a system of field equations valid for any field. With this purpose we will completely discard the concept of normal surfaces and make use of vector lines. Our goal is to demonstrate that for any kind of field M, the right-hand side of Eqs. (I.199) still consists of field generators. Applying the same approach as in the case of the quasi-potential field, we will assume that everywhere, including usual points and surfaces where there may be singularities, the field M satisfies the following equations: curlM = 0 CurlM = 0

divM = 0 DivM=O

(1.206)

From equations curl M = 0

and

Curl M = 0

as well as Stokes' theorem, it follows that the circulation of M along any path must be equal to zero.

Inasmuch as at every point of the vector line both vectors M and dl' have the same direction, that is, M· dl'> 0 we have to conclude that the field M does not have closed lines. At the same time, in accordance with equations divM = 0

and

DivM =0

vector lines of this field cannot be open since they have no terminal points.

1.8 Two Types of Fields and Their Generators: Field Equations

75

Thus, we have proved that the field M satisfying Eqs, (1.205) has neither closed nor open vector lines, and this simply means that the field M is everywhere equal to zero. This analysis leads us again to the conclusion that if at every point space functions W,Ws and Q,Qs are equal to zero, then the field M also vanishes everywhere; for this reason it is natural to suppose that these functions, as in the case of quasi-potential fields, characterize a distribution of sources and vortices. Correspondingly, we will assume that for an arbitrary field we have the same system of equations as in the case of the quasi-potential field; that is, in the differential form curlM=W

divM=Q

CurlM = Ws

DivM= Qs

(1.207)

and the integral form is and

(1.208)

Here M is an arbitrary field, but W,Ws , W * and Q I , Qs , Q * are its vortices and sources, respectively. Here it is appropriate to make several comments. 1. The system of field equations, as in the case of the quasi-potential field, is a "bridge" between the field and its generators. 2. Each field applied in geophysics has a different physical nature, and correspondingly they are governed by different physical laws, such as Newton's, Hooke's, Biot-Savart's, Faraday's, etc. At the same time, the field equations are always derived from physical laws, as indicated below.

IPhysical ' laws I

~

fiieSystem ld equa tiof IOns

3. Usually functions E, E, and Q1 and Qs differ from a corresponding density of field generators by some constant that is independent of the position of the observation point. 4. Fields are often studied in the presence of some medium. Due to this fact some part of generators cannot, in principal, be known before determination of the field. Correspondingly, most of the physical laws such as Coulomb's, Hooke's, Faraday's, etc., which establish a connection between a field and its generators, become useless for field calculation, and the system of field equations turns out to be the only means of solving the problem. At the same time, considering carefully this system, we usually observe the "closed circle problem" since the right-hand side of

76

I

Fields and Their Generators

Eqs. (1.207), (1.208) is not known. In fact, to find the field M we have to know the distribution of its generators, but some part of them cannot, in principle, be known before the field calculation. To overcome this problem a new field is usually introduced that is related to the field M and its generators, with the help of corresponding parameters of a medium. Respectively, instead of the system of two equations, we often obtain a system of four equations with respect to two vector fields. The classical example of such a system is Maxwell's equations. Now we are ready to discuss the third question concerning the uniqueness of a solution of Eqs. (1.207). First of all, from a physical point of view it is almost obvious that if a distribution of field generators is everywhere known, the field is uniquely defined by them. Let us illustrate this fact by one example. Suppose there is a certain distribution of chairs and tables in some room that creates a gravitational field, and outside the room masses are absent. In other words, these chairs and table are the sale generators of the field which, in accordance with Newton's law of attraction, strongly depends on a distance between the observation point p and the masses. If we vary the position of even one chair slightly, but the others remain at rest, then certainly a change of the field will be observed. Of course, at some points located closer to the moved chair, a strong change of the field takes place while far away from this mass there can be a very small change, even unmeasurable by modern sensors. However, it is essential that as soon as there is a new distribution of generators, a new field arises, and correspondingly it is proper to expect that the system of Eqs. (1.207) uniquely defines the field. Let us derive the same result making use of a different approach. Suppose that there are two different fields, Mj(p) and Mip), and that both of them satisfy the system of Eqs. 0.207) with the same right-hand side; that is, curl M, = W(p)

divM, = Q(p)

CurlM l =Ws(p)

DivM j = Qs(p)

and

(1.209)

curlM 2 = W( p)

divM, = Q(p)

Curl M, = Ws ( p)

DivM z = Qs( p)

Now we will form the difference between these two fields.

which is also a vector field.

1.8 Two Types of Fields and Their Generators: Field Equations

77

Taking into account that curl and divergence are linear operators, we have curl M, = 0 div M, = 0 (1.210) curl M, = 0 DivM 3 = 0 Inasmuch as Eqs. (1.210) describe the field M 3 everywhere, we can apply the results derived above and conclude that this field equals zero. In other words, two fields M] and M 2 , caused by the same distribution of generators, coincide with each other. Therefore, we have proved again that the system of Eqs, (1.207) uniquely defines the field provided that its generators are known. everywhere. Our next step is to study the problem of uniqueness in a more complicated case, in which field generators are given only in a certain part of the space. To investigate this problem, let us first suppose that generators are absent in the volume V where the field is studied, but they can be located outside, creating the field M everywhere including observation points of the volume V. Then, taking into account that the functions W, WS ' Q, and Qs are equal to zero within volume V, the system of Eqs. (1.207) becomes homogeneous, and we have curlM = 0 curlM = 0

divM = 0 divM = 0

(1.211)

Thus, the system of equations describing the field behavior within some volume V is homogeneous if its generators are located outside. It is obvious that by changing the distribution of generators, the field M caused by them also varies, but the system of equations still remains uniform within this volume. Correspondingly, we can say that Eqs. (1.211) have an infinite number of solutions and every one of them can be interpreted as a field whose generators are distributed somewhere outside the volume V. Next we will assume that field generators are known inside the volume V, while they are absent outside of it. To some extent, this case is the opposite of the previous one. Then the system of field equations is (a) Inside volume V curlM=W

divM=Q

CurlM=Ws

DivM=Q s

(1.212a)

(b) Outside the volume curlM = 0 CurlM = 0 Here W, WS ' Q, and

divM = 0 DivM = 0

o; are given functions.

(1.212b)

78

I

Fields and Their Generators

It is easy to show that this system uniquely defines the field. In fact, suppose there are two fields, Mt(p) and M 2( p ), satisfying Eqs. (1.212). This means that both of the fields are caused by the same generators located only inside the volume V. Then it is clear that the difference of these fields

satisfies everywhere the homogeneous system curl M, = 0

divM, = 0

Curl M,

DivM 3 = 0

=

0

and therefore or In other words, the system of Eqs. (1.212) uniquely defines the field M. That is hardly surprising since all of its generators are known. Now we are ready to discuss the uniqueness of a solution of the system (1.207) provided that the field generators are known only within the volume V where the field is considered. Unlike the previous case, there can also be generators of the field outside the volume that are not specified, and for this reason it is natural to expect that Eqs. (1.207) do not uniquely define the field M. Indeed, suppose that two fields Mt(p) and Mip) are solutions of this system. Then their difference (1.213) is a solution of the homogeneous system within the volume V. curl M,

=

0

Curl M; = 0

div M,

=

0

DivM 3 = 0

As was shown above, it means that in general the field M 3 differs from zero since its generators can be located somewhere outside. Thus, we have demonstrated that the system of field equations defined within some volume V has an infinite number of solutions that differ from each other by functions describing fields due to generators located outside the volume V. Let us represent the solution of Eqs. (1.207) as a sum. (1.214) where Mi(p) and Me(p) are fields caused by generators inside and outside the volume, respectively. As was shown, the field Mi(p) is uniquely defined, inasmuch as its generators-that is, functions W, Ws ' Q, and Qs

1.8 Two Types of Fields and Their Generators: Field Equations

79

-are known. However, the field Me(p), regardless of distribution of its generators, satisfies a homogeneous system of field equations; this fact explains the nonuniqueness of the solution of Eqs. (1.207). Because this system, defined in some volume, is not sufficient by itself to determine the field, it is natural to raise the following question: What has to be known in addition to the system of field equations in order to determine the field M uniquely? In this section we will not describe this question in detail, but instead, just outline this problem. With this purpose in mind, first we will consider a very simple case when all generators of the field are located within the volume V while the field M due to these generators is studied outside of it. Inasmuch as in this external part generators are absent, the system of field equations is homogeneous,

a

divM = 0

CurlM = 0

DivM =0

curlM =

(1.215)

regardless of the position of generators within the volume V, and correspondingly it has an infinite number of solutions. If we do not know the distribution of field generators inside the volume V but are trying to find the field outside, we have to have some information about the field that can replace the absence of knowledge about its generators. And it is almost obvious that the surface S surrounding the volume V is the most natural location for such information. Then the system of Eqs, (1.215) and certain information about the field behavior on the surface S may provide a unique solution. Later we will demonstrate that in fact the so-called theorem of uniqueness can be formulated in terms of several types of such information on the surface S. Having accepted this fact, we can say that all information providing the uniqueness of the solution of Eqs. 1.215 consists of two parts, namely, (a) Outside the volume V curlM = 0

divM = 0

CurlM = 0

DivM = 0

and (b) Information about the field behavior on the surface S, usually the behavior of some component of the field M, which can be represented as (1.216)

Here M, is either the tangential or normal component of the field and CPt is the given function.

80

I

Fields and Their Generators

Therefore, determination of the field includes two steps. 1. Solution of Eqs. 0.215). As was shown above, an infinite number of vector functions satisfy this system. 2. Choice among these solutions of a field M such that its corresponding component becomes equal to the given function 'Pip) on the boundary surface S.

Taking into account the importance of this last step, the whole process of determination of the field is called a solution of a boundary-value problem. Here it is also appropriate to make several comments. (a) The same analysis can be applied to the case in which the generators are located outside the volume V, but the field is considered inside of V. (b) If a solution of Eqs. (1.215) is considered inside the volume V, surrounded by several surfaces, boundary conditions of either type (1.216) or others have to be defined at every surface. (c) Usually these boundary conditions are derived from an analysis of a physical nature of the problem, and as a rule they characterize a field behavior near the given generators that are known and at infinity. (d) In general, regardless of the type of boundary condition, they do not uniquely define a distribution of generators outside of the volume V. Now let us generalize on the previous case and assume that generators can be distributed everywhere, but that the field M is sought within the volume V, surrounded by the surface S. Then the solution of the value boundary problem, as before, consists of two steps, namely, (a) Solution of the system of field Eq. (1.207) within the volume V, and (b) Selection of such solutions of this system that satisfy the boundary conditions on the surface S. In conclusion to this section let us consider one more subject related to the presentation of the field caused by both types of generators. First, we will represent the field M within some volume V as a sum of three fields, (1.217)

where M 1 and M 2 are fields caused by sources and vortices distributed within the volume V, respectively, while M 3 is the field due to generators located outside this volume. Then, by definition they satisfy the following

1.9 Harmonic Fields

81

systems of field equations:

curl M, = 0

div M, = Q

Curl M, = 0

DivM 1 = Qs

curl M, =W

divM, = 0

CurIM 2 =Ws

DivM 2=O

(1.218)

(1.219)

and

=0

cUrIM 3 = 0

div M,

Curl.M,

DivM 3 = 0

=

0

(1.220)

In the next sections we will study some general features of these fields, and it is natural to begin this analysis from the simplest field M 3 , which of course is a partial case of either field M, or field M 2 • Also it is useful to note that the field M can be presented as M=Mj+Mi +M 3

(1.221)

where Mj is the quasi-potential field Mj =cpgrad U generators of which are located within the volume V; Mi is a field caused by vortices only, located also within this volume; but M 3 is a field due to generators distributed somewhere outside of it. Then, we have curlMj =

wt

divMj = Q

Curl M] = Wl~

DivMj = Qs

curlMi = W2*

divMi = 0

Cur.lMi = W~

DivMi = 0

curl M, = 0

divM, = 0

Curl M, = 0

DivM 3 = 0

where W = W)* + W2* , Ws = W)~ + W~, of the field Mwithin the volume V.

(1.222)

and Q, Qs ,W,Ws are generators

1.9 Harmonic Fields We will study a field M in some volume V assuming that the generators, sources, and vortices of the field are located outside of the surface S that surrounds this volume. Then, as was shown before, the field satisfies the

82

I

Fields and Their Generators

homogeneous system of equations curlM = 0

divM = 0

CurlM = 0

DivM=O

or

(1.223)

~M' t

dl'=O

~M'dS=O s

In the next two sections we will consider fields in the presence of sources and vortices, but it is obvious that results derived from Eqs. (1.223) can be directly applied to such fields, provided that they are studied near points where generators are absent. Now proceeding from the system (1.223) let us describe in detail the behavior of a field M. 1. First, it is clear that inside the volume V vector lines and normal surfaces of the field M do not have terminal points and edge lines, respectively. 2. Inasmuch as field generators are absent within the volume V, the field M does not have singularities; that is, an points are usual ones. In particular, if we imagine some surface within this volume, then near its points the field behaves as a continuous function and correspondingly,

(1.224)

Here M(2) M(2) and M(1) M(1) are the tangential and normal compot' nt' n nents of the field at the front and back sides of the surface, respectively. 3. As follows from the system 0.223) the circulation of the field M equals zero within the volume V, regardless of the contour

~M'dl'=O

(1.225)

t

It is essential that this equality is based solely on the fact that vortices are absent in this part of space, and correspondingly it remains valid also for source fields. We will consider an arbitrary closed contour t inside V (Fig. Li5a). Then the voltage along a closed path t can be represented as the sum of two voltages,

(1.226)

1.9 Harmonic Fields

83

Fig. US (a) Circulation; (b) presentation of Laplace's equation through finite differences; (c) illustration of Green's formula; and (d) illustration of Green's formula.

where

t'=t'1+t'2 Changing the direction of the path t'2 to the opposite one-that is,

t'; -we obtain

or (1.227)

tt

where t'1 and are two arbitrary paths from point a to point b. In other words; the voltage between the two points a and b is path independent and is defined only by the position of these points. To calculate the voltage we have to divide the path of integration into a certain number of elements so that within every element dl' both the magnitude and the direction of field M do not change. Then, taking the dot product and performing summation, we obtain a value for the voltage. If the position of the path is varied, we have to use other values of the

84

I Fields and Their Generators

field M for determining the voltage along a new path with a different length as well as different directions of its elements. For this reason, it is natural to expect that in general the voltage between two points depends essentially on the path along which it is calculated. It is a great surprise then that equality (1.227) shows independence of the voltage of the path of integration for fields described by Eqs. (1.223). For instance, it turns out that the voltage of the gravitational field between two points located at a distance of one meter remains the same whether it is calculated along a straight line with a length of one meter or along an arbitrary path having a length of thousands of kilometers. Let us note that the equality (1.227) can be derived from the fact that the field M is the simplest case of a quasi-potential field, as any path between two points located in the volume V is intersected by the same number of normal surfaces. Now we will demonstrate that the absence of vortices within the volume V also permits us to express the field M through a scalar field U, which facilitates the study of the vector field. With this purpose in mind, we will write the first equation of the system 0.223), curlM=O

(1.228)

in the form

curl, M = _1_ { ah 2 M 2 h!h 2 ax!

_

ah! M 1 } = 0 aX2

Then it is easy to see that a solution of Eq. (1.228) can be written as M

=

-gradU

or

For instance, we have

(1.229)

1.9 Harmonic Fields

85

The scalar function Ui.p) is usually called the potential of the field M, and asa rule it is related to the generators in a simpler way than the field itself. Here it is appropriate to make several comments, namely, (a) The field M can be described with the help of an infinite number of potentials V, which differ from each other by any constant C. (1.230)

where V\(p) and Vip) are an arbitrary pair of potentials. This equality follows from the fact that grad C == O. This ambiguity in the determination of the potential V shows that it does not have any physical meaning, but it is an auxiliary function allowing us to simplify the analysis of the vector field M. (b) The quasi-potential field was introduced as M=q>(p)gradV and correspondingly we can say that a field satisfying Eqs. (1.223) represents the simplest case in which q>(p) = ± l. (c) In general the potential V, as well as the field M, can be a function of the observation point coordinates and of the time of measurement t. (d) As soon as the potential V(p) is known, Eq. (1.229) permits us to find the field M in the volume V by the simple operation of differentiation with respect to coordinates of the point. (e) It is clear that solution of Eqs. (1.228) can be represented as either M = grad V or M = - grad V, and both of these relations are used in this monograph. Now we will show that the introduction of the potential V essentially simplifies the calculation of the voltage. By definition we have

fb M . d/ = - fb grad V . d/ a

a

It is also clear that the dot product of vectors

and

(1.231)

86

I

Fields and Their Generators

equals

au aX l

au aX 2

au aX 3

gradU'dt"= -dx l + -dx 2 + -dx 3 or grad U . dt" = dU

(1.232)

where dU is the differential of the potential U. Substituting Eq. (1.232) into Eq. 0.231) we obtain

t

M ' dt"= -

a

t

(1.233)

dU= U(a) - U(b)

a

Thus, to calculate the voltage it is sufficient to take the difference of potentials at the points a and b. This, of course, is much simpler than an integration of the field M. Let us also note that Eq. (1.233) can be used to determine the potential U at any point of the volume V, provided that the field M is known as well as the value of the potential at some given point. Having introduced the potential U(p), it is natural to derive equations describing its behavior. From the first field equation curlM = 0 we have established that M = -gradU

Then substituting this relation into the second field equation divM = 0 we obtain divgrad U = 0

(1.234)

This process of derivation is shown below.

~I

curl M = 0

j IM =

± grad U

I

!f--------->.!

@J

div M = 0

I

j div grad U = 0

I

Therefore, the system of two field equations in differential form, describing the field behavior at usual points, is replaced by one equation. divgrad U = 0

(1.235)

1.9 Harmonic Fields

87

Making use of the representation of div M and grad U in an orthogonal system of coordinates and denoting the operation div grad

as

V2 = .6

we obtain (1.236) or

a (h zh 3 au) a (h'h 3 au) a (h'h z au ) ax! T ax! + aX2 -,;; aX 2 + aX 3 - , ; ; aX 3 = 0 We have derived one of the most important equations in the theory of fields applied to geophysical methods; it is called the Laplace equation. In particular, in a Cartesian system

a2u

VZU= -

ax 2

a2u

a2u

+ - + - 2 =0 ay 2 az

(1.237)

In a cylindrical system

a au 1 a2u a2u - +r- =0 ar ar + r acp2 JZ2

V 2U= - r -

(1.238)

In a spherical system -

a (R 2sinOau) + -a sin 0au-

JR

aR

ee

a ( 1 au) + acp sin 0 acp =

ee

(1.239) 0

Thus, instead of two partial differential equations of first order containing three unknowns, M!, M 2 , and M 3 , curiM = 0, divM = 0 we have obtained one differential equation of second order with partial derivatives of one unknown, U. Next, we will consider the replacement of equations CuriM = 0 and DivM = 0 by corresponding relations with respect to the potential U; this can be done in the following way. As is well known, the equation M = - grad U means that the component of the field M in any direction t can be presented as

au

Mt -at -

(1.240)

88

I

Fields and Their Generators

Therefore, surface analogies of field equations

M t(2)=MP)

and

can be written in the form

nr» at=at'

au(2)

au(2)

au (1)

--=--

an

(1.241)

an

Here U(2) and u» are values of the potential U at each side of the surface located inside the volume V. It is easy to see that from continuity of the potential U follows continuity of tangential components of the field, and correspondingly Eqs, 0.240 can be slightly simplified.

mr»

au(2)

(1.242)

--=--

an

an

Let us note that continuity of the potential at some surface does not mean continuity of the normal derivative au jan, since its calculation requires the use of potential values at points not located at the surface. Thus, we have obtained the system for the potential U, which is equivalent to the system (1.223):

and U(1)= U(2),

eir»

au(2)

an

an

(1.243) on

S

At the same time, taking into account that within the volume V generators are absent, we can essentially simplify both systems (1.223) and (1.243). In fact, all points inside the volume are usual ones and correspondingly the system of field equations and the potential are curlM = 0,

divM= 0

and

(1.244) LlU=O

Before we formulate the value boundary problem, let us consider the Laplace equation in detail. In accordance with Eq. (1.237) we have a2u LlU= - 2

a2u

a2u

+ - + - 2 =0 ax ay 2 az

89

1.9 Hannonic Fields

That is, in the vicinity of every point of the volume V the sum of the second derivatives along the coordinate lines x, y, z equals zero. At this point it is natural to raise the following question. Does this fact reflect some special features of the potential behavior? To answer this question we will take an arbitrary point p within the volume V and imagine an elementary cube around this point with sides along coordinate lines. The length of each side is 2h (Fig. USb). As is well known, the derivatives of the function at any usual point p can be replaced by finite differences of this function, taken at points located near the point p. For instance, for the first and second derivatives along some line t we have

and 2u

a2 at

=_1 [aU(t+ ~t

at

dt) _ aU(t_ ~t)l 2 at 2

First, applying these relations for derivatives with respect to coordinate x we obtain

_au_(x_,y_,z_) = _1 [u(x + _LlX y z) ax Llx 2 ' ,

_u(x _ _LlX.

y 2 ' ,

z)]

and

(1.245) 2u(x,y,z)

a . ax 2

=_I_[aU(x+ Llx y z)- au(x_ Llx y z)] ~x ax 2 " ax 2 ' ,

Here x, y, z are the coordinates of the cube center, that is, those of the point p. Inasmuch as

au ( ~x ) ax x+T'Y'z

=

1

dX[U(X+LlX,y,Z)-U(x,y,z)]

au(x- Ll ,y,z) 1 2X - " ' - - - - - - ' - = -[U(x, y, z) ax Llx

-

U(x -

~x,

y, z)]

90

I

Fields and Their Generators

we have

aZu(x,y,z) --"""'z--

ax

1

= -_z [U(x+ £\x,y,z) (£\x)

+U(x-£\x,y,z)-2U(x,y,z)] (1.246) where Ui;x + £\x, y, z ) and Ui:x - £\x, y, z ) are values of potential U on opposite faces of the cube perpendicular to axis x. By analogy for the second derivatives with respect to coordinates y and z we have

aZu(x,y,z)

1

ay

(Ay)

_.....:....--z-....:.... = - - Z [U(x, y + Ay, z)

+ U(x, y - Ay, z)

-2U(x,y,z)] (1.247)

aZu(x,y,z) -""':""--2-":""

az

1

= - - Z [U(x,y,z+£\z) + U(x,y,z-Az) (£\z)

.

-2U(x,y,z)] Taking into account that Ax = Ay = Az = h and substituting Eqs. (1.246), (1.247) into the expression for the Laplacian

a2u a2u aZu £\U=-+-+ax 2 ay 2 az 2 we obtain

t

2

V U = -;[ U;-6U(P)] h ;=1 or (1.248)

Here U; is a value of the potential on the ith face of the cube, while is its value at the center of the cube. It is clear that the term 1

6

-LV 6 i=1

I

o:»

1.9 Harmonic .Flelds

91

is the average value of the potential at the point p, 1

Uav(p) = -

6

E Vi

6i~1

Correspondingly, the Laplacian can be rewritten as (1.249) Thus, the Laplacian of a scalar function is a measure of the difference between the average value of the function U'" and its value U at the same point. For example, if the average value exceeds the value of the function, the Laplacian is positive. Now, making use of Eq, 0.237) we obtain the simplest form of the Laplace equation.

Uav(p) - U(p) = 0

(1.250)

Therefore, we can say that if the function U satisfies the Laplace equation, then it possesses a remarkably interesting feature, namely that its average value calculated around some point p is exactly equal to the value of the function at this point. Only a certain class of functions has this feature, and such functions are called harmonic. Correspondingly, we can conclude that the potential U describing a field M M= -gradU is a harmonic function within the volume V, provided that the field generators are located outside of V. Let us also note that in accordance with eq. (1.249) the Laplacian can be considered a measure that shows to what extent some function differs from a harmonic. In accordance with Eq. 0.237) the sum of second derivatives along three mutually perpendicular directions equals zero if the function U is harmonic. At the same time we know that in the one-dimensional case there is also a class of functions y(x) for which the second derivative is equal to zero; that is, (1.251)

and these functions are linear. From this comparison of second derivative behavior, it is natural to assume that harmonic functions are an analogy to linear functions for twoand three-dimensional space and, correspondingly, have similar features.

92

I

Fields and Their Generators

Let us describe some of these that are, in fact, inherent for harmonic functions. 1. It is clear that if values of a linear function are known at terminal points of some interval, it can be calculated at every point inside of this interval. Consequently, if a harmonic function is known at every point of the boundary surfaces surrounding the volume, it can be determined at any point within this volume. 2. If a linear function has equal values at terminal points of the interval, then it has the same value inside of the interval; that is, the linear function is constant. By analogy, if a harmonic function has some constant value at all points of the boundary surface, then it has the same value at any point within the volume. This is a very important feature of harmonic functions and has many applications. In particular, it allows us to give a mathematical proof of the effect of electrostatic screening. 3. A linear function reaches its maximum and a minimum at the terminal points of the interval. The same behavior is observed for harmonic functions, which cannot have their extrema inside of the volume. Otherwise, the average value of the function at some point will not be equal to its value at this point and, correspondingly, the Laplacian would differ from zero.

In conclusion of this comparison, it is appropriate to note that if linear functions are the simplest functions in the one-dimensional case, then harmonic functions are the simplest ones in the two- and three-dimensional cases. In the future we will call a field M satisfying the uniform system of field equations (1.223) a harmonic field, since its potential U is a harmonic function. In Section 1.8 we formulated the boundary-value problem and emphasized the importance of information about the behavior of the field at the boundary surface S surrounding the volume V. Now we will attempt to find boundary conditions on S such that they uniquely define the harmonic field M. With this purpose in mind, we will proceed from Gauss' theorem, which is a natural "bridge" between values of the field inside of the volume and those at the boundary surface. (1.252)

Here n is the unit vector perpendicular to the surface S and directed outward, and X n is the normal component of an arbitrary vector X, which is continuous function within the volume V.

1.9 Harmonic Fields

93

To simplify our derivations we shall make use of the potential V, which satisfies Laplace's equation

v2V =

0

(1.253)

It is obvious that this equation has an infinite number of solutions that can, in particular, correspond to different distributions of generators outside the volume. Let us choose any pair of them, V1(p) and Vip), and form their difference. (1.254) Inasmuch as Laplace's equation is linear, the function V/p) is also a solution. To derive boundary conditions, let us introduce some vector function X( p), which has the form X = V3 grad V3 = V3VV3

(1.255)

Substituting Eq. (1.255) into 0.252) we obtain

J.vV(V

(1.256) dV = ~ V 3 grad, V 3 dS s where grad, U3 is the component of the gradient along the normal n, and 3VV3 )

grad U n

3

aV3 an

=-

(1.257)

It is proper to notice here that the boundary surface S surrounding the volume can consist of several surfaces. As is weI! known, the operator V is a differential operator and correspondingly we have

V(V3 VV3 ) = U 3 V2V3 + VV3 ' VV3 = (VV3 ) 2

(1.258)

since V 2 V3

=

0

Taking into account both Eqs. (1.257), 0.258) we can rewrite Eq. (1.256) as 3 dV= ~ U3 -aV dS (1.259) s an This equality, which in essence is Gauss' theorem, will aI!ow us to formulate the most important boundary conditions; but first let us make two comments.

J.v(VU

3)

2

(a) The integrand of the volume integral in Eq. (1.259) is nonnegative. (b) In equality (1.259), which relates the values of the function inside of the volume V to its values on the boundary surface S, V3 is the difference of two arbitrary solutions of Laplace's equation.

94

I

Fields and Tbeir Generators

Now we are prepared to formulate boundary conditions for the potential of the harmonic field M, which uniquely define it inside of the volume V. With this purpose in mind, suppose that the surface integral on the right-hand side of Eq. 0.259) equals zero. Then

fv(VU3)2 dV= 0 and taking into account that its integrand cannot be negative, we have to conclude that at every point of the volume grad U3 = 0

(1.260)

This means that the derivative of function U3 in any direction t is zero.

aU3

at

=

0

Substituting this equation into Eq, (1.254) we discover that if the surface integral in Eq. (1.259) vanishes, then the derivatives of solutions of Laplace's equation in any direction t are equal to each other

In other words, these solutions can differ by a constant only; that is, (1.261) where C is a constant that is the same for all points of the volume V, including the surface. Note that C can be zero. Now we will define conditions under which the surface integral

au3

,(.. U3 ~

an

dS

(1.262)

vanishes, and correspondingly Eq. 0.261) becomes valid. At least three such conditions are described below.

Case 1 Suppose that the potential Ut.p) is known on the boundary surface; that is, on

S

(1.263)

and we are looking for a solution of Laplace's equation that satisfies the condition (1.263). Let us assume there are two different solutions to this

1.9 Harmonic Fields

95

equation inside of the volume; Uj(p) and U2(p), which coincide on the boundary surface. on

S

Then their difference U3 on this surface becomes equal to zero.

S

on

and consequently, the surface integral in Eq. (1.262) vanishes. Therefore, in accordance with Eq. (1.261), solutions of Laplace's equation satisfying the condition (1.263) can differ from each other by a constant only. However, this constant is known, and it equals zero since on the boundary surface all solutions should coincide. In other words, we have proved that two equations,

and

(1.264)

S

on

uniquely define the potential U as well as the field M, since

M= -gradU Equations (1.264) are the Dirichlet boundary value problem. It is proper to notice that in accordance with Eq. (1.263) we can say that along any direction t tangential to the boundary surface, the component of the field M 1 is also known, since M 1 = -au /Bt. Consequently the boundary value problem can be written in terms of the field M as curlM

=

divM = 0

0

and

(1.265)

acp M = -I

at

S

on

This case vividly illustrates the importance of the boundary condition or

acpj M=-t

at

Indeed, Laplace's equation or the system of field equations have an infinite number of solutions corresponding to different distributions of

96

I

Fields and Their Generators

generators located outside of the volume. Certainly we can mentally picture unlimited variants of the generator distribution and expect an infinite number of different fields within the volume V. In other words, Laplace's equation as well as Eqs, (1.223) provide relatively limited information about the field; namely, they tell us only that generators are absent inside the volume. In contrast, the boundary condition 0.263) is vital inasmuch as it was proved that only one harmonic field satisfies this condition inside V. Here it is appropriate to make two comments. (a) Taking into account that the boundary condition (1.263) uniquely defines the field, it is natural to expect that there is an equation that allows us to find the potential U at every point of the volume if it is known on the boundary surface. As was pointed out above, this fact demonstrates an analogy between linear and harmonic functions. (b) We have proved that Eqs. (1.264) uniquely define the potential U(p). However, it is obvious that uniqueness of the field determination is achieved even if the potential is defined only to within a constant, since M= -VU. Now we will consider one example illustrating efficiency of the use of Dirichlet's problem. Suppose that the potential is constant on the boundary surface and, correspondingly, the value-boundary problem is formulated as

«»

I1U=O U(p) = C

on

S

( 1.266)

As we have proved, there is only one harmonic function, U(p), that satisfies Eqs. 0.266). It turns out that it is very simple to find this function. In fact, let us assume that the potential within the volume is also constant. U(p) = C

in

V

(1.267)

It is easy to see that this assumption is correct. First of all Laplace's equation is a sum of second derivatives that is equal to zero, and therefore the constant function U(p) = C is a solution. At the same time, the boundary condition is automatically satisfied since we have chosen a solution that coincides with the value of the potential on the boundary surface. Inasmuch as both equations of (1.266) are satisfied, our assumption is valid, and the potential U(p) within the volume is also constant, which is equal to that on the boundary surface. It is essential that due to the uniqueness of the solution of Dirichlet's problem, we can say that there is no other solution satisfying Eqs, (1.266). Here we see again an analogy between linear and harmonic functions,

1.9 Harmonic Fields

97

since both of them do not vary within a corresponding interval if they have the same values at terminal points. As follows from the equation M = - grad U, the field M vanishes within the volume where the potential is constant. We can imagine different applications of this example, and in particular, it shows that the principle of electrostatic screening can be proved by solving Dirichlet's problem.

Case 2 Now let us assume that two arbitrary harmonic functions within the volume V, U](p) and Uip), have the same normal derivative on the surface S; that is,

au]

et),

an

an

-=-=ip

(p) z

on

S

(1.268)

where ipip) is a known function. From this equality it instantly follows that the normal derivative of a difference of these solutions vanishes on the boundary surface. on

S

Therefore, the surface integral in Eq. (1.259), as in the previous case, equals zero and correspondingly inside of the volume we have

This means that any solutions of Laplace's equation-for instance, U](p) and Uip)-can differ from each other at every point of the volume V by a constant only if their normal derivatives coincide on the boundary

surface S. Thus, this boundary value problem, which also uniquely defines the field, can be written as

and

(1.269)

on

S

98

I

Fields and Their Generators

or curlM = 0

divM =0

and

(1.270) on

S

and is called Neumann's problem. Unlike the previous case, Eqs. 0.269) define the potential only to within a constant, but of course the field is determined uniquely.

Case 3 We suppose that the boundary surface S is an equipotential; that is,

U(p)

C

=

on

S

(1.271)

In addition, it is assumed that the following integral:

~

au an dS

'P3( p)

=

(1.272)

is known. Here S in the boundary surface. Now we will show that two harmonic functions U\(p) and Uip), satisfying Eqs. 0.271-1.272), can differ from each other by a constant only. Consider again the surface integral in Eq. 0.259).

au3

A:. U3 -

'J1,

an

dS

Inasmuch as the boundary surface is an equipotential surface for both potentials U\ and U2 , their difference, U3(p) = U\(p) - Uip), is also constant on this surface, and consequently we can write

¢,s U aU3 an

3-dS=

~ aU3 U3 - d S san

Then, taking into account Eq. 0.272), we have

3

dS = U3pau dS ~ U3aU3 anan s

=

z } U3{~au\ - dS - pau dS = 0 s an an

and in accordance with Eq. (1.260)

\lU3 =0

1.9 Harmonic Fields

99

or

U2 ( p ) = U1( p ) + c Thus, boundary conditions (1.271), (1.272) define the potential within the volume V up to some constant. Correspondingly, the third value-boundary problem can be presented as

and

(1.273)

where S is an equipotential surface, or curlM = 0, and

divM=O (1.274)

where rp3(P) is the known function and Mn is the normal component of the field that coincides with the magnitude of the field, since on the equipotential surface S the tangential component vanishes. There are many cases when harmonic fields can be found by solving the third boundary value problem. For instance, the determination of the electric field outside conductors, provided that the total charge on every conductor surface is known, reduces to a solution of this problem.

In conclusion, let us summarize the main results derived in this section. 1. Three types of boundary conditions have been determined, and they uniquely define a harmonic field within a volume V. 2. As was pointed out above, the volume can be surrounded by several surfaces, and at every point on them on of these conditions has to be given. 3. The procedure of determination of these conditions, based on the use of Gauss' theorem, is called the theorem of uniqueness. 4. In general, boundary surfaces can have an arbitrary shape as well as location; and here it may be appropriate to distinguish three cases. In the first one, the boundary surfaces surrounding generators are located in the vicinity of the generators; and in essence, boundary conditions on such

100

I

Fields and Their Generators

surfaces replace information about the generator parameters. In the second case, the boundary surface is located far away from all generators; and for this reason, it is natural to assume that field M is very small at points of this surface. Moreover, from an analysis of physical principles of a specific problem, it is usually possible to understand the behavior of the field at great distances from generators, which shows in what manner the field decreases. This information is a boundary condition at infinity. And finally, one more case deserves our attention. Often the boundary surface is chosen in such a way that it corresponds to the surface where measurements of the field are performed; this case arises in solving inverse problems. 5. It is obvious that if the field M satisfies Eqs. (1.223) everywhere, it equals zero. M=O

This result is obvious since this field does not have generators. Correspondingly, in terms of its potential we can say that if a function is harmonic everywhere, it is equal to zero, and again we observe an analogy with linear functions. 1.10 Source Fields

In this section we will study a field M within a volume V, where only sources are located, while outside of the volume, both types of field generators can be distributed. In general, the field M inside V presents a sum of two fields, namely, 1. The field caused by sources located inside the volume, and 2. The harmonic field of external generators.

The main attention will be paid here to the first part of the total field, which it is natural to call the source field. In accordance with Eqs. (1.199) the system of equations of the source field is

~M. t

dl"= 0

fcM. dS = Q S

or curlM = 0

divM=Q

CurlM = 0

DivM=Q s

(1.275)

1.10 Source Fields

101

It is clear that all three forms of the first equation of the system indicate the absence of vortices within the volume V, and correspondingly we can

expect some common features between the source and harmonic fields. Now, making use of the system (1.275), let us describe the most general properties of the source field within the volume V. 1. Vector lines of the field M, unlike those of a harmonic field, have terminal points where sources are located, while edge lines of normal surfaces are absent. 2. Near surface sources, the field M can have singularities. In fact, from the equation

DivM

=

Qs

it follows that the normal component of the field is a discontinuous function (1.276) and this discontinuity is directly proportional to the density of surface sources. In particular, if near some point of this surface sources are absent, the component M; is a continuous function. 3. Because of absence of vortices in the volume, tangential components of the field are always continuous functions regardless of the distribution of surface sources. CurlM

=

0

or

(1.277)

4. In accordance with Eqs. 0.275) the circulation of the field M equals zero for any closed path located within the volume V. For instance, this contour can be partly located inside sources as well as outside of them. 5. As follows from the first equation of the system in the differential form curl M = 0, the field M can be represented as M = -gradU

or

au

M - -at' t-

(1.278)

where U is the potential of the source field. Thus, a source field, in the same manner as a harmonic field, can be described with the help of a scalar function. Such similarity between these

102

I

Fields and Their Generators

fields is not surprising since in both cases vortices are absent within V. Let us make two obvious comments. (a) The potential U is defined from Eq. (1.278) to within some constant. (b) Equation (1.278) is valid only at usual points where the field does not have singularities. 6. Taking into account the fact that the circulation

the voltage of the source field path is independent and, as in the case of a harmonic field, it can be expressed through a difference of potentials. tM' dt'= U(a) - U(b)

(1.279)

II

7. By analogy with the harmonic field we will derive an equation for the potential of the source field. Substituting Eq. 0.278) into the second equation of the system, divM = Q, we obtain

divgradU= -Q

(1.280)

or \12U= -Q

This equation describes the potential behavior at usual points where its first and second derivatives exist; it is called Poisson's equation. It is obvious that outside the sources, Poisson's equation reduces into Laplace's equation. \12U=0

Therefore, we can distinguish two areas within the volume V: The first area does not contain sources and the potential U, satisfying Laplace's equation, is a harmonic function; in the second area, occupied by sources, the potential is a solution of Poisson's equation. In accordance with Eq, 0.248) we can represent Poisson's equation as 6

h 2 [uav(p) - U(p)]

=

-Q

(1.281)

Thus, if the right-hand side of Eq. (1.281) is positive, the average value of the potential around some point p exceeds its value at this point. For instance, such behavior is observed for the potential of the electric field near a negative charge. At the same time, around a gravitational mass as well as around a positive electric charge, the potential of both gravity and electric fields at some point p is greater than its average value near this point.

1.10 Source Fields

103

8. Poisson's equation is one of the most fundamental equations of field theory. Here it is appropriate to notice that source fields are often used in geophysics. Gravitational and electric fields, as well as fields of compressional waves, are typical examples of source fields. 9. In the next chapter it will be shown that the potential of the gravitational field can be represented as

U(p)

=

y

fv

o(q) dV L qp

and at the same time it satisfies outside and inside the masses Laplace's and Poisson's equations, respectively. and Generalizing this result we can say that for any source field, a solution of the equation has the form

1 QdV U(p)=-f47T L q p and describes the potential inside and outside the sources. Of course, there are an infinite number of solutions of Poisson's equations that differ from each other by a potential for a harmonic field. 10. Poisson's equation is equivalent to the field equations curIM = 0

divM

=

Q

written for usual points of the volume. To derive surface analogies of Poisson's equation we will proceed again from Eq. 0.278). Then Eqs. 0.276-1.277) remain valid if the tangential and normal derivatives of the potential U(p) at any surface, located inside the volume V, satisfy the following conditions:

nr» at

aU(2)

at

au(2)

and

nr»

---=-Q

an

an

S

As was shown in the previous section, the first condition can be replaced by a simpler one, UO) = U(2)

since continuity of the potential at the surface results in continuity of the tangential derivative. Thus, the behavior of the potential on the surface

104

I

Fields and Their Generators

located inside of the volume is described by equations

au(2) U(l)

= U(2)

and

su»

---=-Q

an

an

s

(1.282)

11. Taking into account Eqs. (1.280), (1.282) the system of field equations can be replaced by an equivalent system with respect to the potential inside of the volume,

ir» = U(2)

au(2) and

eu»

---=-Q

an

an

(1.283) s

Comparison with the case of harmonic fields shows that this system is more complicated. First, instead of a uniform Laplace equation, we have Poisson's equation with the right-hand side characterizing the distribution of sources within the volume V. Second, in the presence of surface sources the normal derivative of the potential becomes a discontinuous function. From this consideration, it is natural to assume that, as in the case of the harmonic field, both systems (1.275) and (1.283) do not uniquely define the source field. Let us discuss this problem in terms of the field, and afterward we will formulate boundary-value problems with the help of the potential. At the beginning of this section we have represented the total field M inside of the volume as a sum of two fields: (1.284)

Here M j is the field caused by the sources Q and Q s within the volume only, while Me is a harmonic field whose generators are located outside the volume. By definition, the field M j isa solution of the system (1.275). curl M, =0

divM j =Q

Curl M, = 0

DivMj=Qs

(1.285)

Since the given distribution of sources described by the functions Q and Qs generates only one field, we can say that the field M, is uniquely by this system. However, Eqs. (1.275) have other solutions; to demonstrate this, let us refer to the field Me. First of all, it represents a harmonic field in the

1.10 Source Fields

105

volume V, which satisfies the homogeneous system curl Me

=

0

div M, = 0

Curl Me

=

0

Div M,

=

(1.286)

0

Moreover, this field is a continuous function within the volume V, because its generators are located outside V and in essence surface analogies of field equations can be omitted. Now, performing a summation of corresponding equations of both systems (1.285), (1.286) and taking into account that all equations are linear, we see that the total field M is also a solution of Eqs. (1.285). In the previous section we demonstrated that an infinite number of harmonic fields satisfy the homogeneous system (1.286), corresponding to different distributions of generators outside the volume V. Therefore, the system of field equations (1.275) or 0.286) has also an infinite number of solutions that differ from each other by harmonic field Me' In other words, the system 1.283 is not sufficient to define the field uniquely, and we have to formulate boundary conditions on the surface 5 surrounding the volume V. Taking into account that these conditions should uniquely define a harmonic field Me in the volume V, it is natural to make use of the results derived in the previous section and correspondingly to formulate three boundary-values problems. They are 1. Dirichlet's problem

au(2)

U O) = U(2)

nr»

----=

an

an

-Q S

Here 50 is a surface surrounding the volume V, while 5 j is a surface located inside of this volume. 2. Neumann's problem

au(2)

ur» an

---=-Q

an

S

106

I

Fields and Their Generators

3. The third problem

U(I)

= U(2)

aU(2)

etr»

----=

an

an

-Q S

and

U( So)

=

constant

Perhaps it is appropriate to make several comments here. Functions fPl(P), fPip), and fP/p) describe the behavior of the potential of the total field caused by both the internal and external generators. We can formulate the boundary-value problem with respect to the fP2(P), and external field Me' defining corresponding functions fP~(p), fP3( p), providing that in the third problem the boundary surface is an equipotential surface for both fields. This is related to the fact that the potential as well as its derivatives caused by a known distribution of sources, located inside the volume V, can be found without solving the boundary problem everywhere including the boundary surface So' Such considerations show, in fact, that the theorem of uniqueness for the source field can be reduced to that for the harmonic field studied in the previous section. In principle, the field due to known sources distributed inside the volume can be determined by making use of physical laws such as Coulomb's or Newton's, without solving the boundary-value problem. However, determination of the field with the help of the system of field equations also requires boundary conditions that take into account the field caused by external generators. In essence, the theorem of uniqueness formulates all the steps that have to be undertaken to find the field. These steps are 1. A solution of Poisson's equation. 2. A selection among these solutions of such functions that satisfy boundary conditions. 3. A choice among this last group of a function that also satisfies the conditions at interfaces. At the same time it is proper to notice that in modern numerical methods of solution of boundary value problems based on replacement of

1.10 Source Fields

107

differential equations by finite differences, all of these stages are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume is defined by the distribution of the volume and surface sources and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind, we will proceed from Gauss' theorem,

!v div X dV = ~

(1.287)

X . dS So

assuming that the vector function X(q) and its first derivatives exist in the volume V. To simplify the derivations we will express the vector X with the help of two scalar functions cp(q) and I/!(q,p) in the following way:

X = cp(q)VI/!(q,p) -1/!(q,p)Vcp(q)

(1.288)

Here cp and I/! are continuous functions with continuous first and second derivatives, p is an observation point where the potential is determined, and q is an arbitrary point. Substituting Eq, (1.288) into Eq. (1.287) and taking into account that

8I/!

8cp

X . dS = X n dS = cp an -I/! 8n

and divX

=

Vcp . VI/!

+ cpv 21/! - Vcp . VI/! -I/!V 2cp

= cpv 21/J -I/!V 2cp we obtain (1.289) The latter is called the second Green's formula and in essence it represents Gauss' theorem when the vector X is given by Eq. 0.288). In particular, letting I/! = constant we obtain the first Green's formula.

!vV 2cp dV = 'rf..f'so 8cpan dS

(1.290)

Our goal is to express the potential of the field U( p) within the volume V in terms of both the sources inside the volume and values of the potential and its derivatives on the boundary surface So' We will consider several ways of solving this task; with this purpose in mind suppose that

108

I

Fields and Their Generators

the function cp(q) describes the potential U(p) of the source field. Then, taking into account Eq. (1.280) we can rewrite Eq, (1.289) as f U(q)V 2GdV+ fG(q,p)Q(q)dV

v

v

au} =rl- {Ueo - - G - dS ~

an

o

(1.291 )

an

Here G(q, p) = rjJ(q, p) is called the Green's function. It turns out that we can obtain different expressions for the potential U(p) by choosing various functions G(q, p). To illustrate this fact, let us consider several cases.

Case 1 Suppose that the function G(q, p) is 1

G(q,p)

=

L

(1.292)

qp

where L q p is the distance between points q and p. As was shown in the previous section, this Green's function satisfies Laplace's equation everywhere except at the point p; that is, V2-

1

i:

=0

if

q

=1=

(1.293)

p

Inasmuch as the second Green's formula has been derived assuming that singularities of the functions U and G are absent within volume V, we cannot directly use this function G in Eq. (1.291). To avoid this obstacle let us surround the point p by a small spherical surface S * and apply Eq. (1.291) to the volume restricted by both surfaces, So and S*, as shown in Fig. USc. Also at the beginning we assumed that the potential and its first derivatives are continuous functions; that is, surface sources are absent inside the volume. Then, taking into account Eq. (1.293) we have instead of Eq. (1.291);

Q( q) f-dV=~

v

L qp

( So

+~

a l l su;q) U(q)------an L q p L q p an

a 1 U(q)--- ( s, an L qp

I

dS

I

1 au( q) - - - dS L q p an

(1.294)

1.10 Source Fields

109

We will consider the behavior of the last integral as the radius of the spherical surface r tends to zero. Inasmuch as both the potential and its derivatives are continuous functions we have

U{q)

~

U{p)

aU{q)

and

aU{p)

--~---

an

an

That is, functions U(q) and aU(q)jan on the surface S* approach their values at the observation point, respectively. Also from Fig. U5c it follows that the normal and the radius on this surface are opposite to each other, and therefore for points on this surface we have 1

1

L qp

r

a

1

iJ

G=-=-

and

1

1

Then, applying the mean value theorem, the surface integral around the observation point can be represented as

~

a 1 1 au(q) ) U ( q ) - - - - - - dS s, ( an L q p L q p an I

1

aU(q)

= - ( U(q) dS - _A:. - - d S ,2 is. r '!'s. an Inasmuch as

au

-

an

r = - -' gradU = [grad Ul cos s r

we have 1 aU(q) 1 - _A:. - - dS = -I grad U( p) I r'!'s. an r

A:. cos 8r 2 sin BdB x~

s;

d~

r

Therefore, in the limit this integral equals 47TU(p),

~

(

a

1

1

au(q»)

U(q)--- - - - - dS=47TU(p) s; an L q p L q p an

as

r ~ 0 (1.295)

This is the most important result in our derivations since it permits us to

110

I

Fields and Their Generators

obtain an expression for the potential in an explicit form. In fact, substituting Eq. (1.295) into Eq. (1.294) we have

1 Q(q) U(p)=-f-dV+-~

v

47T

--~

1

47T

aU(q) 1 ---dS

1

L qp

47T

So

an

L qp

a 1 U(q)--dS

an

So

L qp

(1.296)

That is, the potential U(p) at any point of the volume V can be calculated if we know the distribution of sources in the volume as well as values of the potential and its normal derivative on the boundary surface So' In particular, if this surface is located at infinity, it is natural to assume that and

aU(q) - - - -.70

an

and therefore

U(p) =

1 Q(q) -f --dV 47T v L

(1.297)

qp

In other words, if the sources are known everywhere the potential is defined by Eq. 0.297). This presentation is often called the fundamental solution of Poisson's equation.

We will consider another example in which sources are absent in the volume V. Then, in accordance with Eq. 0.296) we have

a 1 U(q)--dS 47T So an L q p

--~

1

(1.298)

Thus, the potential at any point of the volume where sources are absent is defined by both values of the potential and its normal derivative on the boundary surface; and they, in effect, replace information about the distribution of generators outside the volume.

1.10 Source Fields

111

At the same time, from the boundary conditions of Dirichlet's and Neumann's problems it follows that the field can be found inside the volume if either the potential U(q) or its normal derivative aU(q) jan is known. To demonstrate this fact we will consider cases where new Green's functions are chosen.

Case 2 Let us introduce a new Green's function that satisfies the following conditions:

1.

V2G=O

2.

G(q,p) ~L

if

q =1= P

1

if

(1.299)

q~p

qp

aG(q,p)

3.

on

=0

an

So

Comparison with the previous case shows that the surface integral around the point p still equals 47TU(p), because in both cases the Green's functions have the same singularity. Then, taking into account that on the boundary surface So

aG -=0

on

instead of Eq, 0.296) we have 1 1 aU(q) U(p) = -4 fQ(q)G(q,p) dV+ -4 rf.. G(q,p)--dS (1.300) 7T

v

7T'1';;0

on

Therefore, we have expressed the potential at any point of the volume V in terms of the distribution of sources and the normal derivative of the potential on the surface So; that of course corresponds to Neumann's boundary value problem. Inasmuch as M

au

n

=--

an

112

I

Fields and Their Generators

we can rewrite Eq. (1.300) as 1

U(p)

=

47TfvQ(q)G(q,p)dV 1 - -f Mn(q)G(q,p) dS 47T

(1.301)

So

In particular, if sources are absent we obtain 1

U(p)=- 47T~Mn(q)G(q,p)dS

(1.302)

o

That is, in this case the potential is defined only by the normal component of the field given at the boundary surface So' As follows from Eqs. (1.299), determination of function G(q, p) is a solution of Neumann's boundary-value problem. To emphasize this, we will write down Green's function as a sum. 1

G(q,p) =

L

(1.303)

+h(q,p)

qp

Here h(q, p) is a harmonic function everywhere within the volume V, and in accordance with Eqs. (1.299) the boundary problem with respect to hi.q, p) is formulated as V 2 h ( q , p)

=

0

a

(1.304)

anh(q,p) =CP2(q), where (1.305)

u»

In other words, determination of the potential by Eq. (1.301) implies a solution of Neumann's problem with respect to the harmonic field h(q, p) and is usually a sufficiently difficult task. Its complexity strongly depends on the shape of the boundary surface. However, there are cases when it is very simple to find hi q, p) as well as the Green's function G(q, p). For instance, suppose that the boundary surface So consists of a plane Sp and a hemispherical surface with infinitely large radius Ssph' Note that the field tends to zero at points of the surface Ssph and the Green's functions also decays at infinity. Here PI is a point

1.10 Source Fields

113

located outside the volume V and is a mirror reflection of the point P with respect to the plane. Correspondingly, the distances L q p and L q p 1 are (1.306) and

since and

Z

PI

=-zP

Inasmuch as the point Pi is located outside the volume V, as Z> 0, the function h(q, Pi) is harmonic and therefore it is a solution of Laplace's equation; that is,

At the boundary plane

a and since

Zq =

= 0 we have

Zq

1

0

At the same time, for the function 'PzCq) we obtain a

'P2(q) = - aZ q

-

1

i;

Zq-Zp

=

u:

zp

=

3

-

-3-

L qp

Comparing the last two equations, we see that function hi.q, p) satisfies the Neumann's boundary problem given by Eqs. (1.304) and, correspondingly, we can calculate the potential of the source field by making use of either Eq. (1.301) or Eq. (1.302), if the Green's function is

1

1

G(q,p)=-+i: L qp I

(1.307)

Certainly it is a very simple function, which allows us to find the potential in one of the half spaces when the normal component of the field is known on the boundary plane; it is often used in gravimetry and magnetometry to

114

I

Fields and Their Generators

calculate the field above the earth's surface (upward continuation). It is appropriate to notice that in this case integration in Eqs. (1.301), 0.302) is performed over the plane Sp only.

Case 3

We can modify Eq, (1.296) in a different way; with this purpose in mind, let us choose a Green's function such that it satisfies the following conditions: 1.

V 2G =0

if 1

2.

G(q,p)

3.

G(q,p)=O

~-

i;

q*p

as

q~p

on

So

(1.308)

Then, the integral containing the normal derivative of the potential on the boundary surface vanishes and instead of Eq. (1.296) we obtain

U(p)

1

=

-lQ(q)G(q,p) dV 47T v

--~

aG

1

41T

U(q)-dS So

an

(1.309)

And in particular, as sources are absent in the volume,

1 U(p) =-~ 41T

aG

U(q)-dS So

an

(1.310)

Determination of the Green's function, as in the previous case, requires a solution of the boundary value problem, which also can be formulated in terms of the harmonic field h. 1

G(q,p) =

L

qp

+h(q,p)

1.10 Source Fields

115

Then, we have the following Dirichlet's problem:

h(q,p)

1

(1.311)

= --

c;

Of course, in general it is a very complicated problem, but it is drastically simplified for some special cases. For instance, in the case of the plane boundary surface Sp, z = 0, and a hemispherical surface with infinitely large radius, the harmonic function is

Here PI is the mirror reflection of the point p with respect to the plane So' Then, on this plane we have

G(q,p) = 0 while the normal derivative of the function G has the form if

z=0

Correspondingly, Eq, (I,31O) can be written as (1.313) It is easy to see that the function

can be expressed as

116

I

Fields and Their Generators

where dw is the solid angle suspended by the surface dS as viewed from the point p. Thus, we can present Eq. (1.313) in the form 1 U(p)=-f U(q)dw 21T s,

(1.314)

Now we will consider one more approach to choosing Green's functions.

Case 4 Suppose that the new Green's function satisfies two conditions, namely, (a) It has a singularity near the observation point p, which is the same as earlier. as

q ---) p

(1.315)

and (b) Unlike the previous cases, the function G satisfies the same equation as the potential U does. (1.316)

In other words, we do not assume any more than that the Green's function is harmonic. Again we will proceed from Eq. (1.289) and taking into account that and the volume integral in this equation vanishes. Then as before, the surface integral consists of two parts. ,f,

eo- - Gau} U - dS

'fJs o { an

+,f,

an

sc

au}

U - - G - dS=O

'fJs * { an

an

1.10 Source Fields

117

Inasmuch as the Green's function has the same singularity as in the previous cases, we obtain

{au

1 U(p) = _A.. G- 47T ~o an

aG} dS uan

(1.317)

Therefore, we have expressed the potential at any point of the volume in terms of its values and those of its normal derivative on the boundary surface So' This result is valid regardless of presence or absence of sources of the field in the volume V. Let us assume that the Green's function has the following form:

feu) G(q,p)=-

(1.318)

-;

where u depends on the distance L qp and

feu)

-s

i

Then, substituting Eq. 0.318) into Eq. (1.317) we have

au

1 U(p) = _A.. G-ds 47T ~o an

1 f'(u) et. - -¢U(q) -----!!£dS 47T 1

+ -~

47T So

L qp

U( q)

an

feu) et: -2-

L qp

---!!£ dS

an

(1.319)

where !'(u) = df(u)jan.

We have considered several types of Green's functions, corresponding to different formulas, allowing us to calculate the potential inside the volume. With this in mind it is appropriate to make several comments illustrating various aspects of this subject. 1. A Green's function can be treated as a potential caused by a certain type of sources. For instance G(q, p) = Ij47TL q p is the potential at the observation point p due to either the elementary mass or a charge with unit magnitude, located at point q.

118

I

Fields and Their Generators

At the same time, the Green's function

(1.320) where

describes the potential of a displacement in a medium as the longitudinal wave propagates with velocity C, caused by an elementary source located at the point q. Here it is appropriate to notice that, substituting Eq, (1.320) into Eq. 0.319), we obtain Kirchoff's formula, which plays an important role in the theory of seismology. 2. In deriving formulas for the potential it was assumed that the field is a continuous function everywhere within the volume V. Now let us consider the case in which sources are distributed on some surface Sj where the normal component of the field is a discontinuous function. Then, surrounding this surface by the surface So; and applying Green's formula, Eq. (1.289), to the volume confined by surfaces So and So;, we obtain an additional integral over Soi' In the limit, as the surface Soi approaches Sj, this integration is reduced to that over the back and front sides of the surface Sj (Fig. 1.15d). Taking into account that the direction of the normal to Soj coincides with the normal of S, on the back side and opposite on the front side, we have

+~

Sj {

aif/l) aifP) acp(l) cp(1) _ _ - cp(2) _ _ -l/J(1)-an an an

acp(2) }

+ l/J(2)_-

dS

(1.321)

an

where cp(1), 1/1(1) and cp(2), 1/1(2) are the values of both functions on the back and front sides of the surface Sj, respectively. Letting cp = U and l/J = G and assuming that both the potential and Green's function are continuous functions on the surface Si, the last

1.10 Source Fields

119

integral in Eq. (1.321) can be simplified, and we have au(2)

aU(1) }

tf..G { - - - dS 't;;; an an

=

-¢,GQsdS

(1.322)

Sj

where G = G(2) = G(1), but M~2) and M~I) are the normal components of the field at either side of the surface Sj, respectively. Thus, in the presence of surface sources we have to add the term given by Eq. (1.322) to the right-hand side of all expressions for the potential derived above. In particular, if the Green's function is harmonic and becomes singular as l/L q p at the point p, we obtain 1

1 -f. Qs(q)G(q,p) dS

U(p) = -4 f.Q(q)G(q,p) dV+ rr v 4rr 1

aU(q)

~

1

+ -4 tf.. --"G(q,p) dS + -4 tf.. U(q) rr't;;o an

rr't;;o

8G(q,p)

an

dS (1.323)

This equation, as well as similar ones, can create the impression that the solution of a boundary-value problem always consists of an integration. However, in general, this is not true, and it is related to the fact that some terms on the right-hand side of Eq. 0.323) remain unknown until the field is calculated. Let us consider this question in more detail. First, let us briefly discuss terms containing the potential and its normal derivative-that is, boundary conditions. As was already pointed out, the volume V, where the field is studied, is usually confined by a surface near primary sources and one at infinity. Correspondingly, knowing from physical consideration the distribution of primary sources and the behavior of the field at infinity, we can assume that the integrals over the surface So in Eq. (1.323) are known. This conclusion is also valid for cases when a part of the boundary surface is not located at infinity, but contacts a medium such that it is easy to formulate boundary conditions. For instance, the normal component of current density and the force due to elastic waves equals zero at a boundary with a free space. Next, we will consider the integrals in Eq. (1.323) that contain terms describing sources. In general, these sources depend on the field M and the properties of the medium. For example, sources arise in the vicinity of

120

I Fields and Their Generators

points where some parameters of a medium vary along the field. Also they can appear at interfaces of media with different parameters. In some cases it is appropriate to represent the source as a sum. and

(1.324)

where Q o characterizes sources that can be specified before the field is calculated, while Q 1 depends on the field. At the same time, let us point out that the total source remains a function of the field. In other words, we are faced with a problem that can be characterized as "the closed circle." Indeed, to find the field we have to know the distribution of sources, but they in turn depend on the field, as is illustrated below:

This means that in principle total sources cannot be specified if the field is not determined, and therefore Eq. (1.323) as well as similar ones are useless in calculating the potential by integration of its right-hand side. Of course, there are some exceptions; for instance, in the case of the gravitational field for geophysical problems, masses can always be defined. Now let us extend this analysis to the system of field equations curlM = 0

divM=Q

CurlM=O

DivM=Qs

(1.325)

which, in general, contains two unknown functions, the field and its sources. From the point of view of calculation this means that the right-hand side of the system is unknown and therefore in this form it is not suitable for solution of the boundary-value problems. However, taking into account that sources depend on their field, we can modify the system in such a way that a new one would either contain an unknown field only or two equations with both unknowns, which allow us to eliminate each of them. A similar problem of "the closed circle" arises in considering the potential. In particular for usual points, we have Poisson's equation

with unknown right-hand side,and correspondingly this also requires some modifications.

1.10 Source Fields

121

Inasmuch as these changes for both the system of field equations and Poisson's equation depend essentially on the physical nature of the field (electric, elastic waves), let us illustrate this procedure by considering two examples.

Example 1 The Electric Field in Dielectrics In this case, the system of field equations is curlE = 0 (1.326)

CurlE = 0 where 80' 8 b and u o ' u b are free and bounded charges, but

are total volume and surface charges. Assuming that due to the electric field there is a displacement of positive charges with respect to negative ones-that is, polarization of the medium takes place-and that this is directly proportional to the electric field, we can show that bounded charges are related with the field by 8b

=

-divaE,

(Tb

= -DivaE

(1.327)

where a is the parameter characterizing the polarization ability of the medium and is called the polarizability. Then, substituting Eqs. (1.327) into Eqs. (1.326) we obtain curlE = 0 CurlE = 0

(1.328)

where E = 1 + a is the dielectric constant of the medium. Thus, we have obtained a new system for the field with a known right-hand side, and it becomes possible to solve this system because two assumptions have been made, namely, (a) In the presence of the field, polarization of the medium occurs. (b) There is a linear relation between the field E and the polarization. It is clear that since free charges can be specified, a new system unlike the original one can be used for determination of the electric field in dielectrics.

122

I

Fields and Their Generators

Now, let us derive the equation for the potential U, corresponding to Eqs. (1.328). From the first equation we have E

=

-gradU

and substituting into the second equation we obtain divt e grad L' )

=

-0 0

or (1.329) Certainly this equation is more complicated than Poisson's equation

but its right-hand side is known, and therefore it can be used to find the potential. Of course, it is very simple to represent Eq. 0.329) as a Poisson equation. In fact, applying the operator V we have

or

00

V6' VU

6

6

V 2U = - - -

=-0

(1.330)

where 0 is the total charge. Again we have obtained Poisson's equation with an unknown right-hand side expressed in terms of the potential and free charge. As concerns surface analogies of the equations for U we have and

aU2

au,

6--6-=-0 2

an

1

an

0

Example 2 The Electric Field in a Conducting Medium Since a constant electric field in a conducting medium is also caused by electric charges, the system of field equations is the same as that in the previous case. curl E = 0

divE

=

0

CurlE = 0

DivE

=

(J'

(1.331 )

but the densities of the free and bounded charges are related to each

1.11 Vortex Fields

123

other and depend on the field E. Correspondingly, the right-hand side of the second equation is unknown, and therefore it is necessary to perform certain modifications of the system to find the field. With this purpose in mind, we have to introduce for consideration another field that accompanies the electric field in a conducting medium, namely the current density j, and make use of two laws. (a) The principle of charge conservation divj

=

0

Divj = 0

and

(1.332)

and (b) Ohm's law j = yE

Here l' is the conductivity of the medium. Then, replacing the second equation of the system by Eq. 0.332) and taking into account Ohm's law, we obtain a new system for the field E only. curlE = 0 div yE = 0 (1.333) CurlE = 0 DivyE = 0 It is clear that the equation for the potential U is

V( yVU) = 0

(1.334)

1.11 Vortex Fields In this section we will consider general features of fields caused by vortices only. For instance, the time-invariant magnetic field, the electromagnetic field due to induced and displacement currents only, and the field of shear waves are examples of the vortex field. In accordance with Eqs. 0.207) the system of equations for this field in some volume V is curlM =W CurIM=Ws

divM = 0 DivM=O

(1.335)

or, in integral form, JM'dS=O

(1.336)

124

I

Fields and Their Generators

That is, the flux of the field through any closed surface inside the volume is always zero, while the voltage between two points in general depends on the path of integration. This is one of the most fundamental differences with the source field, where the voltage is path independent. Of course, we can find lines having the same initial and final points, such that along these lines the voltage remains the same. This occurs if the lines are located in parts of the volume where vortices are absent. As follows from the second equation for the field, the vector lines are always closed, because sources are absent. In general, the field M and its vortices Ware not perpendicular to each other. M· curlM =1= 0 and, correspondingly, normal surfaces do not exist. However, there are exceptions; for instance, as is shown in Section 1.6, the quasi-potential field M

= tp grad

U

can be described with the help of normal surfaces. Now let us evaluate the extent to which the system (1.335) defines the field. With this purpose in mind, we will make use of the same approach as was used in the previous section. Suppose that there are two arbitrary solutions M 1 and M 2 satisfying this system. curl M, =W

div M, = 0

CurIM I =Ws

DivM 1 = 0

(1.337)

and curl M, = W

divM, = 0

CurIM z =Ws

DivM 2 = 0

Then, we will consider the difference between them. M 3 =M z -M 1

(1.338)

As follows from Eqs, (1.337) the field M 3 satisfies the homogeneous system

curl M; = 0

div M, = 0

CurIM 3 = 0

DivM 3

=0

(1.339)

In other words, M 3 does not have generators inside, and therefore it is a harmonic field within the volume V whose generators are located outside. Thus, we can say that system (1.335) defines the field up to a harmonic

1.11 Vortex Fields

125

field. As was shown earlier, this field can be described with the help of a scalar potential a, M 3 = - grad a, and correspondingly, M 2 = M 1 - grad a This analysis demonstrates that Eqs, 0.335) uniquely defines that part of the field caused by generators located inside the volume, as for the case of a source field. At the same time, the system does not determine the harmonic field M 3 , caused by external generators, and therefore it is necessary to introduce boundary conditions. Taking into account the results derived in Section 1.5, we can formulate two boundary-value problems. 1. Dirichlet's problem (a)

curlM =W CurlM =Ws

divM=O DivM =0

and (b) M, = ({)l(P) is known on the boundary surface. Here t is a direction tangential to the boundary surface and ({)l(P) is a given function. 2. Neumann's problem (a) curlM =W divM = 0 CurlM=Ws DivM=O and (b) M n = ({)2(P) is known on the boundary surface, and n is the unit vector normal to the surface. As was shown earlier in the case of harmonic and source fields, the systems of field equations can be replaced by either Laplace's or Poisson's equations for the scalar potential a. However, in general, this replacement cannot be performed for the vortex field since in those parts of the volume where vortices are present, curl M is not zero. At the same time it is possible to replace the system curlM = W

divM

=

0

by one equation, and with this purpose we will make use of the equality div curl A == 0 Therefore, from the second equation of the field, it follows that the vortex field M can be represented as M = curl A

(1.340)

126

I

Fields and Their Generators

The vector A is called the vector potential of the vortex field and is not uniquely defined from Eq, (1.340), inasmuch as we can add any gradient to the vector A but it will still define the same field M. M = curl A = curl(A + grad cp) This ambiguity, which manifests itself even to a greater extent than in the case of the scalar potential, can often be used to simplify the equations for potentials and, therefore, the solution of the forward problem. The transition from the source field to the scalar potential is obvious, since it is much easier to operate with the function U than with the vector field. However, it is not so clear why it is useful to introduce the function A, which is also a vector as well as the field itself. Notice that several reasons can often justify introduction of the vector potential. 1. Sometimes it is possible to describe the field M with the help of one or a maximum of two components of the vector potential, resulting in great simplification. 2. In general, systems of field equations of electromagnetic fields and elastic waves contain four equations with two unknown fields, and it turns out that they can often be described with one or two vector potentials having a small number of components.

Bearing in mind these facts, we will derive an equation for the vector potential A and formulate boundary value problems in terms of this auxiliary function. Substituting Eq. 0.340) into the first field equation we have curl curl A = W (1.341) or VXVxA=W This procedure of replacement of the system of field equations is shown below:

Icurl M = W I

I div M = 0

1 r 1 ~~~IM=CUrlAI wi

I curl curl A =

Inasmuch as curl curl A = grad divA - V 2A

1

I

1.11 Vortex Fields

127

instead of Eq. (1.341) we have V2A = -W + grad divA

(1.342)

Often from consideration of the specific problem we can determine the divergence of the vector potential A and, correspondingly, assume that divA = f3

(1.343)

Here f3 is a known function. For instance, in the case of the time-invariant magnetic field from the Biot-Savart law and the principle of charge conservation it follows that f3;: O. Thus, taking into account Eq. (1.343) we have (1.344)

where WI = W - grad f3

Then, considering components of the vector potential in a Cartesian system of coordinates, we can use the results derived in previous sections and formulate boundary value problems. Further we will consider boundary value problems for A, proceeding directly from Eq, (1.341). First, suppose that inside the volume surface vortices are absent and, correspondingly, Eq. (1.341) is valid everywhere in V. Let us assume that there are two different solutions of this equation, Ai and A 2 ; that is, and

VXVXA 2=W

Therefore, their difference A 3 =A z -AI

satisfies a homogeneous equation (1.345)

Representing this equation as V2A 3 = grad divA 3 and even assuming that divA 3 = 0, we see by analogy with Poisson's equation that Eq, (1.345) has an infinite number of solutions. Now we are prepared to find boundary conditions for the vector potential A, which uniquely define the field. In principle, we will follow the same pattern of derivation as that in the case of the source field. First, let

128

I

Fields and Their Generators

us introduce the vector

x = A3 X V X A3 = A3 X M 3

(1.346)

Applying Gauss theorem to the vector X, we obtain

and performing a differentiation inside the volume integral and taking into account the equality div(a

X

b)

=

b . curl a - a . curl b

we have

Then, in accordance with Eq. (1.345) we obtain (1.348)

It is obvious that if from the boundary conditions it follows that the surface integral on the left-hand side of Eq. (1.348) equals zero, then the field M 3 vanishes inside the volume V and in accordance with Eq. (1.338) the field M is uniquely defined. Now we will determine such boundary conditions for two cases.

Case 1 Suppose that the tangential component of the field M is given at the boundary surface; that is, nXM=nXP

(1.349)

Here P is the given function on the surface So' Then, the difference of two arbitrary solutions with the same tangential component satisfies the following equality on So: n XM 3 = 0

Therefore, taking into account that a . (b X c) = c . (a X b) = b X (c X a)

(1.350)

1.11 Vortex Fields

129

we can conclude that the surface integral in Eq. (1.348) equals zero, and correspondingly at every point of the volume M 3 O. In other words, we have demonstrated that Eq. 0.341) and the boundary condition 0.349) uniquely define the field M.

=

Case 2 Let us assume that the normal component of the field is known at the boundary surface; that is,

n'M=N

(1.351)

where N is a given scalar function. Respectively, for the difference of solutions M 3 on this surface we have n : M 3 =0

(1.352)

That is, the vector M 3 has only a tangential component. This means that its vector potential A 3 is directed along the normal n, and since n '(A 3 X M 3 )

= M3

'(0

XA 3 )

the surface integral in Eq. (1.348) equals zero, and therefore M 3 =0

Thus, we have proved that knowledge of either the tangential or normal component of the field at the boundary surface is sufficient to take into account the field caused by external generators. It is proper to notice that we have proved the theorem of uniqueness by using two different approaches, and, of course, arrived at the same result.

Now we will suppose that there is a surface of singularity of the field S, with vortices Then the system of field equations has the form

w..

curlM =W n X (M(2) -

M(l») = Ws

divM

=

0

n . (M(2) -

M(1»)

=

0

and in terms of the vector potential A the surface analogies of the equations are O'

V' X (A(2) - Nl))

=

0

(1.353)

130

I

Fields and Their Generators

where A(l) and A(2) are values of the vector potential at the back and front sides of the interface, correspondingly. By analogy with the case of the source field we will generalize Eq. (1.348) and obtain

f

+ n . (A(j) X M~l) - A~) X M~2)}

dS

Si

=

fM~dV v

It is clear that the second integral at the left-hand side vanishes if

A(j) X M~l)

= A(~)

X M3

or

Nj) X V' X Nr = N~) X V' X A(~) and this occurs if we require continuity of the two vector functions

A(j) = Aj2)

and

n X V' X A(r

=

n X V' X A(~)

(1.354)

By definition of the function A 3 Eqs, 0.354) hold, provided that both the vector potential A and the tangential component of the field M

nXV'XA obey the following conditions at the interface Sj, where the vortices are located.

1.

A(2) - A(l)

2.

n X (V' X A(2») - n X (V' X A(1»)

=

N, =

Ps

(1.355)

where N, and P are given functions. Comparison with the system of field equations shows that Ps=Ws and this characterizes the distribution of surface vortices. Therefore, in our case a solution of the boundary-value problem consists of three steps, namely,

1. Solution of equation V X V' X A = W. 2. Determination of solutions of this equation that satisfy one of two boundary conditions. 3. Choice among these solutions of functions A that obey Eqs, 0.355).

1.11 Vortex Fields

131

Let us note that this result can be easily generalized to those cases in which there are also lines of singularities. This analysis of boundary-value problems shows that there are relations between the vector potential A, on the other hand, and vortices within the volume V and boundary conditions on the other hand. In principle, there are two approaches to deriving such equations. The first one is based on the representation of A in Cartesian components. A =Axi + Ayj + Azk Inasmuch as the unit vectors i, j, k do not depend on coordinates of the point, each component satisfies Poisson's equation.

and we can make use of the results derived in the previous section for the scalar potential U. At the same time we can derive an expression for the vector potential proceeding directly from Eq, (1.341), and this approach is considered below. Suppose that the vector functions Land N and their first and second derivatives are continuous in volume V and at the surface S. Then, applying Gauss theorem to the vector X, X = L X (\7 X N)

(1.356)

we obtain

fv div(L

X

curl N) dV = ~(L s

X

curl N) . n dS

(1.357)

Taking into account the equality dive a X b) = b X curl a - a X curl b we obtain the vector analog of the first Green's formula (Stratton, 1941).

fv(curl L: curiN - L· curl curl N) dV =

¢(L X curl N) . n dS

(1.358)

In the same manner we can write (Stratton, 1941)

fv(curl L· curiN - N . curl curl L) dV =¢.(N X curlL) . n dS s

(1.359)

132

I

Fields and Their Generators

Then, subtracting Eq. (1.359) from Eq. (1.358) we have the vector analog of the second Green's formula.

fv(N . curl curl L - L . curl curl N) dV = ~(L

X curlN - N X curlL) . n d.S

(1.360)

5

To find the vector potential, we assume here that divA = 0

(1.361)

That is, and introduce the notations L = A(q)

and

N

=

G(q,p)

(1.362)

where q is an arbitrary point within volume and p is the point where the Green's function G has a singularity, l/r. The quantity r is the distance between the points q and p; that is,

As in the case of the source field, the Green's function can be chosen in many ways, but there we consider one function only.

G(q,p)

ao =r

(1.363)

where a o is a vector that does not depend on the position of the point. Inasmuch as the Green's function has a singularity at the point p, we will surround this point by the spherical surface 51 with a small radius r 1• Then, applying Eq, (1.360) to the volume confined by the surfaces 5 and 51 we obtain

fv{G' curl curl A - A· curl curl G} dV =~{A

X

curlG - G X curl A] . n d5

5

+~

{A X curlG - G X curl A} . n d5 51

(1.364)

1.11 Vortex Fields

133

First, consider the volume integral. In accordance with Eq. 0.341) we have

jvG ' curlcurlAdV= jvr~.

WdV

Then, taking into account the equalities curl !paD

=

cp curl a., + grad

X aD

!p

curl curl !paD = grad div !paD - V 2cp a o div !paD =

!p

div a., + a, • grad

(1.365)

tp

and the fact that aD = constant, we obtain

1 curl G = grad - X aD r

curl curl G = grad ( 8 0 ' grad

~

)

and A· curl curl G = A' grad( 30'

grad~

)

since div A = O. Correspondingly, the volume integral can be represented as

fj :0 .W - diV{ =

a,

(aD'

grad~ )A} dV

·f-Wr dV - aD' '~(A' Ys

1

n) grad - dS

r

1

- aD'

~ (A' n) grad - dS Si

r

(1.366)

Now we will study the surface integrals in Eqs. (1.364), (1.366) around the point p; and with this purpose in mind, let us make the following

134

I

Fields and Their Generators

transformations: {A X curl G} . n = {A X (grad

=

a , . {grad

~

~

X a o )}

•

n

X (A X n) }

and MXn (GxcurlA)'n=a o . -r Here M

=

curl A

Then taking into account Eq. (1.366) we can rewrite Eq. (1.364) as W 1 dV = ,{, (A . n) grad - dS fv r 'J;;+s, r

+

f

{grad

+

J

--dS

S+SI

~r

X (A X n)} dS

nXM r

S+S,

(1.367)

On the spherical surface around the point p we have

1 grad -

r

ro

=-

r2

where r o is the unit vector directed to the point p along the radius r. Correspondingly, the integrals around p can be written in the form

1

+-~

r,

(nXM) dS SI

Making use of the identity a X (b X c) = b( a . c) - c( a . b)

(1.368)

1.11 Vortex Fields

135

we can represent the second integrand in Eq. 0.368) as roX(AXn) =(ro'n)A-n(ro'A) =A-ro(A'n) since r 0 have

=

n. Therefore, for the surface integrals around the point p we

1

¢

+ 2" (n X M) dS T1

=

1 2: ~ A dS r 1 51

1

+-

r1

¢ (n X M) dS

Then, making use of the mean value theorem and taking into account the fact that the second integral equals zero, we have (1.369) where NY is the average value of the vector potential. From Eq, (1.369) it follows that in the limit the sum of the surface integrals around point P is equal to 4'7TA(p), and in accordance with Eq. (1.367) we have 1 W(q) 1 n X M A(p)=-f-dV--rf..--dS 4'7T v r 4'7T 'Ys r 1

1

- -rf..(n· A) grad - dS 4'7T 'Ys r - _1 rf.. {( n X A) X grad ~} dS 4'7T 'Ys r

(1.370)

Thus, we have expressed the vector potential in terms of the vortices within the volume V and the values of the field and its potential on the boundary surface. It is obvious that the surface integrals in Eq. (1.370) describe a vector potential caused by vortices located outside the volume V. Here it is appropriate to make the following comments:

1. As was mentioned above, Green's functions can be chosen in different ways. In particular, they can be a solution of the same equation as that for the vector potential, and due to this fact, in some cases the volume integral in Eq. (1.364) vanishes.

136

I

Fields and Their Generators

2. Equation (1.370) has been derived under the assumption that surface vortices are absent. Applying the same approach as in the case of the source field it is not difficult to take into account the presence of surfaces or lines with field singularities. 3. In general, vortices that arise in a medium depend essentially on the field and, correspondingly, they cannot be specified if the field is unknown. In other words, as in the case of the source field, we again experience the so-called closed circle problem, and this fact always requires very serious modifications to the system of field equations.

References Alpin, L.M. (1966). "Theory of Fields." Nedra, Moscow. Stratton, 1.A. (1941). "Electromagnetic Theory." McGraw-Hill.

Chapter II

The Gravitational Field

ILl Newton's Law of Attraction and the Gravitational Field 11.2 Determination of the Gravitational Field Elevation Correction The Bouger Slab Correction Two-Dimensional Model Three-Dimensional Body

11.3 System of Equations of the Gravitational Field and Upward Continuation Integral Form of Equations for Gravitational Field Differential Form of Equations for Gravitational Field

References

II.! Newton's Law of Attraction and the Gravitational Field

The gravitational method is a study of the distribution of Newton's attraction force F caused by all masses of the earth. One part of this force provides the uniform motion of a body around the rotation axis of the earth, and the other part originates the weight force. Thus, F= Fa + P where Fa is the centripetal force directed toward the rotation axis and P is the weight. The directions of these forces are shown in Fig. II.la. It is essential that the centripetal force can easily be taken into account along with the force of the bulk attraction of the earth. Because Fa can be calculated and removed very accurately, we will pay attention only to that part of the attraction force F that obeys Newton's law. To formulate this law, we will suppose that there are two masses, ilm(p) and i1m(q), located in elementary volumes i1V(p) and i1V(q), respectively. The points p and q characterize the position of these volumes. The distance between the points is L p q (Fig. II.1b). It is proper to notice that dimensions of volumes i1V(p) and i1V(q) are much smaller than the distance L q p ; this fact is the most important feature of elernen137

138

II The Gravitational Field

Fig. 11.1 (a) Gravitational force and its components; (b) interaction between elementary masses; (c) arbitrary distribution of masses; and (d) the field caused by an elementary mass.

tary volumes. Then, in accordance with Newton's law of attraction, the elementary mass I1m(q) acts on the elementary mass I1m(p) with a force dF(p) equal to dF(p)

=

-1

I1m( q) Am( p) L3 L qp

(11.1)

qp

where 1 is a coefficient of proportionality, or gravitational constant, which in the International System of Units (SI) is 1 = 6.6710- 11 m 3 kg " ! sec- 2 L q p is the vector

and L~p is the unit vector directed from the point q to the point p along the line connecting these points. AV(q) and I1V(p) are elementary volumes. It is clear that only in such cases the force of attraction of the masses, located in these volumes, does not depend on the position of the

11.1 Newton's Law of Attraction and the Gravitational Field

139

points p and q. From this definition it follows that dimensions of elementary volumes in different problems can change from very small values to extremely large ones. Thus, in accordance with Eq. (II.1) the mass Ilm(p) is subjected to a force dF(p), which is proportional to the product of both masses and inversely proportional to the square of the distance between them, and has a direction opposite to that of L q p . This extremely simple formula describing the basic physical law of gravimetry may need some almost obvious comments, and they are

1. Elementary masses Ilm(q) and Ilm(p) can be different. 2. Newton's law states that due to the presence of any mass Ilm other masses experience an action of the force that tends to attract these masses to Ilm. This effect decreases with an increase of the distance from tsm. 3. Newton's law of attraction is a mathematical description of one of the most fundamental phenomena of nature. Notice, however, that it does not explain the mechanism of transmission of this force through a medium. 4. In accordance with the principle of superposition, the force of attraction between two masses does not depend on the presence of other masses provided that their position in space remains the same. 5. Gravitational masses, unlike electrical charges, have only positive values, and therefore only attraction is observed. 6. In the SI system of units the distance is measured in meters, mass in kilograms, and the force in Newtons. 7. As follows from Eq. (II.1) the force acting on mass Ilm(q) and caused by mass Il mt; p) is

dF(q) =

-r

Ilm(q) Ilm(p) 3

Lp q

Lpq

Inasmuch as

L 3qp -L3pq

and

L oqp -

L Opq

we have dF(q)

=

-dF(p)

Now we are ready to introduce the concept of a gravitational field. Taking into account the fact that the force dF(p) is proportional to the

140

II

The Gravitational Field

elementary mass ~m(p), which is subjected to the action of this force, it is natural to consider their ratio dg( p). dF(p) ~m(q) dg( p) = ~ ( ) = - Y L 3

m p

qp

L qp

(11.2)

or ~m(q)

dg(p)=-y

0

L2

Lqp

qp

The function dg(p) is called the gravitational field at point p, caused by the elementary mass ~m(q). It has the dimensions of force per unit mass; that is,

[g] = mysec ' In practice, the field is usually measured in milligals, which are related to meters per second by cm 1 Gal = 1 -2 = 1000 mGal sec and

1 rnysec '

=

100 gal = 105 mGal

Note that unit "Gal" was introduced in honor of Galileo. It is proper to notice that for a given mass ~m(q), the function dg(p) is a vector field inasmuch as its magnitude and direction depend on the coordinates of the observation point only. As follows from Eq. (II.1) it is a simple matter to find a force acting on any elementary mass when the field is known, and we have dF(p) = ~m(p)

dg(p)

(11.3 )

It is useful to regard Eq. (11.2) as a relation between an elementary mass

and the gravitation field caused by this mass, which exists at any point regardless of presence or absence of other elementary masses at this point. Now let us assume that instead of an elementary mass there is some distribution of masses in volume V (Fig. 1I.1c). To find the field g, caused by all the masses within this volume, we will use the principle of superposition. With this purpose in mind, the volume V is mentally divided into many elementary volumes, the size of which satisfies two conditions, namely, (a) They are small with respect to the distance between this volume and the observation point.

II.t Newton's Law of Attraction and the Gravitational Field

141

(b) The masses are distributed uniformly within every elementary volume. The latter allows us to present its mass, tJ.m, as

tJ.m(q)

=

Seq) dV

(lIA)

where S(q) is the density of the mass, which can vary arbitrarily within the volume V. Inasmuch as the gravitational field caused by the elementary mass is dg( p) = -y

S(q)dV L3

L qp

qp

the total field is the sum of fields due to all masses.

(11.5) This equation describes the gravitational field at any observation point p, whether it is located inside or outside of the masses. It is essential to

note that masses can be specified before we calculate the gravitational field caused by them. This fact dramatically simplifies the determination of the field, and in accordance with Eq, (11.5) it is reduced to numerical integration only. Later, when considering other fields applied in geophysical methods, we will show that determination of fields is usually a much more complicated problem, Now making use of Eq. (1I.s) we will consider several examples illustrating some features of the behavior of the field.

Example 1 Field of an Elementary Mass

We will study the field caused by an elementary mass Srn, located inside of the elementary volume tJ. V in the vicinity of the point q (Fig. IUd). Inasmuch as by definition observation points are located far away with respect to the size of the elementary volume, we can neglect the change of the distance from the observation point to any point inside this volume. Correspondingly, instead of a real elementary mass we can consider the same mass located at point q. This is of course physically impossible, but it is convenient for field calculation. Thus, a point mass is a useful concept when the field is studied at distances greatly exceeding dimensions of the

142

II The Gravitational Field

elementary volume. With this definition of a point mass, the expression for its field is

(11.6) For simplicity we will make use of a Cartesian system of coordinates with its origin at point q. Then, for the vector L q p and its magnitude we have L q p = Lop ";"xi + yj +zk L qp = V _/X2+y2+ Z2

(11.7)

and the gravitational field has a direction opposite to that of the vector Lop. Because x, y, and z are projections of the radius vector Lop on the coordinate axes, the components of the field are m

ym

gx=g·i=-YL3 · L op·i=-L3 x

op

.

gy

= g .J =

m

-

op

.

Y L 3 Lop' J

ym

= -

Op

L3 Y

(II .8)

Op

For illustration, consider the behavior of the field Then, in accordance with Eqs. (II.7), (11.8) we have

III

the x -z plane.

mx (x 2 + h 2 ) 3/ 2 '

g = - y -----;;-= x

since

y= 0

Here h is a constant for the profile characterizing the position of the mass. The behavior of the field components is shown in Fig. II.2a. The horizontal component gx is positive for negative values of x, and it increases with an increase of x, reaches a maximum, and then tends to zero as x ~ O. For positive values of x,

gAx)

=

-gx( -x)

That is, it is an odd function. The vertical component g z has simpler behavior; it is a symmetrical function with respect to x

II.l Newton's Law of Attraction and the Gravitational Field

143

Fig.II.2 (a) Components gx and gz due to an elementary mass; (b) surface masses; (c) the normal component of gn near a surface mass; and (d) field gz near a disk.

and reaches a minimum at x = 0, where the distance from the observation point to the mass is minimal. Negative values of the vertical component are explained by the fact that everywhere along the profile the vector projection of the field on the z-axis has a direction opposite to this axis. For the same reason, the horizontal component gx is positive for negative values of x, and it becomes negative for x > O. Now let us look at these formulas from a geophysical point of view; that is, we will attempt to find the position of the mass and its value. First of all, the change of sign of gx and maximal value of magnitude of the component g z as x = 0, indicate that the mass is located at the z-axis. Then, in accordance with Eqs. (II.9) the ratio of the components is

s,

or X

gz

h=-x gx

(11.10)

That is, this ratio allows us to determine the position of the mass. Finally, using the values of either gz or gx' the mass m can also be calculated. Of course, both parameters m and h can be found from only one of these components. It may be seen that the point mass is an over-simplication of

144

II

The Gravitational Field

the real distribution of masses. It is certainly true, however, with an increase of a distance from masses, distributed within a volume of arbitrary shape and dimensions, their field approaches that of the point mass, regardless of where inside the volume this point mass is placed. This result directly follows from Eq. (II.S), since any mass can be treated as an elementary one if its field is considered far away from the mass. In fact, with an increase of the distance from the observation point to the volume mass we have

(11.11) where qo is an arbitrary point within the volume V, and m is the total mass in this volume. Now, let us look in a little more detail at Eqs. (11.9). Taking the first derivative of the horizontal component gx with respect to x, we find that the abscissas for their extrema are h x=+-

-fi

(11.12)

As is seen from Eqs. (11.9), with an increase of the coordinate x the magnitudes of both components approach each other, and they become equal if x=h

(11.13)

As x increases, the horizontal component becomes dominant; that directly follows from an analysis of the field geometry. It is also proper to notice that Eq. (11.10) demonstrates the relation between the vertical and horizontal components of the field caused by an elementary mass. Proceeding from the principle of superposition we can expect that for an arbitrary mass and an observation point there is also a relation between these components of the field which, of course, is more complex than Eq. (IUD). In this connection let us point out that such relations show that both components of the field measured along a profile contain the same information about the distribution of masses, and this' means that in principle it is sufficient to measure only one of them.

11.1 Newton's Law of Attraction and the GravitationaJ Field

145

As is seen from Eq. (11.9) and was not noticed above, the horizontal component, unlike the vertical one, is an odd function of x and therefore

(11.14) Later we will show that in the general case, when the component gx is not an odd function, this equation still remains valid.

Example 2 The Nonnal Component of the Gravitational Field Due to Plane Surface Masses Suppose that masses are distributed within a plane layer whose thickness is much smaller than the distances from these masses to the observation point (Fig. IL2b). In other words, the distance between the observation point and any point of the elementary volume is practically the same. Taking into account this fact, we can replace this layer by a plane surface with the same mass, located somewhere at the middle of the layer (Fig. lI.2c). Inasmuch as every elementary volume contains the mass

dm =8(q)hdS its distribution on the surface can be described by

dm = a ( q) dS where

(11.15)

u(q) =o(q)h

The function u(q) is called a surface density or density of surface masses. The latter, as well as point masses, is a pure mathematical concept introduced to simplify calculations. Of course, every element of the surface S bears the same mass as a corresponding elementary volume of the layer. Our next step is to find the field caused by these surface masses. First of all, we will distinguish the back and front sides of the surface S and choose the direction of the normal n(q) at every point of the surface from the back to the front side. In accordance with Eq. (11.2) the field caused by an elementary mass on the surface dS is

dm

dg( p)

o dS

= -y L 3 L qp = -yvLqp qp

qp

(11.16)

146

II

The Gravitational Field

For simplicity suppose that the surface is a plane and the masses are distributed uniformly; that is, a = constant. At every point the field can be represented as the sum of two components. (11.17) where nand t are unit vectors perpendicular and tangential to the surface, respectively. As is seen from Fig. 1I.2c the normal component of the field at the point p is

adS

(11.18)

= -Y--zy-Lqp·n qp

where (Lqp,n) is the angle between vectors L qp and n. Applying the principle of superposition we obtain for the normal component of the field due to all surface masses ( dS . L qp gn(p)=-ya), L 3

(11.19)

qp

S

since dS = dS n Taking into account that L qp = -L pq and L~p =L~q, normal component of the field

we have for the

(11.20) As follows from Eq. (1.44) the integral at the right-hand side of this equation describes the solid angle w(p). ( dS . L pq L3

w(p) ").

S

(11.21)

pq

Thus, the normal component of the gravitational field caused by masses uniformly distributed on the plane surface can be expressed as (11.22) Making use of the results derived in Chapter I, we will describe the behavior of the normal component gn' With an increase of the distance from the surface, the solid angle becomes smaller, and therefore the normal component of the field decreases. On the other hand, in approach-

11.1 Newton's Law of Attraction and the Gravitational Field

147

ing the surface the solid angle tends to its limiting values: - 27T and + 27T on the front and back sides of the surface, respectively. Thus, for the normal component gn on the surface S, we have g

-27T'}'lT

= n

{

27T'}'lT

on the front side on the back side

(11.23)

and correspondingly, the discontinuity of the normal component of the field on the surface mass is

where g~ and g;; are values of the field at the front and back sides of the surface. As we shall see in the next example, the field is everywhere a continuous function, but this discontinuity arises due to replacement of volume masses by surface ones. Let us consider two special cases. (a) A plane surface, infinite in extent. Then the solid angle under which the plane is seen from the front and back sides does not depend on the position of the point p and is equal to =+= 2 7T, respectively. In other words, plane surface masses with infinite extension and constant density create a uniform field in each half space. (b) A plane surface in the form of a disk with radius a and the normal component measured along the z-axis (Fig. 1I.2d). In this case, making use of Eqs. (1.48) we have g zC z) =

=+=

27T '}' (1 - cos a )u

(11.24)

where [z]

With an increase of the distance z, the angle a tends to zero, and therefore the field gz decreases. Expanding the radical (Z2 + a 2 ) 1/ 2 in a series,

148

II

The Gravitational Field

we obtain

or 2

m ( 1 -3-a- ) g7(Z)Z+Y-

Z2

4

Z2

if

z» a

where m is the total mass of the disk. From this equation it is very simple to evaluate the minimal distance z at which the field of masses located on the disk coincides with that of a point mass. In approaching the disk the field tends to its limit, equal +27TyO', which corresponds to the field due to the infinite plane surface with constant density 0'. Applying again the expansion of the same radical, we have if

a»lzl

Example 3 Field Caused by a Volume Distribution of Masses in a Layer with Thickness h and Density 8 (Fig. lI.3a) First, introduce a Cartesian system of coordinates with its origin at the middle of the layer and the axis z directed perpendicular to its surface. Let us note that the layer has infinite extension along the x and y axes. At the beginning, suppose that the observation point is located outside the layer, that is, [z] > h/2. Then we mentally divide the layer into many thin layers which are in turn replaced by a system of plane surfaces with the density 0' = 8 Sh, where Sh is the thickness of the elementary layer. Taking into account the infinite extension of the surfaces, the solid angle under which they are seen does not depend on the position of the observation point and equals either - 27T or + 27T. Correspondingly, each plane surface creates the same field. if

h

z>2

and if

h z < -2

11.1 Newton's Law of Attraction and the Gravitational Field

149

Fig. 11.3 (a) Model of a homogeneous layer; (b) observation point inside a layer; (c) field behavior, gz; and (d) system of layers.

Therefore, after summation the total field due to all masses in the layer is if

gz= -2rryoh

h z> - 2

and

(II.25) gz

=

2rryoh

if

h z< - -2

It is interesting to notice that these formulas are used to calculate the

Bouguer correction in gravimetry. Now we will study the gravitational field inside a layer when the coordinate of the observation point z satisfies the condition

First, suppose that z > O. Then, the total field can be presented as a sum of two fields: one of them is caused by masses with thickness equal (h /2) - z, and the second one is caused by masses in the rest of the layer

150

II

The Gravitational Field

having the thickness z + h/2 (Fig. 1I.3b). In accordance with Eqs. (I1.25) these fields are

and

(11.26)

Correspondingly, for the total field we have if

h h - -2 -
and by analogy, if

h --
(11.27)

The behavior of the field due to masses within the layer is shown in Fig. lI.3c. Thus, for negative values of z the field is positive, since the masses in the upper part of the layer create a field along the z-axis, and this attraction prevails over the effect due to the masses located below the observation point. At the middle of the layer, where z = 0, the field is equal to zero. Of course, every elementary mass of the layer generates a field at the plane z = 0 in accordance with Newton's law, Eq. 0I.2); but due to symmetry the total field turns out to be zero. For positive values of z the field has the opposite direction, and its magnitude increases linearly with an increase of z. As follows from Eqs. (11.25)-(11.27) the field changes as a continuous function at the layer boundaries. One more feature of the field behavior is worth noting. Inasmuch as the layer has infinite extension in horizontal planes, the distribution of masses possesses axial symmetry with respect to any line parallel to axis z that passes through an observation point. For this reason it is always possible to find two elementary masses such that the tangential component of the field caused by them is equal to zero. Respectively, the field due to all masses of the layer has only a normal component, described by Eqs. 0I.25)-0I.27). Now let us consider the field inside a layer when it is surrounded by two other layers with densities 0l(Z) and 0iz), which are arbitrary functions of z (Fig. IUd). In accordance with Eqs. (11.25-11.27) the field in the

11.1 Newton's Law of Attraction and the Gravitational Field

151

middle layer can be represented as if

h h - -
- 2

where /3 is the constant density of the middle layer, while g lz and gzz are the fields caused by the masses of the upper and lower layers, and both these fields do not change within the middle layer. Suppose we would like to find the density /3. Then, after determination of the field at two points of the layer we have

Therefore,

and /3

=

gzC z,) -qzCzz) 47TY(ZZ-ZI)

Note that this approach is used in measuring the density of rocks in a borehole.

Example 4 The Field Caused by Thin Spherical Shell with Density
Taking into account the spherical shape of the body (Fig. lI.4a) we will choose a spherical system of coordinates R, e, 'P with its origin at the center of the shell. We can in general expect three components of the field, gR' go, and g'P' directed along coordinate lines R, 8, and 'P, respectively. However due to the symmetry, it is a simple matter to show that components go and g'P are everywhere equal to zero. In fact, coordinate planes e = constant and 'P = constant are planes of symmetry, and therefore it is always possible for any elementary mass to find another symmetrically located with respect to it so that the field caused by both masses does not have components go and g'P' Correspondingly, the field of a mass shell has only a radial component, gR' First, we will study the field

152

II

The Gravitational Field

Fig. 11.4 (a) Measuring the field inside a shell; (b) measuring the field outside a shell; (c) field behavior due to shell masses; and (d) spherical mass with radius a and density 8.

inside the shell at any point p caused by two elementary masses (T dS j and (T dS z, as is shown in Fig. BAa. In accordance with Eq. (If.Z) these fields are

«as, dg j R

=

dg Z R

-'Y~,

(TdS z

=

-'Y~

pq2

pql

Inasmuch as

as,

as,

dw

L;ql = L;q2 = cos a (dw is the solid angle) and the fields have opposite directions, the total

field caused by these two elementary masses is equal to zero. Considering the spherical shell as a system of such pairs, we conclude that the field inside a uniform spherical shell is zero. Now we will consider the field outside the shell (Fig. BAb). First let us determine the field caused by the mass within the elementary ring with radius x and width r dip, located on the shell surface. Each elementary

11.1 Newton's Law of Attraction and the Gravitational Field

153

mass of the ring creates a field at the point p with magnitude dg

=

a r dtp dt -Y--L-::2-

where r dc: dt is the area of the elementary mass, and L is the distance from the mass to the observation point. As is seen from Fig. BAb its radial component is dg R

=

dg cos a

R - r cos 'P = dg-~-

L

or a r dip dt

dg R

=

-

Y

3

(

sin e

= -

L

R - r cos 'P)

where x

r

Since all elements of the ring are located at the same distance from the observation point, we have the following expression for the radial component due to the ring mass:

Replacing L and x with L = (r 2 + R 2 - 2rR cos 'P)1/2, x = r sin 'P, and integrating, we obtain for the field caused by all masses of the shell 7T sin 'P ( R - r cos 'P ) gR=-yu27Tr21 '12 d'P o (r 2 + R 2 - 2rR cos 'P

r

or

where =

sin 'P dip 1o (r 2+R 2-2rRcos'P)

3/2

12 =

'P cos 'P 'P 1o (r 2+Rsin2-2-rRcos'P)

3/2

7T

II

7T

d

(IL28)

154

II The Gravitational Field

We will use the notations b = 2rR Then we have sin 'P dip

1a (a - b cos 'P ) tr

II

=

3/2

_ (Tr sin 'P cos 'P dsp

12 -

l,

a (a - b cos 'P )

3/2

Introducing the variable z

=

a - b cos 'P

we obtain dz

=

b sin 'P de:

and

=

2{ II } (a - b) (a + b)

b

1/ 2

1/2 -

For the second integral,

=

-

-

2

{a

b? va

+b

-

a va - b

+ Va + b - va - b }

Inasmuch as (a+b) we have

1/2

=R+r

and

(a-b) 1/2 =R-r

11.1 Newton's Law of Attraction and the Gravitational Field

155

Whence

(II .29) where m is the total mass on the spherical surface.

Thus, masses located on the shell with density a generate a field, which outside of this surface coincides with that of a point source with the same mass, placed at the center of the shell. The behavior of the field gR' inside and outside is shown in Fig. HAc. As follows from this analysis the normal component gR' as in the case of the plane surface, has different values at each side of the shell. In accordance with Eqs. (11.23) and (11.29) this discontinuity is the same for both of these surfaces. This coincidence is not accidental, and it can be shown that for arbitrary surface masses the difference of normal components of the gravitational field is equal to - 47TYU(q); here ai.q) is the surface density at vicinity of point q where the field is studied. The next example is a natural generalization of this case.

Example 5 Gravitational Field of the Sphere (Fig. HAd)

Applying the same approach as in the previous example we can conclude that the field has a radial component gR only; that is, go = gCP = O. To calculate this component, let us mentally divide the sphere, which has the radius a and volume density B, into a system of thin spherical layers with surface density o = BdR, where dR is the layer thickness. First, suppose that the observation point is located outside the sphere R > a. Then applying the principle of superposition and making use of Eq. (11.29) we have 47TBy

gR=

-~

a

fa r

2dr=

47Ta 3 B m - - 3 - y R 2 = -y R 2

if

R"?r (11.30)

where m = 17Ta3B is the total mass of the sphere. If the observation point is inside the sphere and its coordinate is R, then as follows from the previous example, all spherical shells that have radius r exceeding R do not contribute to the gravitational field at the

156

II

The Gravitational Field

Fig. II.S. (a) Field due to spherical mass; (b) point p is the center of an elementary spherical mass; (c) the secondary field; and (d) one-dimensional model of a medium.

observation point. Correspondingly, making use of Eq. (11.29) we obtain

47TO

gR

=

-"7 fa

R

2

47T

r dr = -

3"oR

(II.31)

Thus, inside the sphere the field magnitude increases linearly from zero to a maximum value on the sphere surface, equal to (47T j3)yoR, and then it decreases inversely proportional to the distance R (Fig. II.5a). It is a simple matter to generalize this result to the case in which the density 0 is a function of the radius. Then it is clear that inside the sphere the field decreases approaching its center, while outside the sphere it behaves as the field of the point source. Let us emphasize that the main part of the gravitational field of the earth has this behavior. It is proper to notice that as in the case of the layer, the field inside the sphere has everywhere finite values, and it is a continuous function. We will consider this fact proceeding from Eq. With this purpose in mind, we will represent the total field at point p as a sum of two fields: The first part is caused by mass within the elementary spherical volume with its center at point p, and the second part is due to other masses (Fig. II.5b). The latter are located at distances equal to or exceeding the

m.su

11.2 Determination of the Gravitational Field

157

spherical radius p, and therefore their field has a finite value at the point p and is a continuous function near this point. At the same time, in accordance with Eq. (11.31), the first part of the field is equal to zero at point p. Thus, the total field has a finite value at any point inside the mass. Bearing in mind that on the surface of the elementary spherical mass its field is directly proportional to radius p, Eq. (11.31), we can conclude that with a decrease of this radius the contribution of the first part of the field becomes negligible; this results in continuity of the total field. Note that this consideration can also be applied to an arbitrary distribution of masses. As follows from Eq. (11.30) the behavior of the horizontal and vertical components gx and qz along some profile outside the sphere is the same as that for the point source studied in the first example. This means, in particular, that both the direction and magnitude of the 'field do not change if the radius of the sphere and its density vary in such a way that the total mass remains the same. Of course, this is true only if the observation point is located outside the sphere. This consideration shows that measuring the gravitational field along some profile or system of profiles we can determine only the total mass of the sphere-that is, the product }7TOa 3, as well as the position of the sphere center-but it is impossible to find out separately the density of masses 0 and the radius of sphere a.

This example vividly demonstrates the simplest case of nonuniqueness in which different distributions of masses generate exactly the same field as for measurements performed outside the masses. Certainly, this is a negative factor that imposes some limitations in determining the mass distribution. Later we will discuss more complicated types of such ambiguity in various geophysical methods. In this section we have obtained some insight into the field behavior by making use of Newton's law of attraction and the principle of superposition. Now we consider their application to the theory of the gravitational method. IL2 Determination of the Gravitational Field In this section we will continue the study of the behavior of the field, but main attention will be paid to calculation of the gravitational field caused by masses located beneath the earth's surface.

158

II

The Gravitational Field

Taking into account the fact that the field is defined by a distribution of masses, it is appropriate to give some information about the density of rocks. Oil Water Sand, wet Sand, dry Coal English chalk Sandstone Rock salt Keuper marl Limestone (compact) Quartzite Gneiss

900 1000 1950-2050 1400-1650 1200-1500 1940 1800-2700 2100-2400 2230-2600 2600-2700 2600-2700 2700

Granite Anhydrite Diabase Basalt Gabbro Zinc blende Chalcopyri te Chromite Pyrrhotite Pyrite Haematite Magnetite Galena

2500-2700 2960 2500-3200 2700-3200 2700-3500 4000 4200 4500-48 0 4600 5000 5100 5200 7500

Rock density in kg/m 3 [after Parasnis (I 979)]

As is seen from this table, the densities of sedimentary rocks vary within a relatively small range, and they have lower density than those of igneous and metamorphic rocks. For example, the density difference between rock salt and sandstone has a maximum of 600 kg/m 3 or approximately 30% of their values. The gravitational field of the earth, as is well known, depends on several factors, such as 1. the latitude of the observation point

2. 3. 4. 5.

tides elevation topography of the area where the field is measured lateral changes of rock density

Taking into account these factors, we will present the total gravitational field as G= gN+ g

(II.32)

where gN is the normal field, which depends on the latitude, and g is the anomalous field, which depends on the elevation, topography, and lateral changes of rock densities. Inasmuch as our main purpose is the study of the anomalous field, let us only briefly describe the normal field. The earth's surface can be

11.2 Determination of the Gravitational Field

159

presented as approximately spheroidal, slightly enlarged at the equator and flattened at the poles: The ratio between the equatorial and polar radii is around 1.003. At every point of this surface, the normal field gN is perpendicular to the spheroid and can be described by the formula

(11.33) where go = 978,0318 Gal, which is the normal field at the equator, a = 0.0053024, {3 = - 0.0000058, and cp is the latitude. About the normal field let us make several comments. 1. The spheroid surface can be considered as the first approximation to the geoid surface, and the equation describing this spheroid surface can be derived assuming that the spheroid is a fluid that rotates around its polar axis. The extreme differences between the spheroid and geoid surfaces are -105 m and +73 m. 2. Equation (11.33) describes the field caused by masses of the spheroid and the centripetal force. Notice that if the earth were a nonrotating ' sphere, the gravitational field would not depend on the latitude. 3. The change of the field due to the attraction of the moon and sun (tides) is usually very smooth and has a maximum value of approximately 0.3 mGal. Assuming that the normal field is known, we will concentrate our attention on the anomalous or secondary field only.

Inasmuch as the vertical component of the field gz is the main subject of measurements in the gravitational method, we will further study only this component. First we will compare magnitudes of normal and anomalous fields. With this purpose in mind, we will consider several numerical examples. 1. Suppose that the earth is a sphere with radius R = 6370 km and density 8 = 5500 kg/m 3• Then in accordance with Eq, (IUl) the field on its surface is g

= 6.671O- ll j1T 55006370 = 9.8 mysec? =

980 Gal

=

0.9810 6 mGal

This is the normal field. 2. Here we will evaluate the contribution of sediments assuming that their density and thickness are 0 = 2000 kg/m 3 and h = 5 km, respectively. Replacing this spherical layer by a surface with density (J

= Bh

=

2000 kgyrn:' 5 km = 107 kg/m 2

160

II

The Gravitational Field

and making use of Eq. (11.29) we obtain for the field magnitude g

=

4rry(T

=

4rr 6.6710- 1110 7

= 0.8410- 2 mysec" = 0.84 Gal In other words, sediments contribute very little to the normal field. 3. Now we will calculate the field on the z-axis caused by a disk (Fig. Il.Zd), Suppose that the radius and the thickness of the disk are 1 km and 0.5 km, respectively, and that the density is 1000 kg zrn:'. Such a model can be considered the first approximation to some basement structure. Then letting z = 2 km and making use of Eq. (11.24) we have

g

= 2rryoh(l- cos a) = 2rr' 6.67 '10- 1110 3 . 500( 1-

=

2.210- 5 mysec '

=

2.210- 3 Gal

=

~

)

2.2 mGal

That is, this effect is extremely small in comparison with the normal field. In fact, in the practice of gravimetry even smaller signals are measured. One more example follows. 4. Suppose that the radius of the sphere is 1 m and its density is 1000 kgyrn:'. Then, the field on its surface is g

= ~rroyR

= ~rr'

10 3 . 6.6710- 11110 5 mGal ::::; 0.03 mGal

This example can serve as an illustration of the effect of some boulder located near an observation point where the field is measured. Before we start to derive formulas for the vertical component of the field, let us in the most general form outline the main features of interpretation of gravitational data. To simplify this subject we will begin with the simplest model and gradually approach more complicated ones. First suppose that measurements of the gravitational field are performed along a profile shown in Fig. II.5c. Here 0o(z) and 0 are densities of the surrounding medium and a body, respectively. We assume that the density 00 is in general a function of coordinate z, and in particular, the medium surrounding the body can be described by a horizontally layered model. It is clear that in absence of the body, the gravitational field does not change along the profile and, correspondingly, the anomaly vanishes.

II.2 Determination of the Gravitational Field

161

Fig. II.6 (a) Equivalent model of nonuniform medium; (b) elevation correction; (c) Bouguer correction; and (d) field calculations due to dimensional model.

In the presence of a body with a different density, however, the secondary field arises and its magnitude becomes greater as the density difference ~o = 0 - 00 increases. This simple analysis shows that we can mentally present the original model of the medium as a combination of two simpler models, namely,

1. A half space with density oiz), which does not create a secondary field (Fig. II.5d). 2. A body with density ~o = 0 - 0o(z) surrounded by free space, which generates the anomalous or secondary field (Fig. II.6a). Undoubtedly this model is simpler than the original one, and in calculating the secondary field by Newton's law, Eq. (11.5), it allows us to perform the necessary integration only within the body volume. Two more obvious conclusions follow:

1. On the surface of a horizontally layered medium the field does not change; that is, information about the medium due to measurements at one point and along a profile is the same. However, the field depends on many parameters of this model, such as the density and thickness of layers,

162

II

The Gravitational Field

and therefore their determination from the gravimetry becomes impossible. 2. To apply the gravitational method there must be lateral changes of rock density. Now we are ready to discuss some aspects of interpretation in gravimetry. Suppose that from measurements along a profile or a system of profiles, the anomalous field is known (Fig. II.6a). Then, the main purpose of interpretation is to determine the location, shape, dimensions, and density of the subsurface bodies. This task is often called the inverse problem of the gravitational field theory, since it is necessary to find a distribution of masses when the field caused by them is known along some profile or in some area. It is essential that the field is not known in a volume occupied by masses, since measurements are always performed at some distance from them and thus interpretation usually becomes a rather complicated problem. In accordance with Newton's law the field can be represented at every observation point as a sum of fields caused by elementary masses of the body; and their contributions depend on the size and location of the corresponding volumes. In particular, those masses within the body located relatively far away from the observation point only slightly affect the field magnitude. Strictly speaking, at every observation point the field is subjected to the influence of all parameters of the body, although to different extents, and their relative effect varies from point to point since they have different positions with respect to the body. Thus, in principle, all information about masses is contained in the measured field. Taking into account this simple but fundamental fact, let us formulate the main steps of interpretation. First, we will make some assumptions about the distribution of masses, and correspondingly ascribe values to parameters of the body that characterize its geometry and density contrast. Such a step is usually called the first guess or the first approximation, and it is mainly based on specific geological information. For example, if the gravimetry is carried out for detecting salt domes, there is usually some information about the density of both the surrounding rocks and the salt domes, as well as their shape and location. Of course, the difference between the first approximation and the factual values of the body parameters can vary significantly depending on our knowledge of the geology. The second step of interpretation consists of calculating the vertical component of the field along the profile, using the parameters of the first approximation and comparing the measured and calculated fields. Coincidence of these fields suggests that the chosen parameters of the model are

II.2 Determination of the Gravitational Field

163

close to the real ones. If there is a difference between the measured and calculated fields, all parameters of the first approximation or some of them are changed in such a way that a better fit to these fields is achieved. Thus, we obtain a second approximation of the mass distribution. Of course, in those cases when even this new set of parameters does not provide satisfactory matching of fields, this process of calculation has to be continued. As we see from this process, every step of interpretation requires application of Newton's law. Let us note that this procedure, based on the use of Newton's law, is often called the solution of the forward problem of the gravitational field. In summary we can say that the process of interpretation for the simplest case, shown in Fig. Il.Sc, includes two steps, namely, 1. Formulation of the first approximation for parameters of the body, the "first guess"; and 2. A solution of the forward problem that includes changing the parameters of a model to provide a better fit between measured and calculated fields.

Inasmuch as in the gravity interpretation process every step is reasonably well defined, we may arrive at the impression that this procedure of interpretation is straightforward and does not contain any complications. Unfortunately it is not true even in the case when the field is caused by only masses of the body-that is, when we deal with only the useful signal -that "noise" is absent. First, suppose that both calculated and measured fields are known with infinitely high accuracy. Then, by sequentially repeating the solution of the forward problem we can obtain a set of parameters such that difference between these fields will be extremely small. In this connection, the following question arises. Does it means that by providing an unrealistically ideal fit between the measured and calculated fields, it is always possible to determine with infinitely small error the shape, dimensions, and density as well as the location of masses that create the given field? As the theory shows, the answer is negative and in general a solution of the inverse problem is not unique; that is, different distributions of masses can create exactly the same field along a profile or a system of them. In other words, in general, different bodies can generate a field that provides exact fitting with the measured field. The simplest example of such nonuniqueness is the case when the field is caused by different spheres with the same mass and' common center but with different densities and radii. This phenomenon is hardly obvious. In fact, Newton's law of attraction tells us that a change of mass distribution should result in a change of the

164

II

The Gravitational Field

field. However, from nonuniqueness it follows that different mass distributions can create outside them exactly the same field, regardless of the accuracy of measurement; that is, it is impossible to detect the difference between fields generated by such masses. Nonuniqueness is really an amazing fact that is more natural to treat as a paradox than as an obvious consequence of gravitational field behavior. Now let us look at this subject from a practical point of view and imagine that nonuniqueness is always observed when the inverse problem of the gravitational field is solved. Then, it is clear that interpretation of gravitational data would always be impossible. In fact, having determined parameters of a body that generates the given field, we have to also assume that due to nonuniqueness there are other distributions of masses creating exactly the same field. Certainly we can say that such ambiguity would be a disaster for application of the gravitational method. Fortunately, nonuniqueness does not always manifest itself and there are such types of mass distribution for which the solution of the inverse problem for them becomes unique. Moreover, some of these cases are of great practical interest because they allow with sufficient accuracy to approximate a real distribution of masses. For instance, the inverse problem is unique if a body can be presented as a prism or a system of prisms even if their density is unknown. Another example corresponds to a more general shape of the body when every ray drawn from any point within its volume intersects the body surface only once. It turns out that in these cases of so-called star-shaped bodies the inverse problem is also unique if the density is known. Further, we will pay attention to only such distributions of masses for which the solution of the inverse problem is unique. Now we are ready to make the next step and discuss some aspects of interpretation for real conditions when the gravitational field is measured with some error; that is, the numbers that describe the field are accurate to some number of digits only. From this fact it follows that the accuracy of the field calculation can be practically the same as that of the measured field, and because of it there is always a difference between these fields. For this reason any attempt to achieve fitting of the calculated and measured fields with an accuracy exceeding that of their determination has no meaning. Taking into account the fact that the difference between the measured and calculated fields has always a finite value, which can sometimes constitute several percent of the field, let us consider the influence of this factor on the interpretation. As was pointed out, the vertical component of the field can be represented at every observation point as a sum of fields

11.2 Determination of the Gravitational Field

165

caused by different masses within the body, and their contributions essentially depend on the location and distance of these masses from the observation point. In particular, masses located closer give a larger contribution, while remote parts of the body produce smaller effects. It is obvious that there are always such masses within the body that their contribution to the total field is so small that with the given accuracy of measuring it cannot be detected. For instance, we can imagine such changes of a shape, dimensions, location of the body, as well as the density of some of its parts, that the measured field would remain the same. In other words, due to errors in determination of the secondary field there can be an unlimited number of different distributions of masses that generate practically the same field at observation points. Inasmuch as the secondary field is caused by all masses within the body -that is, an integrated effect is measured-some changes of masses in relatively remote parts of the body can be significant; but their contribution to the field would still remain small. At the same time similar changes in those parts of the body closer to observation points will result in much larger changes of the field. For this reason, in performing an interpretation it is appropriate to distinguish at least two groups of parameters describing the distribution of masses, namely; 1. Parameters that have a sufficiently strong effect on the field; that is, relatively small changes of their values produce a change of the field that can be detected. 2. Parameters that have a noticeable influence on the field only if their values are significantly changed. It simply means that they cannot be defined from a field measured with some error. Therefore, we can say that an interpretation or a solution of the inverse problem consists of determining the first group of parameters of the body even though usually they incompletely characterize the distribution of masses. It is clear that this "stable" group of parameters describes a model of the body that differs to some extent from the actual one, but both of them also have common parameters. For instance, these can be the depth to the top of the body or the product of its thickness and its density, etc. Before we continue let us make several comments, which allow us to summarize out discussion and outline some features of a solution of the inverse problem. They are 1. In general the inverse problem of the gravitational field is not unique. 2. At the same time, there are types of bodies for which a solution of the inverse problem is unique. In particular, it is true for a body that can

166

II

The Gravitational Field

be presented as a prism or a system of prisms, and this fact is one of the theoretical foundations of interpretation of gravitational data. 3. In considering the problem of uniqueness it is assumed that the field is known exactly. 4. Interpretation is performed within a class of models for which the problem is unique. 5. The most important factor, which in essence defines all features of the interpretation, is the fact that the measured field caused by some distribution of masses is known with some error. Because of this, uniqueness itself does not guarantee that the error in determination of some parameter has a finite value. In fact, this error can be infinitely large. Such inverse problems are called ill-posed or unstable ones. 6. In general, inverse problems in gravimetry, as well as in other geophysical methods, are ill-posed. To illustrate this fact, let us write the following relation between the change of the field Jl g and that of a body parameter JlPi'

Here k, is the coefficient of proportionality for the ith parameter of the body, and Jlg is the change of the field within a certain range caused by the presence of an error, which of course varies from point to point. In the case of an ill-posed problem an upper limit of coefficient value k i , in principle, cannot be established. In other words, even unlimited change of some parameters of the body cannot produce a noticeable variation of the field, and as a result of this, it is impossible to define a range within which these parameters vary. Such ambiguity in determination of parameters of a body is an obstacle that can be to some extent compared with non uniqueness. To overcome this problem the interpretation is usually performed within a much narrower class of models of the body, where the parameter k , has a finite upper limit; that is, the inverse problem becomes stable or well posed. For instance, if an approximation of the real distribution of masses is performed with the help of different prisms having the same number of sides, the inverse problem turns out to be well posed. 7. The transition from an ill-posed problem to a well-posed one is called the regularization of the inverse problem, and it is of great practical importance. 8. With an increase in the number of model parameters the approximation of a real distribution of masses can in principle be better. However, the error with which some of these parameters are determined also increases.

II.2 Determination of the Gravitational Field

167

9. It is obvious that the interpretation of gravitational data is useful if the parameters of a model, approximating a real distribution of masses, are defined within a range of values, that is sufficient from a practical point of view. Usually a choice of such a group of parameters is automatically selected making use of the corresponding algorithm of the solution of the inverse problem. 10. The interpretation of gravitational data is greatly facilitated by the presence of additional information about sources of the field, usually derived from geology and other geophysical methods. Now taking into account the obvious fact that a decrease of the error in field determination is of great importance, let us consider factors which, along with the error of measurement and interpolation between observation points, define the accuracy of separation of the useful signal from the secondary field. One factor is related to the change of a position of an observation point with respect to the masses that create the normal field. If, for example, measurements are performed at different distances from the earth's surface, the normal field varies, in accordance with Eq. (11.29). To separate the secondary field, which is usually much smaller than the normal field, its change has to be taken into account, and such a procedure is called the elevation correction. The second factor arises due to the presence of surface structures, since masses within them generate a field that can also vary from point to point. For these reasons, it is necessary to determine this effect and then remove it from the measured field. This procedure is called the terrain correction; for very gentle topography it reduces to the Bouguer's correction. A third factor also affects the anomalous field, and it is the field caused by lateral changes of density other than those that are sources of the useful signal. This part of the field is usually called the geological noise, and it is mainly caused by sources located relatively close to the earth's surface. It is not simple to remove their influence. One of the methods allowing us to reduce the geological noise will be described in the next section. Certainly, the second factor can also be treated as geological noise. It is clear that the accuracy with which all these factors can be taken into account, along with that of measurement, defines the accuracy of determination of the useful signal that in turn affects the quality of interpretation. In connection with the first two factors, notice that procedures for corrections do not involve a change of positions of observation points. Now we will demonstrate the application of Newton's law in calculating these corrections and the useful signal.

168

II The Gravitational Field

Elevation Correction

As is seen from Fig. I1.6b, points "1" and "2" are located at different elevations with respect to the earth's surface, and correspondingly the normal field is different at these points. It is clear that at point "2" this field is smaller, and to take into account this change it is sufficient to assume that the earth is a sphere. Then its gravitational field at both points is the same as that due to a point source. Making use of Eq. (11.29) we obtain the difference of vertical components of the fields due to an elevation change.

where Rand h are the earth's radius and the elevation, respectively. Inasmuch as in reality h/R « 1, we have

R2

1

h 1+ ( R

)2

=

h)-2 2h (1+ R ::::: 1 - R

and therefore (II.34) Having assumed that the earth's radius R and the normal gravitational field gz are R

=

6378 km

and

s, = 980 Gal

we obtain ~g,

h:::::0.3086~

mGal

(11.35)

If, for example, point "2" has higher elevation than point "I" we add to the measured field g/R + h), to compensate a decrease of the field due to the elevation. Often, the anomalous field has a magnitude around 1 mGal or smaller. In such a case the elevation has to be known within a few centimeters. Let us notice that the standard correction formula is derived using the spheroid model.

11.2 Determination of the Gravitational Field

169

The Bouguer Slab Correction

Again we will compare the fields at points "1" and "2," assuming that the latter is located above some layer of rocks that is absent around point "1" (Fig. II.6c). It is clear that its presence leads to an increase of the field at point "2." Unlike the previous case, where the difference of normal fields was considered, here we pay attention to the change of the secondary field. If we assume that the influence of finite horizontal dimensions of the layer on the field at point "2" is negligible, we can use the results from Section 11.1 (Example 3), and in accordance with Eq, m.23) the field caused by the layer with thickness Sh is b.g z = 21TyO b.h

(II.36)

For instance, letting S = 2600 kqyrrr', we have b.g z b.h

=:: 0.11

mGaIjm

Equation 01.36) describes the Bouguer slab correction and, as well as Eq. (I1.35), is commonly used in the practice of gravimetry. Of course, if the layer model is not able to describe adequately the effect caused by the topography, more complicated formulas are used. Next, we will derive formulas based on Newton's law.

g(p) =

1v b.o(L3q) dV L

-y

qp

(II.37)

qp

To calculate the vertical component of the field caused by some distribution of masses beneath the earth's surface, we will start with a two-dimensional case. Before we discuss the reduction of geological noise, let us derive some equations allowing us to simplify the calculation of the useful signal. Two-Dimensional Model

Suppose that a body is strongly elongated in some direction and, correspondingly, that it can be treated as two-dimensional. In other words, an increase of the dimension of the body in this direction does not practically change the field at observation points. Thus, we will consider a two-dimensional body with an arbitrary cross section and introduce a Cartesian system of coordinates x, y, z, as is shown in Fig. II.6d, so that the body is elongated along the y-axis, It is clear that at any plane y = constant the

170

II

The Gravitational Field

behavior of the field is the same. To carry out calculations we will perform two procedures, namely, 1. The cross section of the body is mentally divided into many elementary areas, and correspondingly we can treat the model as a system of many elementary prisms. Dimensions of every elementary cross section are much smaller than the distance between an observation point and any point in this area. 2. An elementary prism is replaced by an infinitely thin line directed along axis y, and it bears the same mass per unit length as that of the prism.

These two steps have allowed us to replace the two-dimensional body by a system of infinitely thin lines, which are parallel to each other, and the distribution of mass on them is defined from the equality dm

=

il8(q) dS(q) dy =A(q) dy

since A(q)

=

il8(q) dS(q)

( 11.38)

where A(q) is the linear density of mass on the line passing through point q, and dS(q) is the area of the elementary cross section of the prism. Let us note that the density is a function of the coordinates x and z, but it does not depend on y. Derivation of the formula for the gravitational field caused by an infinitely thin line with the density A is very simple, and it is illustrated by Fig. II.7a. We will consider the field at the plane y = O. Due to the symmetry of the mass distribution, we can always find a pair of elementary masses A dy and - A dy, which do not create a field component g y directed along the y-axis, and respectively the field generated by all elements of the line has only component g" located in the plane y = O. Here r is the coordinate of the cylindrical system with origin at point 0*, and the line is directed along its axis. As is seen from Fig. II.7a, the component dg r generated by the elementary mass located at the distance dy is A dy dg; = -y R 2

or

•

r R

=

dy -yAr R3

Il.2 Detennination of the Gravitational Field

171

Fig.II.7 (a) Field caused by a linear mass; (b) model of a surface mass; (c) cross section of a 3-D body; and (d) the solid angle subtended by an elementary layer.

We will assume for a moment that the line length is 2 t'. Then, integrating fields caused by all elements, we obtain

dy

+t

s, =

-')'Ar

Introducing the new variable

f- t

(2

Y

+r

2)3/2

ip,

y

= r

tan

ip

we have dy

= r

sec 2

ip dip

and ')'A gr = - r

f'P

O

-'Po

cos

2')'A

ip dip

= - - - sin r

ipo

172

II

The Gravitational Field

where (

tan 'Po

= -

r

Since tan 'P sin e

=

(1 + tan? 'P)

1/2

we obtain 2y,1 g

(

=-r

r

((2+ r 2)1/2

(11.39)

Because it is assumed that ( » r, let us represent the latter in the form of a series.

(II 040)

The first term of this series describes the field caused by the infinitely long line, and in this case, 2y,1 g =--r r

(11041)

Comparison of the last two equations allows us to determine the error that occurs when we replace a line of finite length with an infinitely long one, and then apply this result to a real elongated body. For instance, if the length of the line is 10 times greater than the distance from an observation point, this error is less than one-half percent. As follows from Fig. 1I.6d for the vertical component of the field, we have

In particular, on the earth's surface where zp = 0,

(II 042) and

(II 043)

II.2 Determination of the Gravitational Field

173

Equation (H.42) describes the vertical component of the field caused by masses within an elementary prism. Correspondingly, for the field gz due to a two-dimensional body, we have

_ j

gz-2y

Ao(q) dS(q)

S

r

if

Zq

2

zp = 0

(II.44)

Thus, instead of a volume integral, the field is represented as a surface integral, which simplifies calculations. If the function Ao(q) is constant, we have

gzCp) = 2yAo

Zq d S

js - 2 r

(II.4S)

or

For those cases when the cross section of the body has a relatively simple shape, the integrals on the right-hand side of Eq. (lIAS) are usually expressed as elementary functions-for instance, the cylinder, the thin rod, the sheet, the fault, the thick prism, etc. However, in more complicated cases, determination of the field is performed by numerical integration of Eq, (11.45). To further simplify calculations, suppose that the two-dimensional body is oriented along the y-axis, and its thickness is much smaller than the distance between the body and observation points; that is, h(x) «rq p

where hex) is the body thickness. Both hex) and the density 0 can be a function of the coordinates x and z. Then it is obvious that if the body is replaced by a two-dimensional strip (Fig. II.7b) bearing the same mass and placed somewhere inside the body the field would not change at the observation point. Now Eq. (II.44) is greatly simplified and we have

gz(p)

= 2YZq

j

X 2Q

x!q

where m = x p - x q ,

Zq


2+

dxq

2 Zq

(11.46)

is the strip coordinate along the z-axis, !T(x q ) = A8(x q ) h( x q )

is the surface density of masses within the strip, and x Iq and x 2q are coordinates of terminal points of the strip. Note that such a model is

174

II

The Gravitational Field

useful in calculating, for example, the gravitational field caused by twodimensional structures (anticlines, depressions) on the surface of the basement, wherein their amplitudes are small with respect to the sediment thickness. Now we will make one more simplification and assume that the surface density a is constant. Then we have

Applying again the substitution m

= Zq

tan 'P

the field g z due to masses of the strip is expressed as (II .47)

where if

xlq
In the limiting case when the strip becomes a plane, we again obtain the formulas for the Bouguer slab correction. In fact, in this case 'PI = 7T" /2, 'Pz = -7T" /2, and therefore

Suppose that the observation point is located in the plane x p = 0 and x 1q = -X Zq =X. Then, we can rewrite Eq. (11.47) in the form

Ixi

gz(x) =4yhdotan- l -

Zq

Assuming that Ixl» Zq we can expand tan-1lxl!zq in a series. [x]

tan - I -

7T" ::::: -

2

Zq

Zq -

Ixl

1

z~

+- -3 3 x

and then we obtain

z~ ... } g (p) ::::: 2Y7T"h do 1 - -Zq + - 2 -3 z

(

Ixl

37T"

Ixl

It is clear that this equation allows us to evaluate the difference between

11.2 Determination of the Gravitational Field

175

the fields caused by a plane and a strip with a finite width. Of course, this evaluation can be done directly proceeding from Eq, (lIA7). Now we will show that, making use of this equation, we can calculate the gravitational field caused by masses in a two-dimensional body with an arbitrary cross section. With this purpose in mind, let us mentally divide the body cross section into a sufficient number of relatively thin layers with thickness h, (Fig. H.6d). Then, applying the principle of superposition and Eq. (1.47) for every elementary layer, we have N

gz( p)

=

2y

L h, .6.0;( 'PI; -

(liAS)

'P2;)

;=1

Here gz(p) is the anomalous field caused by all masses of the body, hi = Z i + 1 - z;, Z i + 1 and z, are vertical coordinates of the bottom and the top of the i-layer, N is the number of elementary layers, and 'Pli = tan

_IXp-X li

ZOi

where ZO;

=

2

are horizontal coordinates of terminal points of i-layer, and .6.0; characterizes the distribution of masses in an elementary layer. In particular, if .6.0; is constant within the body, we have

X2;' Xli

N

gz(p) =2y.6.0

L

(Zi+I- Zl)('P2;-'Pli)

(HA9)

i=1

Note that with an increase in the number of elementary layers, the accuracy of field determination increases too. Three-Dimensional Body

Suppose that the gravitational field is caused by masses in a three-dimensional body (Fig. 1I.7c). In principle, the field can be calculated from Newton's law, Eq. (I1.5); but even with fast computers numerical integration over the volume for many observation points requires a lot of time, and it is natural to apply methods that allow us to simplify this integration. Two of them are described in this section. By analogy with the previous case let us represent the three-dimensional body as a system of elementary layers located in horizontal planes whose thickness is much smaller than

176

II

The Gravitational Field

the distances from them to observation points (Fig. 1I.7d). In such a case, every layer can be replaced by a horizontal surface of finite dimensions with the density u(q) = Llo(q)h(q)

Correspondingly, the vertical component of the field caused by all masses of the body can be presented as a sum of fields giz due to elementary surface masses.

(11.50) As was shown in the first section, the field generated by surface masses is expressed through the solid angle wi(p) under which the surface is seen from the observation point p. Assuming that the density difference, Llui(q), is constant at the ith surface and making use of Eq. (11.22), we have

(11.51 ) here

Thus, for the field g z we obtain N

gAp)

L

='}'

LloihiWi(P)

i=l

or N

gAp)

='}'

L

LlO;(Zi+l- Z;)Wi ( p )

(11.52)

i=1

where hi = Z i + 1 - z, is the thickness of the elementary layer. In particular, if the function Lloi is constant within the body we finally have N

gzCp) ='}'Llo

L

(Zi+l- Z ;) Wi(

P)

(11.53)

i~1

Correspondingly, determination of the vertical component of the field due to masses in a three-dimensional body consists of calculating a set of solid angles; this procedure is described in Chapter I.

IL2 Determination of the Gravitational Field

177

The other approach allowing us to simplify the field calculation is based on the use of Eq, 0.89). q

1v gradTdV=~TdS

s

where S is the surface surrounding the volume of the body, dS = dSn, and n is the unit vector normal to the surface element and directed outward. T is a continuous function in the volume v. Assuming that the function ~8 is constant and taking into account Eq. (1.80),

we can rewrite Eq. (II.37) as

g(p)

=

Lqp -3-

1-c: dV

-'Y~'Y

1

q =

-'Y~81

v

(II.54)

grad-dV L qp

where the index "q" means that derivatives are taken with respect to the point q. Now making use of Eq. 0.89) we obtain

g(p)

=

-'Yl18~-

dS

v;

Respectively, for the vertical component of the field we have

where k is the unit vector directed along the z-axis. Inasmuch as dS . k = dS cos f3 where f3 is the angle between nand k depending on point q of the surface, we have

gz(p)

=

-'Y~8~--

dS(q)

s L qp

cosf3(q)

(II.55)

17S

II

The Gravitational Field

Thus, instead of the volume integral Eq, (II.5), we have derived an expression for the field that requires an integration only over the surface. The formulas described in these two sections allow us in many cases to simplify the solution of the forward problem in calculating the useful signal. They are also used to take into account the topography effect, the correction for the change of elevation of observation points, and to introduce the Bouguer slab correction. At the same time, as we stated earlier, there is one more factor that strongly affects the quality of interpretation. Geological noise is mainly caused by the lateral change of rock density near the earth's surface. Of course, separation of the geological noise and the useful signal cannot be done without some error, and very often the latter ultimately defines the degree of ambiguity of interpretation. If we had some reasonable information about the distribution of masses, which characterizes the geological noise, then the use of Newton's law would be the most natural way to evaluate its contribution. However, such information is usually absent, and correspondingly it is impractical to perform this separation by solving forward problems. Note that sources of the useful signal are located, as a rule, deeper than sources of the geological noise, and this fact results in a difference in geometries of these two parts of the anomalous field. For this reason, the reduction of the geological noise is based on study of geometry of the field caused by sources located at different distances from observation points; in the next section we will describe the theoretical basis of one such approach. To accomplish this task it is necessary to derive a system of equations of the gravitational field, introduce its potential, and make use of Green's formula (Chapter 1). 11.3 System of Equations of the Gravitational Field and Upward Continuation

As is demonstrated in Chapter I, field equations show the relationship between a field and its generators. In the case of the gravitational field there is only one type of generator, namely sources (massesj-i-the vortex type of generator is absent. Proceeding from this concept, we will derive the system of equations for the field. First, we shall consider an elementary mass located at point q and calculate the flux of this field through an elementary surface at point p, as is shown in Fig. II.Sa. Applying Newton's law we have dm L q p ' dS g • dS = - y 3 = - y dm dw (II.56) L qp

11.3 Systems of Equations of the Gravitational Field Upward Continuation

179

Fig. 11.8 (a) Flux through an elementary surface; (b) flux of surface masses; (c) evaluation of mass generating secondary field; and (d) voltage due to an elementary mass.

where dto is the solid angle under which the surface dS is seen from point q. It is obvious that the flux through an arbitrary surface S presents a sum of elementary fluxes, and therefore

1s

g . dS

= -

y dm w

where w is the solid angle under which surface S is seen from the point q. In particular, the flux through an arbitrary closed surface surrounding the elementary mass dm is

~ g : dS = -4'lT')' dm s

(11.57)

since the solid angle under which the closed surface is seen from the point

180

II

The Gravitational Field

q is always equal to 47T, regardless of the surface shape and the position of the point q in the volume surrounded by the surface S.

Now making use of the principle of superposition and assuming that inside the volume V there is an arbitrary distribution of masses, we obtain

~ g : dS = -47T'}'m s

(11.58)

where m is the total mass in the volume V. Equation (11.58) is called the second equation of the gravitational field in the integral form, and in this regard let us make two comments. 1. The theory of fields described in Chapter I shows that the flux of any field through a closed surface characterizes the quantity of sources in the volume surrounded by the surface S. Therefore, it is natural that the mass m is present in the right-hand side of this equation. At the same time the coefficient - 47T'}' follows directly from Newton's law, and its value is defined by the system of units. 2. Masses located outside the volume have an influence on the field everywhere, including points of the surface S surrounding this volume. At the same time the field caused by these masses does not contribute to the flux: through this surface. This fact is proved in Chapter I for any field regardless of its nature; but it also follows from Eq. (11.56), since the solid angle under which a closed surface is seen from a point located outside it is always equal to zero. This is a remarkable fact that is difficult to predict if we do not know that the flux of the field through any closed surface is defined only by masses within the volume surrounded by this surface.

Assuming a volume distribution of masses characterized by density 8, we will present Eq. (11.58) as

(11.59) Now we are ready to derive the differential form of this equation. In accordance with Gauss' theorem, Eq. 0.138), we have

f

~ g . dS = div g dV = s v

- 47T'}'

fv8 dV

(11.60)

Inasmuch as this equality holds for any volume, the integrands are also equal. div g =

-47T'}'

8

(11.61)

11.3 Systems of Equations of the Gravitational Field Upward Continuation

181

This is the differential form of the second equation of the gravitational field, which is valid for regular points, where the first derivatives of the field g exist. In particular, outside of masses this equation is essentially simplified, and we have div g = 0

(II.62)

In reality there are always only volume distributions of masses with finite values of 8(a). However, as was demonstrated in the first section, for certain conditions it is useful to introduce surface masses with density ai.q). In such cases, making use of Eq, (I.135), it is easy to derive a surface analogy of the second equation. In fact, we shall assume that there is a surface distribution of masses shown in Fig. Il.Sb, and imagine a cylindrical surface around point q. Then, applying Eq, (II.58) we have gz·dSz+gj·dS+

f g·dS= -47TyudS Sf

where

as,=dSn,

dS I = -dSn

Sf is the lateral surface of the cylinder, n is the unit vector directed from the back to the front side of the surface, and tr dS is the elementary mass inside the cylinder. In the limit, when the cylinder height tends to zero, we obtain

(II.63) where gZn and gin are normal components of the field at either side of the surface. Therefore, the difference of the normal components of the field near the surface mass is defined by the surface density at the same point. Equation (II.63) represents the surface analogy of Eq. (Il.Sl) in the vicinity of points where singularity in the field behavior is observed. Thus, we have derived three forms of the second equation of the gravitational field. ¢g. dS = -47Tym

div g =

-

47Ty8

(II.64)

Before we derive the next equation of the field, let us illustrate one rpplication of Eq, (II.58) in interpretation of gravitational data. Suppose

182

II

The Gravitational Field

that measurements of the field are performed over some areas and corresponding corrections are introduced. Also, the useful signal gz practically vanishes at the boundaries of this area (Fig. II.8c). Then, the half space is a volume where all sources of this field are located. This volume is surrounded by the area of measurement and a half-spherical surface So with relatively large radius where the field can be considered to be that of the point source. Correspondingly the flux through this surface is

f

g : dS

-21T"ym

=

So

For this reason, the flux through a closed surface surrounding this volume is expressed in terms of a surface integral over the observation area only. Therefore, we obtain

(11.65) since

s : dS =

-gz dS

Thus, we have found the total mass causing the useful signal provided that geological noise is absent, and it is equal to (II .66) Now we shall derive the first equation of the gravitational field making use of two approaches. The first one is based on results described in Chapter I where it is shown that the circulation of any field characterizes the amount of vortex generators. Since the gravitational field is caused by sources (masses) only, we can instantly write all three forms of the first equation of the field. curl g = 0

(11.67)

where n is the normal to the surface and q2 and ql are fields from the front and back sides of the surface, respectively. The second approach is based on Newton's law, and we will describe it in detail. Suppose that there is an elementary mass at point q, and

11.3 Systems of Equations of the Gravitational Field Upward Continuation

183

consider the voltage

between two points along the path bb' and b'a, where bb' is an arc and b' a is a displacement along the radius (Fig. 1I.8d). In the case of the elementary mass it can be presented as

r g' d/ I,rb g . d/ + l,r g' d/

l,

=

I

b

b

=

-

ra

y dm l.

b,

b,

dt

-2L qp

(11.68)

r dL = -ydm ,

Jb -L2q p-

since along the arc bb I the field g and displacement d/ are perpendicular to each other and, correspondingly, dot product g . d/ is zero, while along the path b I a displacements dl' and dL coincide. Performing the integration in Eq. (II.68) and taking into account that L q b l = L q b we obtain

1 g'd/=ydm [1-L a

b

qb

1

- -1 L qa

(11.69)

Now we shall represent an arbitrary path between points b and a as a system of elementary arcs and small displacements in a radial direction (Fig. II.9a). Then, taking into account the fact that integration along arcs does not give a contribution to the voltage, we obtain

1)

+ ... __ i;

=

y dm

{_I __1} i-; i.;

(11.70)

where L q b I and L q b 1+1 are distances from an elementary mass to terminal points of corresponding displacements along a radial direction. As follows from Eqs. (11.69), (I1.70) the voltage does not change when the path of integration varies, but it depends on the position of the terminal points

184

II

The Gravitational Field

Fig. 11.9 (a) Voltage along an arbitrary path; (b) circulation of the gravitational field; and (c) continuity of tangential components.

(a, b). In other words, the voltage of the gravitational field is path

independent. This well-known result directly follows from Newton's law and reflects the fact that only masses generate the gravitational field. I think it is natural to be impressed by this amazing feature of the field. Indeed, suppose there are two points at a distance 1 m apart. Calculating the voltage between two points along a straight line with length 1 m, we obtain its value. Then, let us choose a completely different path between the same points, which has a length of thousands of kilometers and goes through mountains, valleys, oceans. Of course, the field g varies in magnitude and direction from point to point of this path. But what is really remarkable is the fact that in both cases the voltage remains the same. Only one step is left to derive the first equation of (II.67). We will consider two arbitrary paths 2 1 and 2 2 between points a and b (Fig. 1I.9b), and due to independence of the voltage of the path, we have

1 g·d.!= 1 g·d.! .2"1

.2"2

or

f

acb

god.!=

f

adb

god.!

11.3 Systems of Equations of the Gravitational Field Upward Continuation

185

Inasmuch as a change of the direction dl" to the opposite one results in a change of the sign of the voltage, we can write

f

g'dl'=-L g'dl'

acb

bda

or

f

g : dl' +

acb

1 s : dl'

=

0

bda

Finally we have (11.71)

where 2' is an arbitrary closed path. Thus, we have proved that the voltage along a closed path (circulation) is always zero for the gravitational field, and Eq, (11.71) is called the first equation in the integral form. Of course, this result is valid for any closed path that can, in particular, pass through media with different densities. Let us emphasize again that Eq, (11.71) does not require a proof as soon as it is known that the gravitational field is caused by sources (masses) only. Now applying Stokes' theorem at the vicinity of regular points of the field, we have

¢C g . dl' = 'Sf'

I. curl g . dS = 0 S

or curl g = 0

(11.72)

where S is an arbitrary surface bounded by the contour 2', and the directions dl" and dS are related to each other by the right-hand rule. Equation (II. 72) represents the first equation of the gravitational field in differential form, which is valid at points inside and outside masses where the first derivatives of the field exist. If we suppose that there are also surface masses, then it is necessary to derive a surface analogy of Eq. (11.72). This is related to the fact that this equation cannot be applied near masses where the normal component of the field, gn' is a discontinuous function, Eq. (11.63). To derive this analogy let us calculate the voltage along the path shown in Fig. II.9c. Making use of Eq, (I1.71) we obtain (11.73)

186

II

The Gravitational Field

since the displacements de' are small, and the integrals can be replaced by dot products of the field and the displacement while the voltage along path dh, perpendicular to the surface, vanishes when dh tends to zero. Taking into account that

dt; = -dl; we obtain or (11.74) where g 11 and g 21 are the tangential components of the field. Equation (11.74) is the surface analogy of the first equation of the gravitational field, and it shows that the tangential component of the field g is a continuous function. Thus, we have derived three forms of the first equation. curl g = 0 and all of them contain the same basic information, namely, that gravitational field is caused by masses. Now we are prepared to present the system of equations of gravitational field, derived from Newton's law, as well as of course, principle of superposition; both integral and differential forms of system are shown below.

the the the this

Integral Form of Equations for Gravitational Field

Newton's law

g (p)

=

( 8(q)L q p dV -y JT 3 V

L qp

/ f g' d/ ~ 0 I

1_ 11

--'

Here y is the gravitational constant and m is the mass located inside the volume surrounded by the surface S.

187

11.3 Systems of Equations of the Gravitational Field Upward Continuation

Arrows show that both equations, which are valid everywhere, are derived from Newton's law. Differential Form of Equations for Gravitational Field

Newton's law g(p)

B(q) = -y 1-3-Lqp

v L qp

dV

(11.75)

I

div g =

curl g = 0

gZn - gln =

on S

-

47TyB

Div s = '-47TyB

I

on S

Here (J" is the surface density of mass, and the last pair of equations describe the field behavior in the vicinity of points where surface masses are present. Inasmuch as the field equations are derived from Newton's law and the principle of superposition they do not contain more information about the field behavior than these laws themselves. However, they allow us to understand better some of the features of the field and, in particular, to develop a method for reduction of geological noise. With this purpose in mind, let us first of all introduce a new scalar function called the potential of the gravitational field. It can be done making use of the first equation in differential form. As is shown in Chapter I, the solution of the equation curl g = 0 is g = grad U

(11.76)

and this result is verified by direct substitution of Eq. (11.76) into Eq. (lI.72). Thus, we have expressed the vector field g through a scalar function, U(p), with the help of a relatively simple operator, Eq. (1.64).

1 au g= - - i l h , aXl

1 au

1 au

h z ax z

h 3 aX3

+ - - i z + - - i3

(11.77)

where hl' hz, and h 3 are metric coefficients; xi' x 2, x 3 are coordinates of

188

II

The Gravitational Field

the observation point; and iI' i 2, and i 3 are unit vectors of the coordinate system. It is clear that Eq, (II.76) defines the potential U up to a constant; that is, an infinite number of potentials describe the same field g. For this reason it is natural to treat the potential as an auxiliary function introduced with only one purpose, namely to simplify the analysis of the more complicated vector field g. Inasmuch as the field g is expressed through the potential U, it is proper to derive an equation describing its behavior. We already used the first equation, curl g = 0, to introduce the potential; now, substituting Eq. (11.76) into the second field equation, Eq. (11.61), we obtain div grad U = -4'lTrc5 or

(II.78)

Thus, we have obtained Poisson's equation, which in accordance with Eq, (1.236), can be represented in an orthogonal system as

At the same time, outside masses Eq. (II.78) is simplified, and we obtain Laplace's equation. (II.79) Both relations (11.78) and (II.79) describe the behavior of the potential at regular points where the first and second equations of the field are valid. To characterize the behavior of the potential near surface masses, let us make use of Eq. (11.76) according to which any component of the field along some direction t is equal to the derivative of the potential in this direction; that is, (II.80) where t is a line along any direction. Thus, instead of surface analogies of field equations (11.74) and (11.63) we obtain

aU2

eu,

at

at

---=0

(II.81 )

189

11.3 Systems of Equations of the Gravitational Field Upward Continuation

and

auz auz

-

an

-

-

an

=

-47TyCT

(II .82)

where U, and Uz are values of the potential at the back and front sides of the surface, respectively. It is obvious that continuity of tangential derivatives of the potential follows from continuity of the potential itself, and correspondingly Eq. (II.8l) can be replaced by (II.83) Thus, the behavior of the potential is described by the system of equations given below. Newton's law (II.84)

IVzU= -47TyO I

IVzU= 0 I

if 8 * 0

0=0

auz

-

on S

an

-

eo.

-

an

=

-47TyO"

on S Now we shall find the relation between the potential U and masses. First consider an elementary mass dm = 0 dV. In accordance with Newton's law we have g( p) = -

dm y-3-Lqp L qp

p

1

= Y dm gradL qp

(II.85)

since

where the index p means that the gradient is considered in the vicinity of the point p. Comparing Eqs. (II.76) and (II.85) we can conclude that the function U corresponding to the field caused by elementary mass 0 dV located at

190

II

The Gravitational Field

point g is m U(p)=8-+C i;

(II.86)

since if gradients of two functions are equal, then the functions themselves differ in general by a constant. Taking into account the fact that the field g caused by mass dm tends to zero at infinity, it is natural to assume that its potential also vanishes as L q p ~ 00. Then from Eq. (11.86) it follows that C = 0, and we obtain U(p)

m =

(II.8?)

y-;

Now applying the principle of superposition we derive an expression for the potential U caused by a volume distribution of masses.

U( p)

=

y

fv

8(q) dV

(II .88)

L qp

Comparison of Eqs. (11.5) and (11.88) clearly shows that the potential is related to masses in a much simpler way than g, and this fact is one of the reasons for its introduction. If, along with volume masses, all other types of masses are considered, we have

U(p)=y

[fv

8 ( q ) dV L qp

"m 1.A(q)dt] (II.89) + j(T(q)dS +,-,-+ j

S

L qp

j~ I L q p

Sf'

L qp

where m j is a point mass and A(q) is the linear density. Let me show one more application of the potential, and with this purpose in mind, we will consider the change of this function in the vicinity of some point dll. As is well known, this can be represented as (II.90) where dt p dtz' and dt3 are elementary displacements along the coordinate lines x p x z, and x 3 •

I1.3 Systems of Equations of the Gravitational Field Upward Continuation

191

It is easy to see that the right-hand side of this equation can be written as a dot product of two vectors, namely,

and

(II.91)

Here

Therefore,

dU = dl' . grad U = g . dl'

(II.92)

After integration of this equation along an arbitrary path with terminal points a and b we obtain

U(a) - U(b)

=

fa g : dl'

(II.93)

b

Thus, the voltage along some path is expressed by the difference of potentials at terminal points of this path. Certainly it is much simpler to take a difference of the scalar at two points, Ui a) - U(b), than to perform an integration, f:g . dl; and this fact demonstrates another advantage of using the potential. Let us return to Poisson's and Laplace's equations, which describe a behavior of the potential inside and outside masses. and

f1U = -47TyO

f1U=O

At the same time we have already derived an explicit expression for the potential U that allows us to find this function if masses are known [Eq. (II.88)]. This means that

U( p)

=

Y

fv

o(q)dV

L qp

is the solution of Poisson's equation inside of masses, and it satisfies Laplace's equation outside of them. The potential is also useful to establish relations between different components of the gravitational field. In accordance with Eq. (II.76) we

192

II

The Gravitational Field

have and and taking into account the fact that /

Lqp=V(xp-X q)

2

+ ( Yp-Y q) 2 +(Zp-Zq) 2

we obtain from Eq. (11.88)

s, =

y

v; - v,

jV 0 -L qp 3

dV,

and

(11.94)

Of course, these equations follow directly from Newton's law, too. Then, taking corresponding derivatives, we have

(11.95) The equations indicate that the information contained in one component of the field about its sources cannot be in principle increased by measuring other components. In fact, this result was established earlier in considering the field of an elementary mass. Now we will demonstrate another merit of the potential that is important for interpretation of the gravitational data. Suppose that the anomalous field measured on the earth's surface consists of two parts: One is caused by relatively deep structures and represents the useful signal, while the other is generated by lateral changes of rock density near this surface and characterizes the geological noise (Fig. II.10a). Let us assume for a moment that the field is measured at different distances above the earth's surface. Then it is obvious that with an increase of the distance both parts of the field begin to decrease. However, there is one important difference in their behavior; namely, the useful signal decreases more slowly because distances between deep structures and observation points alter by a relatively small amount, while the signal caused by the geological noise varies more rapidly because sources of this field are closer to the observation points. Consequently, the contribution of the geological noise to the anomalous field decreases with an increase of an elevation of the observation point. This tendency is observed until

11.3 Systems of Equations of the Gravitational Field Upward Continuation

193

Fig. n.IO (a) Useful signal and geological noise, and (b) analytical continuation upward.

dependence of both fields on the distance to their sources becomes practically the same. Taking into account this behavior of the useful signal and the geological noise, we will describe a method to calculate the field in the upper half space when its values on the earth's surface are known. It is clear that such a procedure will allow us to reduce the influence of the geological noise. First of all we know that in the upper half space, where masses are absent, the potential U satisfies Laplace's equation.

.lU=o Suppose that on the earth's surface the vertical component of the gravitational field, that is, the derivative au/az is known. Inasmuch as only the anomalous field is considered, at sufficiently large elevations the field becomes small and, correspondingly, we can assume that the potential U is equal to zero on some half-spherical surface with a relatively large radius R (Fig. 11.10b). Thus, we have a volume V, surrounded by the earth's surface, where the derivative au/az is known, and the hemispherical surface with radius R, where the potential is zero, and our task is to find the potential U as well as the field g at every point of this volume. This means that we have to solve Dirichlet's value-boundary problem (Chapter 0, which uniquely defines both the potential U and the field g. This can be written as the problem of field determination within the volume V if the following is known: 1. Above the earth

194

II

The Gravitational Field

2. On the earth's surface

where gz is a known function 3. V ~ 0, as r ~ 00 To derive formulas allowing us to calculate the field in the volume V, we will make use of the second Green's formula, Eq. (1.289).

(11.96) where both functions cp(q) and l/J(q) are continuous together with their first derivatives, and n is the unit vector directed outside the volume. In essence, Eq. (11.96) is Gauss' formula and consequently it establishes a relation between values of scalar function inside the volume V on one hand and those of functions and their derivatives on the surface S on the other hand. For this reason it is natural to apply Eq. (11.96) to find the potential U in the upper half space above the earth's surface. Following very closely the derivatives in Chapter I, we assume that function l/J(q) is the potential of the gravitational field Ut.q) and introduce the notations

l/J(q)=U(q)

and

cp(q)

=

G(q)

Then, Eq. (II.96) can be rewritten as

- G av) dS fvU V G dV = rr.'.fs (vaG an an 2

(11.97)

since V2 V = 0 in the upper half space. Our task is to derive an explicit expression for the potential U proceeding from Eq. (II.97); that is, we have to take the function V out of integrals in this equation. To realize this, let us choose a function G that satisfies the following conditions: 1. Everywhere inside the volume V it is a solution of Laplace's equa-

tion V 2G =0

except at the observation point p. 2. In approaching point p function G has a singularity l/L p q ; that is, 1

G(q,p) ~L' qp

195

11.3 Systems of Equations of the Gravitational Field Upward Continuation

3. With an increase of the radius R of the hemispherical surface (Fig. II. lflb), the function G decreases at least inversely proportional to distance L q p • In other words, the function G, which is often called Green's function, is harmonic except at the point p, where the potential is calculated. Inasmuch as Gt.q, p) has a singularity at the point p, we can apply Green's formula 0.97), provided that the point p is surrounded by a surface S * with very small radius r 1 (Fig. II. l Ob). Then, applying this equation to volume V surrounded by surfaces Sand S * and taking into account the fact that V 2G = 0, we obtain

au ) dS+,.{.. (aG au ) dS=O so ,.{. U--GU--Gan 'f:.,. an an

(11.98)

'f:., ( an

since the volume integral vanishes. We will represent the first term of this equation as

ec

au) an

,.{.. U - - G - ds=f

'f:., ( an

Su

(aG au) (aG au ) U - - G - <:! U - - G - dS an an SR an an (11.99)

where So is the earth's surface, the z-axis is directed downward, and SR is the hemispherical surface with radius R. Since with increasing radius R both functions U and G decrease as IjR, their first derivatives aujaR and aG jaR tend to zero as 1jR 2 • Correspondingly, the integrand of the second integral at the right-hand side of Eq. (11.99) decreases as IjR 3 , and making use of the mean value theorem we obtain

ec - Gau) c - dS --'> - f

f SR (uaR

aR

R

3

SR

dS

=

c

-21TR 3

2

--'>

R

a

as

R

--'> 00

where C is some constant. Therefore, Eq. (11.98) can be rewritten in the form

f s,

au } dS+,.{.. {aG au } dS=O eo U--G{U--Gaz az 'f:.,u an an

(II .100)

Now we will consider the integral over the spherical surface S * around the point p with radius r l' which in the limit tends to zero. Let us make

196

II

The Gravitational Field

two comments about the behavior of the integrand, namely, 1. The potential U(q) and its normal derivative on the spherical surface

au au -=--=-g an ar r have finite values since they describe a real field. 2. With a decrease of radius r function G(q, p) behaves as l/r, and its derivative aG/an = -aG/ar increases proportionally to l/r 2 • Then applying again the mean value theorem, we obtain

rf..

ea au eo au) U - - G - dS=-rf.. U-dS+J G-dS an an 'f'.s. ar s; ar

'f'.s. (

1 1 aU(p) = U(p) 2417rf + - --41Trf = 417U(p) r1

r1

Br

as

r 1 ~ 0 (II.lOl)

Thus, due to the fact that the chosen Green's function has a singularity l/L q p at the point p, we have been able to take function U(p) out from the integrand and obtain its expression in an explicit form. In fact, from Eqs. (11.100), (11.101) we have

417U(P)=J {U(q) So

aG(q,p)

az

au(q)} -G(q,p)-- dS

az

or

U(p)

=

1 -4 17

} J { G(q,p)gzCq) - U(q) aG(q,p) a dS z

(II.102)

~

since

au

s, = a; Therefore, we have derived a formula for calculation of the potential of the field everywhere in the upper half space if both the potential U and the vertical component of the gravitational field are known at the earth's surface SQ. Having taken the derivative from both sides of Eq. (11.102) we obtain for the vertical component of the gravitational field at the point p (II.103)

11.3 Systems of Equations of the Gravitational Field Upward Continuation

197

where zp and Zq indicate that derivatives of the Green's function are taken with respect to the coordinate Z near points p and q, respectively. We can imagine an infinite number of Green's functions satisfying the conditions formulated earlier. The simplest of these is

1 G=-

(11.104)

c;

Indeed, it satisfies Laplace's equation everywhere except at the point p, since it describes up to a constant the potential of a unit mass located at the point p. Also, it has a singularity at this point and it provides a zero value of the surface integral over the hemisphere when its radius R tends to infinity. Correspondingly, we can write

However, this equation is impractical since the potential U is not measured on the earth's surface, and therefore we have to choose a Green's function such that its derivative aGjaz on the earth's surface would be zero. In this case Eq. (11.102) is greatly simplified.

1

U(p)

=

4

1T

f G(q,p)gz(q) dS

(11.106)

So

where gz(q) is the measured vertical component of the field on the earth's surface, and Gi q, p) is an unknown function that satisfies the following conditions:

1. In the upper space, z < 0, Green's function G is a solution of the Laplace equation

everywhere except at the observation point p. 2. It has a singularity of type IjL q p , that is, in approaching point p,

G(q,p)

1 ~-;

198

II

The Gravitational Field

3. At the hemispherical surface it decreases at least as I/R with an increase of radius R; and finally, 4. At the earth's surface the derivative aG(q, p)/az vanishes. aG

-=0

an

As was demonstrated in Chapter I, determination of the function G satisfying all these conditions presents in essence a solution of the boundary value problem; and in accordance with the theorem of uniqueness, these conditions uniquely define the function G(q, p). In general, a solution of this problem is a complicated task, but there is one practical important case of the plane surface So when it is very simple to find the Green's function. Let us introduce the point s, which is a mirror reflection of the point p with respect to the plane of the earth's surface (Fig. 1I.10b) and consider the function GI(p, s, q) equal to

(11.107) where q is a point at the earth's surface, and

L qs

= {(

x q - x s)

2

2

+ ( Yq - Ys) + ( Z q - Z s)

2} 1/2

Taking the derivative aGIIaz, we obtain aG I aZ q

=

_ Zq-Zp

_

L~p

Inasmuch as at every point at the earth's surface L qp

=

Zq =

0 and zp = -zs' but

L qs

the derivative aGI/az q is equal to zero. It is obvious that the other three conditions are also met. Therefore, in accordance with Eq. (11.106) we have

References

199

The latter allows us to calculate the vertical component of the field in the upper half space when it is known at the earth's surface. Correspondingly, this transformation is called upward continuation and is used to reduce the influence of geological noise.

References Garland, G.D. (1979). "Introduction to Geophysics." W.B. Saunders, Philadelphia. Grant, F.S., and West, G.P. (1965). "Interpretation Theory in Applied Geophysics." McGraw-Hill, New York. Green, R. (1986). The use of the subtended solid angle for calculating the magnetic anomaly over structurally complex bodies. Geoexploration 2461-69. Parasnis, D.S. (1979). Principles of Applied Geophysics 3e. Chapman and Hall.

Chapter III

Electric Fields

111.1 Coulomb's Law Normal Component of the Electric Field Caused by a Planar Charge Distribution Effect of a Conductor Placed in a Free Space and Situated within an Electric Field

111.2 System of Equations for the Time-Invariant Electric Field Potential Integral Form of Equations for Time-Invariant Electric Field Differential Form of Equations for Time-Invariant Electric Field A Conductor Situated in Free Space

III.3 IlIA III.5 III.6 111.7 111.8 III.9 III.10

The Electric Field in the Presence of Dielectrics Electric Current, Conductivity, and Ohm's Law Electric Charges in a Conducting Medium Resistance The Extraneous Field and Its Electromotive Force The Work of Coulomb and Extraneous Forces, Joule's Law Determination of the Electric Field in a Conducting Medium Behavior of the Electric Field in a Conducting Medium References

In this chapter we will develop the theory of electric fields, used in different electrical methods. This theory is based on Coulomb's law, Ohm's law, and the principle of charge conservation. Let us begin with Coulomb's law.

III.I Coulomb's Law

Experimental investigations carried out by Coulomb in the 19th century showed that the force acting on an elementary charge situated at the point p due to the presence of an elementary charge situated at the point q is

m.l

Coulomb's Law

201

described by an extremely simple expression.

F(p)

de(q)de(p)

I = --

3

Lqp

47TEO

L qp

(III.I)

where L q p is the vector

Here L q p is the distance between points q and p; L~p is a unit vector directed along the line connecting points q and p; and EO is a constant known as the dielectric permeability or electrical permittivity of free space. In the International System of Units (SI) this constant is 1'0=

I --10- 9 Fyrn 367T

By definition elementary charges occupy volumes much smaller than the distance L q p between them, and

de(q)

=

8(q) dV,

de ( p)

=

8( p) dV

where 8 is the volume density of charges. Equation (III.1) can also be written as

F(p)

I

de(q) deep)

= --

2

L qp

47TE O

0

Lqp

(111.2)

and it is obvious that

F(p)

=

-F(q)

The electric force of interaction between two elementary charges is directly proportional to the product of the charge strengths and inversely proportional to the square of the distance between them. Unlike gravitational mass, electric charges can be both positive and negative, and for this reason the electric force F( p) has the same direction as the unit vector L~p when charges have the same sign, and it has the opposite direction when the product of charges is negative (Fig. III.1a). This simple expression is valid of course only as long as the distance between charges is far

202

III

Electric Fields

a

b p

q

c

q

d

Fig. 111.1 (a) Interaction between elementary charges; (b) electric field of volume charges; (c) normal component of charges on a plane surface; and (d) normal component of charges on an arbitrary surface.

greater than the dimensions of the volume within which the charges are situated. To define the electric force of interaction between charges when one of them is contained in a volume having a dimension comparable to the distance between charges, we must make use of the principle of superposition, as was done in the case of the gravitational field. According to this principle the force of interaction between two charges is independent of the presence of other charges. Using this principle, the force between an elementary charge at point p, dee p), and an arbitrary volume distribution can be written as (Fig. HUb) _ de(p) (o(q) dV

F(p) -

4

17£0

if

V

L3

qp

Lqp

(III.3)

Extending this approach to a more general case in which all types of charges are present (volume, surface, linear, and point charges) and again applying the principle of superposition, we obtain the following expression for the electric force of interaction between an elementary charge deep)

III.I Coulomb's Law

203

and a completely arbitrary distribution of charges. _ deC p) [j8(q) dV j I(q) dS F(p)- 47TE L3 L qp + S L3 L qp o v qp qp

+ j,

L

A( q ) dt . "ei(q) 3 L qp + z: -3- Lqp L qp L qp

1

(IlIA)

where 8 dV, I dS, A dt are elementary volume, surface, and linear charges with densities 8, I, and A, respectively, and e, is a point charge, that is, an elementary charge regarded as if it is located at some point. Unlike volume charges, the others are mathematical models of real distributions of charges that in many cases drastically simplify calculation and analysis. At this point we will define the strength of the electric field E( p) as being the ratio between the force of electrical interaction F and the size of the elementary charge (considered to be a test charge) at the point p,

(IlLS) For convenience the strength of the electric field is usually referred to merely by the term "electric field." It does not have the same dimensions as the force, but in the SI system of units it is measured in volts per meter. The electric field E can be thought of as the electric force acting on a test charge de, inserted into a region of interest, and normalized by this charge. Under this action a positive charge moves in the direction of this field and a negative charge moves in the opposite direction. Of course, if the electric field is known, it is a simple matter using Eqs. (IlLS) to calculate the force of interaction F. As follows from Eq. (IlI.1), charges having opposite signs attract each other while charges with the same sign, unlike gravitational masses, repel. In accordance with Eq, (IliA) the expression for the electric field can be written as 1 [8dV IdS E(p)=--j-3- Lqp+!.-3- Lqp 4'17"EO v L qp S L qp

(III .6) If the distribution of charges is given, the function E depends only on coordinates of an observation point p. Because of this the function E, in

204

III

Electric Fields

the same manner as the acceleration g, is termed a "field." Here it is appropriate to make the following comments: 1. Electric charges are the sole sources of an electric field that does not vary with time. 2. Coulomb's law describes the dependence of this field on charges. It is essential to note that the electric field caused by a given distribution of charges is independent of the physical properties of the medium. In other words, the electric field due to the same distribution of charges remains the same whether it is considered in free space or in a nonuniform medium. This follows from the fact that neither the dielectric constant nor the conductivity of a medium are present in Eq, 011.6). 3. Coulomb's law, like Newton's law, was not derived from other equations; and in this sense it is the fundamental physical law that describes the behavior of the constant electrical field. Consequently, the basic field equations will be obtained from Coulomb's law. 4. Under certain conditions of great importance for geophysical applications, Coulomb's law remains valid even when electric fields change with time. 5. As in the case of the gravitational field when distribution of electric charges is known, calculation of the field E, using Eq. (III.6), presents no serious difficulties. However, unlike the gravitational field, in most practical cases it is impossible to know all the charges prior to calculation, and correspondingly Coulomb's law becomes useless from a practical point of view. Now we shall consider two examples of fields caused by specific distributions of charges.

Normal Component of the Electric Field Caused by a Planar Charge Distribution Suppose that there is a distribution of charges with density !,(q) on the plane surface as shown in Fig. 1I1.1c. Introduce the vector dS =dSn where n is the unit vector directed away from the back side of the plane (1) toward the front side of the plane (2) on which the charge is distributed. We consider only the normal component of the field, that is, the component perpendicular to the surface. In accordance with Coulomb's law, as expressed by Eq. (111.6), every elementary charge !,(q) dS creates a

111.1 Coulomb's Law

205

field described by the equation 1 I(q) dS dEep) = -4- L 3 L qp 7TE O

(111.7)

qp

Therefore, the normal component of the field is

(111.8) Here (L qp, n) is the angle between the directions of L~p and n. It is clear that the product dS L qp cos(L qp, n) can be written as a dot product as follows:

(III.9) since L q p = - L p q • Thus, the normal component of the electric field can be written as

dEn(p)

1

dS . L p q

47Te o

L pq

= - -

3

I(q)

Inasmuch as the expression

represents the solid angle dw(p), subtended by the element dS from the point p, we have (1II.10)

In a similar fashion, for the normal component caused by all surface charges we obtain (111.11) In particular, if the charge is distributed uniformly on the surface CI is

206

II1

Electric Fields

constant), we have

(III.12) where w(p) is the solid angle subtended by the surface S when viewed from the point p, As was shown in Chapter I, the solid angle is either positive or negative depending on whether the back or the front side of the surface is viewed. With increasing distance from the surface S the solid angle decreases, and correspondingly the normal component of the field becomes smaller. In the opposite case, when the point p is considered to approach the plane surface S, the solid angle increases and in the limit becomes equal to - 271" and + 271" when the observation point p is located on either the front side (2) or the back side (1) of the plane surface, respectively. Thus we have the following expressions for the normal component of the electric field on either side of the surface:

I

E(I)= - n 2£0

(III.13)

These two expressions, as well as similar ones for the case of the gravitational field, indicate that the normal component of the electric field is a discontinuous function across the surface S. Let us examine this behavior of the normal component in some detail. The normal component of the electric field can be written as the sum of two terms.

(III.14) where E~ is the part of the normal component caused by the elementary charge I(q) dS located in the immediate vicinity of the point q, and E~-q is. the part of the normal component contributed by all of the other surface charges. It is clear that

where ws-q(p) is the solid angle, subtended by the plane surface S without the element of the surface dS(q), as viewed from the point p. Letting the point p approach the elementary area dS(q), the solid angle subtended by the rest of the surface, WS-q(p), tends to zero, and the normal component is defined only by the charge located on the elementary surface dS(q). as

p

~q

III.I Coulomb's Law

207

At the same time the solid angle subtended by the surface element dS(q), no matter how small that area is when viewed from an infinitely small distance from point q, tends to ± 27T. as

p

~q

Therefore, the normal component of the field on either side of the surface is determined only by the elementary charge located in the immediate vicinity of the point q.

(111.15) The difference in sign of the fields on either side of the surface reflects the fundamental fact that the electric field vector shows the direction along which an elementary positive charge will move under the force of the field. Thus, the discontinuity of the normal component of the field, as an observation point passes through the surface, is caused only by the elementary charge located near this point. For example, if there is a hole in the surface, the normal component on either side of the surface is E~-q, and therefore the field is continuous along a line passing through the hole. We can generalize these results to the case in which the surface carrying the charge is not planar. Making use of the same approach based on the principle of superposition and the definition of solid angles, we arrive at the following expressions for the normal components on either side of a surface:

(III.16) if

p

~q

In contrast to the previous case, the normal component E~-q(q) caused by charges located at the surface but outside the element dS(q) is not necessarily zero at the point q (Fig. IIUd). However, we can readily recognize a very important feature of this part of the field. Inasmuch as these charges are located at some distance from the point q, their contribution to the field is a continuous function when the observation point p passes through the element dS(q), and therefore

(111.17)

208

III

Electric Fields

Correspondingly, Eq. (I1I.16) can be written as

(IIU8)

This means that the discontinuity in the normal component, as before, is (III.19) and it is caused only by charges located within the elementary surface dS(q). It should be stressed that Eq. (I1I.19) is a fundamental equation describing electromagnetic field behavior, and it is valid for any rate of change of the field with time. In essence, we might say even though we risk getting ahead of ourselves that Eq, (I1I.19) is a surface analogy of Maxwell's third equation. Unlike the gravitational field, the surface distribution of charges plays an essential role. In fact, in real conditions of electrical methods practically only this type of charge distribution occurs. For this reason, regardless of how surface charges arise, Eqs. (HU8), (III.19) will be used often in this chapter. Effect of a Conductor Placed in a Free Space and Situated within an Electric Field

We will now consider a second example illustrating electrostatic induction. Suppose that a conductive body of arbitrary shape is situated within the region of influence of an electric field Eo, as shown in Fig. IIL2a. Under the action of the field the positive and negative charges residing inside the conductor move in opposite directions. As a consequence of this movement electric charges develop on both sides of the conductor. In doing so, these charges create a secondary electric field directed in opposition to the primary field inside the conductor. The induced surface charges distribute themselves in such a way that the total electric field inside the conductor will disappear; that is, (I1I.20) where E, is the electric field strength within the conductor. This process is

III.l Coulomb's Law

a

b

c=:>e2

209

~e3

air

earthsurface

f_,tiJt41\iI:'li;W4@UiiM!'lIlilJkliiil§_

c

d

a Fig. 111.2

(a) Electrostatic induction; (b) influence of charges in air; (c) flux of surface charges: and (d) voltage of the electric field.

termed "electrostatic induction." At this point it is important to make two comments.

1. In our description of this phenomenon we have given a very approximate picture of the process in which only the electrostatic field is considered to be present. In fact, the process of accumulation of surface charges involves other phenomena of the electromagnetic field including, in particular, a change of the magnetic field with time. Very often this process lasts a relatively short time, and afterward the constant electric field is governed by Coulomb's law. 2. The phenomenon of electrostatic induction is observed in any conductive medium regardless of the electric resistivity, provided that the conductor is surrounded by an insulator and that the sources of the electric field Eo are located outside of the conductor. For example, the conductive body could be composed of metal or an electrolytic solution, minerals, or rocks. One can show that the magnitude of the resistivity plays a role in determining the time required for the electric field inside the conductor to disappear, but it does not change the final

210

III

Electric Fields

result of the electrostatic induction, namely that the internal electric field will go to zero (E j == 0). It should be obvious that the secondary electric field contributed by the surface charges can be defined from the equation

(I1I.21) where !,(q) is a surface density of charges. Consequently, condition (I1I.20) can be rewritten as (I1I.22) where Eo is the primary field due to charges located outside the conductor, S is the conductor surface, and q is an arbitrary point on this surface. For instance, if a single point charge e is situated outside the conductor at the point a, its electric field at any point b inside the conductor is I

EoCb)

e

= - - - 3 - Lab

(111.23)

47TE O Lab

That is, it is the same as if the conductor were absent. As was stressed above, the electric field caused by a given system of charges does not depend on the electrical properties of the medium; and if the field changes, this means that charge distribution has been changed, or new charges have arisen. In our case, positive and negative charges appear on the surface of the conductor. At the same time, the total charge es of the conductor, which is not charged, remains zero. (I1I.24) In other words, when the electric field Eo is absent, within every element dV and dS there is an equal amount of positive and negative charge so that the presence of such a conductor itself does not create an electric field inside or outside the conductor. In contrast, due to the presence of the electric field Eo, positive and negative charges occupy different parts of the conductor surface and consequently the secondary electric field E, arises, even though the total charge e s is still equal to zero. If we knew the density of surface charges, calculation of the electrical field outside the conductor could be easily performed from Eq. (III.2l). However, the distribution of charges caused by electrostatic induction is

III.I Coulomb's Law

211

not known beforehand, and this fact reflects the fundamental difference between solutions to the forward problems of the gravitational and electric fields. Inasmuch as Coulomb's law cannot be used we are forced to develop special methods for field calculations that do not require a knowledge of charge distribution. Now we will describe one approach considering the effect of electrostatic induction. With this purpose in mind, we will derive the so-called integral equation with respect to surface density I(q). Then, knowing this function, we can make use of Coulomb's law, Eq. (III.21), and calculate the secondary field E s ' First, from Eq. (111.19) we have

E~(p)

-E~(p)

I(p)

=

(III.2S)

EO

where E~ and E~ are normal components of the electric field from the external and internal sides of the conductor surface near the point p, respectively. Because of the electrostatic induction the electric field inside the conductor vanishes, and therefore E~ == O. As a consequence, Eq. (III.2S) is simplified.

E~(p)

=

I(p)

(III.26)

EO

Applying the principle of superposition, let us write three terms.

E~

as the sum of (111.27)

where E~ is the normal component of the primary field at the point p, Eg is the normal component of the field caused by the elementary surface charge I( p) dS situated in the immediate vicinity of the point p, and E~-P is the normal component contributed by the rest of the surface charge. In accordance with Eq. (nUS), I(p) EP=-21'0

n

where n is the unit vector directed outward from the conductor surface. Inasmuch as ES-P

1 = --

4 7TEO

f -I(q) -L L S

3

qp

qp

dS

if

q"*p

212

III

Electric Fields

we have ES-P n

=

1 I(q)(L· --f qp 4'lTeo s L~p

n )

p dS

(111.28)

where n p is the unit normal vector at the point p and L qp • up is the dot product of vectors L qp and np' L qp• n p = L qp cos(L qp, np)

Collecting all terms in Eq. (I1I.27) and taking into account Eq. (III.26) we obtain

I(p) - - =E~+ EO

I(p) 1 - - + --fI(q)K(p,q) dS 2eo 4'lTE O s

or (111.29)

where

and 2EOE2 are known functions. Equation (III.29) presents a Fredholm integral equation of the second kind with respect to the unknown density at any point p on the conductor surface. In practice we can conceptually replace the surface of the conductor with a system of small cells, within each of which the charge density is practically constant. In doing so, the integral equation (III.29) can be rewritten as an approximation. if

q"* p (11I.30)

Having written this equation for every cell, we obtain a system of N linear equations with N unknown terms. When the charge density is known, then, using Eq, (III.2l) the secondary field is calculated in terms of the surface integral. I hope that this example vividly demonstrates how determination of the field becomes much more complicated with respect to the use of Coulomb's law, if some of the sources are unknown. This is the main reason, unlike the case of the gravitational field, for deriving the system of field equations and considering the boundary-value problems.

Ill.2 System of Equations for the Time-Invariant Electric Field Potential

213

Let us look one more time at electrostatic induction. Suppose that we consider a model of a conducting earth that consists of a sequence of layers each characterized by its own resistivity (Fig. III.2b). Above the surface of the earth electric charges due to atmospheric processes are present. In accordance with Coulomb's law they create the same field in the conducting medium as would have been observed if that volume were free space. But because of electrostatic induction, induced charges appear on the earth's surface that exactly compensate the primary field inside the conducting medium. Because of surface charges that accumulate on the interface between the upper half space and the conducting medium, the charges situated above the earth's surface do not have any effect on the electric field within the earth, and therefore, the constant field within the conducting medium can only be caused by sources existing within the medium. Fortunately, due to electrostatic induction, electrical methods can be used in geophysics, based on measuring the constant electrical field. Unlike the conducting medium, charges induced on the earth's surface create an electric field in the upper space that has only a vertical component; and near the earth's surface it is equal to 100v1m = 105mv1m, which is much greater than the horizontal component of the field, measured on the earth's surface by electrical methods. 111.2 System of Equations for the Time-Invariant Electric Field Potential Due to the mathematical similarity of Coulomb's and Newton's laws we will follow exactly the same path in deriving the equations for the timeinvariant electric field as in the case of the gravitational field. Let us start from the second equation and with this purpose in mind, we will consider an elementary charge de at the point q and its field. 1

E( p) = 4'7Tc

dee q)

u- L o

qp

qp

Then the flux through an elementary surface dS(p), caused by this field, is de(q) L q p ' dS

E·dS=------:.:.~-

4'7Tco

L~p

or 1 E' dS = --de(q) dw(q)

4'7Tco

(IIL31 )

214

III Electric Fields

where dw{q) is the solid angle under which the surface dS is viewed from point q. Consequently, for the flux through a closed surface surrounding the charge dei.q), we obtain

~E.dS=-

de(q)

s

eo

Now applying the principle of superposition we generalize this equation for an arbitrary distribution of charges inside the closed surface S.

~E's dS=~[f8dV+eo

fIdS+ fAdt'+Ie i ]

v

S

(III.32)

L

where 8, I, A are the volume, surface, and linear density of charges; ej is a point charge; and all of these are situated inside of the surface S. The flux of the field caused by charges located outside of the surface is zero and, as in the case of the gravitational field, this follows from the behavior of the solid angle. Inasmuch as in the vicinity of regular points only volume charges are present, Eq. (1II.32) is simplified and we have

~E.dS=S

1

f 8dV eo v

(111.33)

This is traditionally called the second equation of the field in integral form, describing the field at regular points. Taking into account that inside the volume V we can expect positive as well as negative charges, the flux of the electric field through a closed surface, unlike the gravitational field, can be zero in spite of the presence of charges in the volume. To characterize in more detail the distribution of charges, we will make use of Gauss' theorem. Then we obtain

~E.dS= S

fv divEdV=-eo1 fv 8dV

Consequently, 8 divE= eo

(III.34)

This is the second field equation in differential form, valid only for regular points. Now we will suppose that there is a surface distribution of charges (Fig. III.2c). As was shown in the previous section, the normal component

111.2 System of Equations for the Time-Invariant Electric Field Potential

215

of the electric field is a discontinuous function of the spatial variables in passing through a surface charge. Correspondingly, the derivative JEn/Jn does not exist, and therefore Eq. (III.34) cannot be used. Then, applying Eq. (111.32) to an elementary cylindrical surface enclosing a small piece of the interface (Fig. III.2c) and keeping only the term containing the surface charge I dS, we obtain the third form of the second field equation.

(111.35) By starting with Coulomb's law we have obtained three forms of the second equation.

s

divE= - ,

(III.36)

EO

Each of these characterizes the relation between charges and the electric field. In particular, if in the vicinity of some point charges are absent, we have

a

divE =

E(2) =E(I)

and

n

n

Next we will derive the first equation of the electric field. Inasmuch as both the electric and gravitational fields are caused by sources only, we can make use of the results of Chapter I and write the first equations of the electric field. curlE =

a

However, taking into account the importance of the concept of voltage in the theory of electric fields, let us derive the above equations. First of all, the integral

t

E· dl'=

ay

t

Edtcos a

(III.37)

Q2'

is the voltage between points a and b measured along some arbitrary path 2' (Fig. Ill.Zd) and caused by the electric field. Here a is the angle between the electric field vector and the tangent to the path 2' at every point. It is clear that the product E . dl'is the elementary work performed by the electric field in transporting a unit positive charge along the displacement d.l. This product has the dimensions of work per unit charge, and in the practical system of units has dimensions of volts. Therefore, the integral in Eq. (111.37) represents the work or voltage done in carrying a

216

III

Electric Fields

a

b

b

a

~

de(q)

c

a

d

Lacb

Fig. IlL3 (a) Voltage along a radius-vector; (b) voltage along a radius-vector and arc; (c) voltage along an arbitrary path; and (d) circulation.

charge between two points a and b. In the general case of an alternating electromagnetic field, for a given function E this integral depends on the particular path of integration 2' that is chosen, and on the terminal points a and b of the path. Starting from Coulomb's law we will show that the voltage of the electric field caused by charges only is independent of the path of integration, as is that of the gravitational field. Assume that the source for the field is a single elementary charge de, then its electric field is

E(p)

1

dee q)

= ----3-Lqp 41TE O

L qp

If both terminal points a and b are situated at the same radius vector L q p and the path of integration is along this radius (Fig. III.3a), the voltage between these points is very easily calculated.

217

III.2 System of Equations for the Time-Invariant Electric Field Potential

because d/'· L q p = dtL q p cos 0 = L q p dL Carrying out the integration as indicated, we obtain (III.38) Now suppose that the points a and b are situated on two different radius vectors L q a and L q b , as shown in Fig. III.3b. Let us choose a path .2"1' which consists of two parts. The first part is a simple arc ab', and the second element of the path is along the radius vector L q b . In this case the voltage can be written as

V=

1

--

dee q)

[I

41TE O

arcab'

d/'·3L q p L qp

+

Jb d/'· L 1 qp

3

b'

L qp

The integral along the arc ab' is clearly zero since the dot product is

d/'· L q p = 0 Thus, the voltage between points a and b is again equal to

(III.39) That is, it remains the same in spite of the fact that the path of integration has been changed. If instead of the path .2", we consider an arbitrary path .2"2' it should be clear that this path could be represented as a sum of arcs and elements of radius vectors, as is seen in Fig. III.3c. All the integrals along simple arcs are zero, while the sum of integrals along the radius vectors is

[1

1]

dee --q) - ----41TEO L qa L qb

and this is equal to the voltage along the path .2"t. We have established the second fundamental characteristic of the electric field, namely, that the voltage between two points does not depend on the particular path along which integration is carried out, but is

218

III

Electric Fields

determined by the terminal points only. This fact can be written as

fb

E . dl'=

fb

0.2'1

E . dl' = .. , =

t

E' dl'

( III AD)

Q..:lfn

aY2

Making use of the principle of superposition, this result can be generalized to a field caused by any distribution of charges. It must be stressed again that this result is valid only for the electric field caused by constant electric charges; it cannot, in general, be applied to time-varying fields. Let us also notice that unlike the gravitational field, the voltage plays a much more important role because the basic measuring device, the voltmeter, measures the value of the integral

that is, voltage, and if within this path the field E does not vary, then we are able from the voltage to find the electric field. The independence of the voltage of the path of integration can be written in another form. Consider a closed contour .2', as is shown in Fig. III.3d, as consisting of two other contours, .2'acb and .2'bda' In accordance with Eq. (IIIAO) we have

f

E' dl'=

acb

f

E' dl'

(IIIA1)

adb

In these integrals the element dl'is directed from a to b. Changing the direction in the integral on the right-hand side of Eq. (III.4l), we can write

f

E' dl' =

-

acb

f

E· dl'

bda

That is,

f

acb

E· dl' +

f

E· dl' = 0

bda

or (IIIA2) Thus, the voltage along an arbitrary closed path is zero. Sometimes the quantity ¢E . dl'is called the "circulation" of the electric field or the electromotive force. The path .2' can have an arbitrary shape, and it can intersect media with various physical properties (Fig. III Aa). In particular, it can be completely contained within a con-

111.2 System of Equations for the Time-Invariant Electric Field Potential

a

b ...-

•

219

_

l

.

(,~#g:'

./ .....•....

_

Ii

1111

./

c

d

Fig. IlIA (a) Circulation through a conducting medium; (b) continuity of tangential components; (c) electric dipole; and (d) double layer.

ducting medium. Because the electromotive force caused by the electric charges is zero, Coulomb's force E C cannot alone cause an electric current; and this is the reason that non-Coulomb forces must be introduced to provide a current flow. This question will be examined in detail in the next section. Equation (111.42) is the first equation for the electric field in integral form. Applying Stokes' theorem for regular points of the medium, we have

i E' d/= f curl E> dS = 0 :2'

S

or curlE = 0

(IIIA3)

The latter is the differential form of the first equation, showing that the electric field is not generated by vortices, and it applies at those points where the first derivatives of the electric field exist. To obtain a differential form of Eq. (111.42) near surface charges where the normal component En is discontinuous we will apply this equation along the elementary path shown in Fig. II1.4b. Considering that elements

220

III

Electric Fields

dl" and dz" are separated by a distance dh, which tends to zero, we obtain E . dl'''

+ E . dl" + E . dh = 0

or E(2)

t

dt -

E(1)

t

de'

=

°

and finally or

(III.44)

That is, tangential components of the field are continuous as the path passes across a surface charge. Let us note that there is one case, namely a nonuniform double layer, in which the tangential component E, is a discontinuous function. We have now derived three forms of the first equation based on Coulomb's law.

¢E. dl'= 0,

curiE = 0,

E(2)

t

=

E(1)

t

(III.4S)

Each of them expresses the same fact; that is, the electromotive force caused by electric charges is zero, or in other words, the voltage between two arbitrary points does not depend on the path of integration. We must make an important comment about Eqs. 011.45). The first two relationships are not valid when the field is time-varying, since the second type of generators (vortex), that is, the change of magnetic field intensity with time, is not taken into account. On the other hand, the surface analogy for these equations is valid for any electromagnetic field. This reflects the fact that in the development of this particular form of the equation, it was assumed that the area surrounded by the integration path was zero, and therefore the flux of the magnetic field through this area vanished. Let us note one further feature of the field. Although the equations ~E· dl'= 0 and curlE = 0 are not valid for time-varying electromagnetic fields, this does not mean that Coulomb's law is always inapplicable in such cases. Now we are ready to write the system of equations for the timeinvariant electric field in two forms.

III.2 System of Equations for the Time-Invariant Electric Field Potential

221

Integral Form of Equations for Time-Invariant Electric Field (III, 46)

Coulomb's law

II

~E. s

dS =

~

So

where e is the total charge inside the volume V, surrounded by the surface S; and .2' and S are an arbitrary contour and a surface, respectively, which can intersect media with different electrical properties.

Differential Form of Equations for Time-Inoariant Electric Field Coulomb's law

I

curlE

=

0

II

divE

8 =-

So

(I1IA7) Here it is appropriate to make the following comments:

1. Any electric field caused by charges satisfies Eqs, (IIIA6), (I1IA7); that is, it is itself a solution of the system of field equations in the integral and differential forms. 2. These equations contain the same information about the field as Coulomb's law, but as will be shown later they allow us to find the field in those very practical cases when Coulomb's law turns out to be useless. 3. Comparison with the system of equations for the gravitational field shows the two systems are identical, and this happens because these fields are caused by sources only. Moreover, in accordance with Newton's and Coulomb's law, in both cases the field possesses the same dependence on

222

III

Electric Fields

distance provided that masses and charges are distributed in a similar manner. 4. Systems (III.46) and (III.47) are valid everywhere, and correspondingly they correctly describe the electric field in the presence of any conducting and polarizable medium. In particular, the time-invariant electric field, measured in all electrical methods used in geophysics, is itself a solution to these systems. 5. In spite of the identity of the systems for the gravitational and electric fields, stilI there is one fundamental difference. In the case of the gravitational field the right-hand side of the second equation-that is, the density of masses-can usually be specified, while the density of electric charges is usually unknown. There is, however, one exception: when the electric field of a given distribution of charges is considered in free space and conductors and dielectrics are absent. In such a case, to calculate the electric field we can use Coulomb's law directly in the same manner as Newton's law is applied in solving forward problems of the gravitational field. But this case has hardly any practical interest in applied geophysics. Considering the electric field in the presence of conductors and dielectrics, we cannot in principle specify the density of charges before the electric field is calculated. The example of the previous section describing electrostatic induction has been designed to illustrate this problem. Thus, the system of field equations, for instance Eq. (III.47), contains several unknowns such as the electric field E and the density of charges 0 and k. Because of this we will be forced to replace this system by a different one that contains only one unknown, namely the electric field E. Now let us, proceeding from the first equation curl E = 0, introduce a scalar function U. E = -gradU

(III.48)

since curl grad U == 0 The scalar function U is called the potential of the electric field. In accordance with Eq. (III.48) the electric field E coincides with the direction of maximum decrease in potential, and any component of the field can be expressed in terms of the potential as follows:

au

E r : - -at

(III.49)

Now we will demonstrate three main reasons why it is useful to introduce the potential U. First, we will write an expression for the voltage

III.2 System of Equations for the Time-Invariant Electric Field Potential

223

using the potential. It is clear that

au

dU = -

at

dt = dl'" grad U = - E . dl"

(111.50)

where dl" = dti o and i o is a unit vector. Integrating the last of these terms along any path between two arbitrary points and taking into account the fact that the voltage is path independent, we obtain

t

E ' dl"= -

p

t

dU= U(p) - U(b)

(111.51 )

p

That is, the voltage of the electric field along any path with terminal points p and b can be written as the difference of the potential between these

points. Therefore, knowing the potential, it is very simple to calculate the voltage, and this is the first reason for the introduction of this function. Next we will use Eq. (IlL51) to define the potential caused by an arbitrary distribution of charges. From this equation we have

U(p) = U(b) + JbE' dl"

(III.52)

p

It is obvious that at great distances from charges the field E is very small and therefore the potential also vanishes. Then, letting b equal infinity in

Eq. (IlL52) and assuming that the potential at this distance is zero, we have

f'E'

U(p) =

(III .53)

dl"

p

Suppose that the source of the electric field is a single elementary charge de, situated at the point q. By using Eq. (IlLS) and Eq. (III.53) we obtain de

00

U(p)=-J 47TEO

P

dL

de

2 =--

L

47TE OL q p

(111.54)

Making use of the principle of superposition for an arbitrary distribution of volume, surface, linear, and point charges, we arrive at the following expression for the potential: 1 [ /5 dV "i dS A dt e. ] U ( p ) = - j - + j - + l - + L : - (III.55) 47TE O v L qp S L qp '2' L q p L qp

224

III

Electric Fields

Comparison with Eq. (III.6) clearly shows that the potential U is related to the charges in a much simpler way than the electric field. This simplicity is the second reason for using the potential. It is obvious that the electric field can be easily found applying Eq. (I1IA9), if the potential is known. Let us consider two examples.

Example 1 The Potential and the Field of an Electric Dipole

Suppose we have two elementary charges equal in magnitude and of different sign, as is shown in Fig. IlIAc. Then, the potential U due to these charges is (III .56) where ql and q2 are points in the vicinity of which the negative and positive charges with magnitude e are located. We will consider the field only at distances greatly exceeding the distance between charges; that is, the displacement dt of one charge with respect to another is much smaller than the distance L q p , where q is midpoint between the charges. (I1I.57) In this case the system of charges is called an electric dipole. Taking into account the inequality (111.57), the difference in the right-hand side of Eq. (111.56) can be replaced by 1

1

L q 1P

-;

a

1

---=--dt

at i ;

and in accordance with Eq. (111.50) we have U(p)

=

e ale q 1 - - - - d t = --dl"grad417"£0

at L q p

417"£0

L qp

where the index "q" means that only the point q changes. Since

(I1I.58)

111.2 System of Equations for the Time-Invariant Electric Field Potential

225

Eq. (III.58) can be presented as 1

M' L q p

41TB O

L qp

U(p) = - -

3

(III.59)

where M = e dl' is the moment of the electrical dipole, which is a vector showing the direction of displacement of the positive charge with respect to the negative one, and whose magnitude equals the product of the positive charge and the distance between them, de. As follows from Eq. (III.59) we have

U(p)

M cos e

1 = - .-

2

41TB O

L qp

(III.60)

where () is the angle between the direction of the moment M and that of the radius vector from the dipole center q to the observation point p. In accordance with Eq, (III.60) the potential of the electric dipole decays more rapidly along the radius vector than the potential due to a single charge and essentially depends on the angle e. In particular, if e < 1T/2 it is positive, since the positive charge is closer to the observation point; at the equatorial plane e = 1T/2, it is equal to zero, because at all points of this plane a distance to both charges is the same; and finally, when e> 1T/2, the potential is negative. Now let us introduce a spherical system of coordinates R, e, cp with z-axis directed along the dipole moment. Then the potential U can be written as 1

U(p)

= -41Teo

M cos e

R2

and taking into account Eq, (III.49) the components of the electric field are 2M cos (J

E ----R - 4rrB o R3 '

M sin e E ----(J 4rre R3 o

(III.61)

Thus, the electric field of the dipole decays more rapidly than that of the single charge. Along z-axis, as follows from Coulomb's law, the field is directed parallel to the dipole moment, regardless of whether the angle (J is equal to 0 or tr. In the equatorial plane the field has a direction opposite to that of the electric dipole. It is interesting to note that the direction of the vector E along a given radius vector e = constant, remains

226

III

Electric Fields

the same. In fact, from Eq, OIl.6l) we have tan cp

Eo = -

ER

1 = -

2

tan (J

(III.62)

where cp is the angle between the field E and radius vector R. In deriving formulas for the potential and the field of the electric dipole we have made only two assumptions, namely, 1. The sum of charges is equal to zero, and 2. The field is considered at distances that greatly exceed the displacement between the charges. For this reason we can say that Eqs. (111.60), (III.6l) describe the potential and the field of an arbitrary but neutral system of charges, provided that the observation points are located far away with respect to the dimensions of the volume where charges are situated. For instance, due to electrostatic induction, charges of both signs appear on a conductor surface, and with an increase in distance the electric field of these charges tends to that of the electric dipole. It is essential to note that this behavior of the field takes place regardless of the shape and size of the conductor.

Example 2 The Potential and the Field of a Double Layer Let us imagine that positive and negative charges with density and

(III.63)

are distributed on two surfaces S + and S _, respectively, and that the separation t between them is much smaller than the distance from the observation point to these surfaces (Fig. IlIAd). As follows from Eq, (IIl.63) every pair of surface elements located opposite to each other have charges with the same magnitude and different signs. Such a system of charges is termed a double layer, and double layers are widely applied in the theory of self-potential and induced polarization methods of electrical prospecting. For convenience we will choose some surface S, which is located between the surfaces S + and S _, whose normal n points in the direction from the surface S _ to the surface S +. Let us consider some point q of

IIl.2 System of Equations for the Time-Invariant Electric Field Potential

227

this surface and two corresponding charges located at distance t from each other. de 2 = de = ~(q)

de, = -de =

and

dS

-~(q)

dS

It is clear that these charges form an electric dipole with a moment dM equal to dM

or

dM=de/

dS tn

= ~(q)

and dM

7J( q) dS

=

(III.64)

where

TJ(q) =!.(q)t(q)

(III.65)

which characterizes the density of dipole moments, and correspondingly it is called the double-layer density. In accordance with Eq. (III.59) the potential of the electric field caused by this dipole is 1

dU(p)=-4- TJ ( q ) 7TEO

dS· L q p L3 qp

or 1

dU(p)=---7J(q) 47TE o

dS· L p q 3

L qp

( 111.66)

where dS = dS n is an element of the middle surface S. In essence, we have replaced the real distribution of charges at both surfaces S _ and S + by the mathematical model of the double layer placed on the surface S. It is obvious that if in both cases the density TJ(q) remains the same, and the observation points are located far away, this replacement does not alter the field. Now making use of the results derived in Chapter I we can rewrite Eq. (111.66) as

dU(p)

= -

7J

-dw(p)

(III.67)

41TE O

where dcoi p) is an elementary solid angle. Therefore, for the potential caused by charges of a double layer we obtain 1 U(p) = - -fTJ(q) dos (III.68) 41TE O S

Taking into account the fact that the procedure for calculation of the solid

228

III

Electric Fields

angle is known, Eq, 011.68) allows us to find the potential and, correspondingly, the electric field caused by an arbitrary double layer. In particular, if the double layer is uniform, that is, 1) = constant, then we have

U(p)

1)

=

(III.69)

-4-w(P)

-

7TSo

where w(p) is the solid angle subtended by the layer surface S when viewed from the point p. In the case where the uniform double layer is closed, we have for points in the volume confined by the surface S, and outside, .

U'(p)

1)

and

=So

(III .70)

respectively, and since the potential does not vary, the electric field equals zero. Next we will study the behavior of the potential and the electric field near the double-layer surface, when its density 1)(q) is an arbitrary function. The potential U(p) at every point can be represented as the sum of two terms. (I1I.71)

where uq(p) is the potential caused by the element dS(q) of the double layer in the vicinity of the point q, and US-q(p) is the potential due to the rest of the layer. When the observation point approaches the point q the solid angle subtended by the surface dS(q) tends to +27T, and correspondingly, for the potential caused by this element we have U( ) = p

+ 1)(p) -

2

if

p

~q

So

Therefore, the potential U at the front and back sides of the double layer is

if p

~q

Inasmuch as all charges located outside the element dS are at some distance from the point p, the potential US-P is a continuous function in

111.2 System of Equations for the Time-Invariant Electric Field Potential

229

the vicinity of the point p; that is,

Therefore, the difference of the potential on either side of the double layer is (111.72) Thus, the potential is a discontinuous function across the double layer, and this discontinuity is defined by the density 7J near the point of observation. Let us notice that such behavior of the potential is an exception, and it is inherent only to the model of the double layer. If we consider the potential between surfaces S _ and S + this. discontinuity disappears. Having taken the derivative at both sides of Eq. (1II.72) in any direction t tangential to the double layer, we obtain

or

(III.73) That is, the tangential component of the electric field is a discontinuous function at those points of the double layer where the density 7J is not constant. Unlike the case where the charges are distributed on one surface, the normal component of the field due to a double layer turns out to be a continuous function. Indeed, in accordance with Eq. (lIUS), two elementary surfaces with charges - I(p) dS and I(p) dS create inside the double layer a field equal to I(p)/e o, but outside at both surfaces S _ and S + it is zero. At the same time other elements of the double layer, being at finite distances from the point p, generate a field that is a continuous function near this point. Thus, the normal component of the electric field has the same values on both sides of the double layer. (111.74)

230

rn

Electric Fields

As follows from Eqs. (IlI.S!) and (IlI.72) we have (IlI.75) where E i is the field inside the double layer, and the indices" +" and " - " mean that the points belong to the surfaces S + and S _, respectively.

Now let us consider the third and perhaps the most important reason justifying the introduction of the potential U. Earlier, from the first field equation curl E = 0 we obtained E

=

-gradU

Substituting this into the second equation we see that the potential satisfies Poisson's equation.

(III.76) The procedure for derivation of this equation is shown below

s

I curiE = 0 I

divE = -

EO

1 1

I

E = - grad U

I---+~=====~_~

j

-s- -,

divgrad U = V' 2 U = - EO

In particular, in the vicinity of points where charges are absent, the potential obeys the Laplace equation. (IlI.77)

111.2 System of Equations for the Time-Invariant Electric Field Potential

a

231

b U = constant

c

d U = constant

U =constant

Fig. 111.5 (a) Potential behavior near surface charges; (b) potential behavior on conductor surface; (c) electrostatic screening; and (d) charges inside a conducting shell.

The analogy with the potential of the gravitational field is obvious and follows from the fact that both fields are caused by sources only. Equations (III.76), (I1I.77) are valid at regular points only. For this reason let us study the potential behavior near surface charges, where the normal component of the field is a discontinuous function. In accordance with Eq. (III.53) the potential on either side of the surface (Fig. IIL5a) is and or (III.78) Taking into account the fact that the field on both sides of the surface is finite, but the distance between points PI and Pz is vanishingly small, the difference in potential from both sides tends to zero. Therefore, the potential of the electric field across any surface carrying a charge with

232

density

III

~

Electric Fields

is a continuous function. ( 1II.79)

Now we are ready to replace the system of field equations by another system, which describes the behavior of the potential of the electric field. Coulomb's law

(1II.80)

B

aU

aU

an

an

-2 - -l on

S

Here it is appropriate to make several comments. 1. System (I1I.80) consists of three parts; one of them is Poisson's equation, which describes the behavior of the potential at usual points where the first derivatives of the field exist, and the two others characterize the potential in the vicinity of surface changes. Note that there is one exception, namely the surface of a double layer, where this system cannot be applied. 2. In accordance with Eq. (III.55), the potential caused by a volume distribution of charges is

U(p)

1

j8(q) dV L qp

=47TE O V

(III .81)

On the other hand, the potential is described at regular points by Eqs. (1II.76), (I1I.77). Therefore, the expression given in Eq. (1II.8l) is the solution of Poisson's and Laplace's equations inside and outside of volume charges, respectively. 3. As was shown in Chapter I, Poisson's equation can be written as 6

8

- 2 [U( p) - U av] = h EO

(1II.82)

III.2 System of Equations for the Time-Invariant Electric Field Potential

233

since

where 2h is the length of the side of an elementary cubic volume surrounding the point p, and U(p) and are the value of the potential and its average value at this point, respectively. As follows from Eq. 011.82), outside of charges,

or:»

U(p)

=

U'"

or

That is, the potential is a harmonic function. In the vicinity of points where the charge is positive, the potential exceeds the average value. U(p»u av while at places where negative charges are distributed, the opposite relation holds. U(p)

< o-:

4. Conditions A and B of the system 011.80) provide continuity of the tangential component and discontinuity of the normal component of the electric field across the surface charges, respectively. 5. The system (III.80) is identical to that for the potential of the gravitational field, and transition from the electric field E to the scalar field U often essentially simplifies the solution of the forward problems of the theory of electrical methods. This fact constitutes the third merit that justifies the use of potential U. However, introduction of the potential itself does not allow us to remove the basic difficulties of field determination, such as absence of knowledge of induced charges if the electric field is unknown. In this and the next sections we will consider several cases that vividly illustrate this problem. Let us start from the following model of a medium.

A Conductor Situated in Free Space Suppose an arbitrary conductor is placed in the electric field Eo caused by a given distribution of charges with density [) in free space (Fig. III.5b). Due to electrostatic induction, the electric field E, vanishes inside of the conductor. Correspondingly, the derivative of the potential U, in any

234

III

Electric Fields

direction within the conductor is zero.

au

_'=0

at

That is, the potential U, does not vary, and in particular, the conductor surface is an equipotential surface. on

S

(III.83)

where C is a constant, usually unknown. Let us write the potential as the sum of two terms. U= U o + Us where U O is the potential caused by a given distribution of charges, (III.84) and Us is the potential due to surface charges with density 2, which appear on the conductor surface. Us = _1_j2(q) dS 47TSo v L q p

(III.85)

where q is a point on the conductor surface. It is useful to make three comments. 1. Surface charges with density 2 are distributed in such a way that the electric field within the conductor disappears. 2. Equations (nI.84), (nI.8S) describe the potential everywhere as if the conductor were absent. 3. Of course, these equations are similar, but there is one essential difference, namely, the sources of the field Uo are given, while the density 2(q) is unknown. Considering the electrostatic induction in the previous section we have been able to establish a relation only between the field at the external side of the conductor surface E~ and the density 2(q).

(I1I.86)

but E~=E~(q)

where E~(q)

and E~(q)

+E~(q)

are the normal components of the field on the

III.2 System of Equations for the Time-Invariant Electric Field Potential

235

external side of the conductor surface caused by the given and unknown distributions of charges, respectively. As follows from Eq. OII.85) the potential Us can be found if the density of induced charges I is determined. On the other hand, in accordance with Eq. OII.86) these charges can be specified, provided that the electric field ES

=

-grad Us

is known. Thus, as was stressed in the previous section, we are faced with the problem of "the closed circle," which vividly shows that the field calculation cannot be performed by using Coulomb's law. This is the general problem inherent for practically all fields in applied electrical methods. Consequently, to find the electric field E we have to make use of a completely different approach based on the theorem of uniqueness, and formulate boundary-value problems. Since the field inside a conductor is known (E i == 0), our attention is paid only to the potential in a volume of free space, confined by a conductor surface S, and a spherical surface with a very large radius that in the limit tends to infinity. As we know from Chapter I the electric field is uniquely defined if the potential U satisfies the following conditions: 1. Outside the conductor or conductors, the function U is a solution of Poisson's equation in every point of free space.

where 0 is the volume density of charges, which is specified. Of course, in the vicinity of points where the density is zero, the potential is a harmonic function.

2. Since both fields, caused by the volume and surface charges, decrease with distance the condition at infinity is U--+O

as

L qp --+00

3. At the conductor surface the potential must be constant and, depending on the type of available information we have, also must have

236

III

Electric Fields

either or

or

where 'Pl(q) and 'Piq) are given functions; Q is the total charge on the conductor surface; and in the case of a neutral conductor, Q = O. There are various approaches to solving the forward problem, including the trial-and-error method. In particular, we have illustrated the idea of the integral equation method, when electrostatic induction was considered. Certainly all of these methods are much more complicated than the direct application of Coulomb's law, but it is natural "retribution," since an essential part of the source, namely the induced charges on the conductor surface, is unknown until the field is calculated. Now we will demonstrate the use of the theorem of uniqueness in describing the electrostatic shielding effect. First suppose that some volume V is surrounded by a conducting surface S-for instance, a metal foil -and that the sources of the field Eo are located outside this volume (Fig. III.5c). Due to electrostatic induction, positive and negative charges arise on the surface, and they are distributed in such a way that the potential does not change within the conductor. Our task is to find the field inside the volume V. Taking into account the fact that charges are absent in this volume, but the surface S is an equipotential surface, we have

\12U=0 U=c

(III.87)

on

S

where C is an unknown constant. In accordance with the theorem of uniqueness (Chapter I) we have formulated Dirichlet's problem, and so Eqs. 011.87) uniquely define the field, even if the constant is unknown. We will look for a solution with the help of the trial-and-error method. Suppose that within the volume V the potential is also constant, which equals C; that is,

u=C

m V

(111.88)

TII.2 System of Equations for the Time-Invariant Electric Field Potential

237

It is clear that this function U automatically satisfies the boundary condition as well as Laplace's equation, since even the first derivative from constant is zero. Therefore we have found the potential inside the volume and there is no other solution to this problem. Inasmuch as the potential is constant, the electric field in the volume, surrounded by an arbitrary conductive surface, vanishes. £=0

(111.89)

In other words, we have described the effect of electrostatic screening and proved it by making use of the theorem of uniqueness. Next assume that the sources of the field Eo are surrounded by a conducting surface S, but we are interested in the behavior of the field outside of this surface (Fig. III.5d). Again, due to electrostatic induction, charges of both signs appear at the internal and external sides of the surface S, which becomes an equipotential surface. Taking into account the fact that the electric field within a conductor is zero, the total flux of the field through any closed surface S, of the conductor is also zero.

~E'

dS = 0

Si

This means that the induced charges at the internal surface of the conductor are equal in magnitude and opposite in sign to the charges located in the volume V. Correspondingly, at the external surface the same charge as that in the volume V appears. It is obvious that this system of sources creates a field outside of the conducting surface that is not zero. In other words, electrostatic shielding does not work in both directions. Let us show this again using the theorem of uniqueness. In this case the potential outside of the conductor satisfies the following conditions: 1. At regular points

2. On the conducting surface

U=c 3. At infinity the potential tends to zero U~O

It is obvious that any constant value of the potential does not simultaneously satisfy both boundary conditions if C i= 0, and consequently the potential changes from point to point. Therefore the electric field differs

238

III

Electric Fields

from zero although charges in the volume V are surrounded by the conducting surface. At the same time, in accordance with the theorem of uniqueness, if the potential U on the surface S becomes equal to zero due to grounding, the potential vanishes everywhere outside of the surface S.

U=o

(111.90)

From this consideration it also follows that, surrounding some volume V by two conducting surfaces that are connected with each other, we perform the electrostatic shielding of this area from the field caused by external sources even without grounding. This happens due to the contact between these surfaces, since the potential becomes the same on both of them. Correspondingly, inside the volume the potential also equals a constant that coincides with that on surfaces.

111.3 The Electric Field in the Presence of Dielectrics

In the previous section we have shown that any conductor, placed in an electric field, produces a change of the field. This happens because every element of the conductor contains an unlimited amount of charge, which under the action of an electric force moves freely through the conductor and, due to electrostatic induction, surface charges appear. Now we will consider another type of medium, which is also able to change the field. This medium is called a dielectric, and its influence can be approximately described in the following way. One would think that every element of the dielectric volume does not have free charges that can move significant distances, as in a conductor, and each element contains an equal amount of positive and negative charges, bound to each other. This description is based on the fact that either molecules or areas around ions, located at lattice nodes, are a neutral system of charges. Due to the external electric field, bound charges slightly change their position; but unlike free charges, they do not leave their elementary volume. Due to the external electric field, positive and negative charges move in opposite directions, and therefore an elementary volume acquires a dipole moment; this phenomenon is called polarization. Now we are ready to describe the field caused by polarization within each elementary volume. In fact, performing a summation of these fields over the entire volume of the dielectric we obtain the secondary field which, together with the primary or external field, forms the total observed field.

ID.3 The Electric Field in the Presence of Dielectrics

239

Before we continue, let us make one comment, namely, the dielectric is not antipodal to the conductor. In reality every medium possesses both conducting and polarizable features and it is only for simplicity and to emphasize the polarization effect that the presence of free charges is not taken into account here. In fact, the insulator is the antipode to the conductor. To find the secondary field, first we will consider an elementary volume dV, which contains an equal amount of positive and negative charge, slightly displaced from each other and therefore having a dipole moment dM. dM=PdV

(111.91 )

where P is the polarization vector or, shortly, polarization. This is the dipole moment of the elementary volume referred to in this volume; that is, it is the density of dipole moments.

dM P=dV The greater the amount of charge and the greater the distance between the positive and negative charges, the greater the polarization vector directed toward the positive charges. In most cases we can assume that the polarization P is directly proportional to the field E; that is, P=aE

(III.92)

where a is the polarizability, which characterizes the ability of a medium to display polarization-in other words, the number of dipoles oriented along the field. On the right-hand side of Eq. (III.92) E is the total field caused by all sources, including dipoles in neighboring parts of the dielectric. Correspondingly, it consists of the primary and secondary fields (III.93) Taking into account the fact that charges within every elementary volume dV(q) create a field equivalent to that of an electric dipole and making use of Eq. (III.59), the potential dU/p) outside of the polarized element is (III.94) where L q p is the distance from the point q that defines the position of the element dV to the observation point p. Therefore, the polarized dielectric

240

III

Electric Fields

in the volume V generates an electric field E, whose potential is

Us(p)

=

1 P(q) . L --f L 3

4'77"£0

V

qp

dV

(111.95)

qp

This expression is relatively simple, but it cannot be used for calculation of the potential of the secondary field. On the one hand, in accordance with Eq. (111.95), to find the potential U; we have to know the polarization vector P. On the other hand, the latter is defined by the total field, Eq. (111.92), including the secondary field, E, = - grad u". Thus, we are again faced with "the closed circle" problem illustrated in the previous section. Correspondingly, the system of field equations (I1I.47), describing the constant electric field everywhere including dielectrics, contains three unknowns: the electric field E and the sources 0 and 2. Because of this interdependence we have to modify this system in such a way that a new system has only one unknown, namely the field E. With this purpose in mind, we will perform some simple transformations on the right-hand side of Eq. (111.95) and introduce a new vector, the electrical induction, as well as bound charges. Let us represent the potential due to polarization as

Us(p)

=

1 q 1 -1.P(q). V - d V 4'77"£0 v L qp

(111.96)

since

Taking into account that q 1 q P 1 q P(q)·V-=V---VP i.: -: -:

instead of Eq. (III.96) we have

Us(p) =

P 1 divP -1 fv div-dV-1. -dV L L 4'77"£0

qp

4'77"£0

V

(III.97)

qp

To simplify this equation we are going to make use of Gauss' theorem and replace the first integral by a surface integral. Also, let us suppose that in the volume V, surrounded by the surface S, there is some surface Sj where either the polarizabiIity a or the field E or both of them are discontinuous functions, and therefore the polarization vector P is also

241

111.3 The Electric Field in the Presence of Dielectrics

b

c

d

-E

e--- --0 --0 ;' e--- --01, e-----0 e--- --0 \. Fig. III.6 (a) Polarization in a dielectric; (b) negative bounded charges; (c) positive bounded charges; and (d) movement of charges in a conductor.

discontinuous. Having surrounded S, by the "safety" surface 50 (Fig. III.6a), we obtain the volume VI which is bound by surfaces 5 and 50 and where Gauss' theorem can be applied. Then, in the limit the surface 50 coincides with the front and back sides of 5 i , but the volume VI tends to the original volume V. Therefore, the volume of integration remains the same, but the surface of integration includes the surface 5 and both sides of 5 i , where the vector P is a discontinuous function. Now applying Gauss' theorem to the first volume integral in Eq. OII.97) we obtain

1 ( P . dS 1 +--},--+-417"£0

S

L qp

417"£0

1- div P dV V

L qp

(III.98)

where p(1) and p(2) are the values of the polarization vector at the back and front sides of the surface 5 i , respectively. Taking into account that in performing the integration over the surfaces 51 and 52 the normals have to be directed outside the volume (Fig. III.6a),

242

III

Electric Fields

we have I

-f

47TE O S,

pO) .

I

dS

L qp

+-f

p(2) .

47TE O S2

dS

L qp

I

+ p(2) • 02

pO) • 01

=-f 47TE O

s,

I

dS

L qp

(III.99)

pO) _ p(2)

=-f n 47TE O

s,

Lqp

n dS

where 01 = n and 02 = -0, and n is the normal directed from the back to the front side of the surface Sj' Now we will show that the integral over the surface S, which envelopes all possible dielectrics, vanishes. In fact, at infinity the electric field and therefore the polarization vector P decreases at a rate inversely proportional to the square of the distance. Consequently, taking a spherical surface S of very large radius R, and making use of the mean value theorem, we have as

R

~

00

where C is some constant. Thus, we have derived a new expression for the potential presence of dielectrics. I

- div P

Us(p)=-f 47TEO v L q p

1

dV+-f

47TE O S

- Div P L qp

dS

III

the

(III.100)

where (III.lOI) is the difference of the normal component of the polarization vector at either side of the surface of singularity Sj' Considering the volume V as the whole space, we include all dielectrics as well as those parts of the volume where dielectrics are absent; and it is obvious that integration over such parts does not affect the field, since

p=o.

Both Eqs. OII.95) and (III.lOO) describe the same potential, caused by polarized dielectrics; however, the latter turns out to be much more useful for understanding the charge distribution within dielectrics as well as for formulation of boundary-value problems. First let us write down expres-

III.3 The Electric Field in the Presence of Dielectrics

243

sions for the potential caused by free charges and those that arise due to polarization.

1 - div P 1 - Div P U=-f dV+-{ dS s 41TEO V L qp 41TEO Js Lqp

(III.102)

The striking resemblance between these equations is obvious. From their comparison we can conclude that, due to polarization, volume and surface charges appear, and their density is defined by

Ob= -divP,

lb= -DivP=ppl_ppl

(III.103)

We use the index "b" to distinguish these charges from the free ones Do and lo. Let us now investigate these polarization-induced charges in more detail. The presence of this type of charges has been established analytically due to the transformation of Eq. (111.95), but we can arrive at the same conclusion from a physical point of view. Indeed, if the polarizability varies from point to point, the density of dipole moments also changes, and correspondingly we can expect the appearance of charges. This phenomenon is illustrated in Fig. III.6b and c, when either positive or negative charges arise. It is especially obvious when the boundary between the dielectric and free space is considered. These charges, unlike free ones, can only slightly change their position, and this is why they are called bound charges. The different extent of movement of the free and bound charges is the sole difference between them, but what is most essential is the fact that they both create an electric field in the same way, Eqs. (H1.102). Thus, in accordance with Eqs, 011.102), (HU03) we can say that due to the polarization the bound charges arise in a dielectric volume and at its interfaces, and these charges create an electric field E, that obeys Coulomb's law. Correspondingly, the potential and the electric field caused by the free and bound charges are

1 0 dV 1 l dS U(p)=-f-+-f41TE o v L q p 41TE O Sj L q p and

(III.104)

244

III

Electric Fields

where the total densities of charges are (III.lOS) and 80 and ~o are volume and surface densities of free charges that are assumed to be known. In general, in the vicinity of some points, there can be only free charges-for instance, outside of dielectrics-while at other points inside dielectrics either bound charges or both can exist. As soon as bound charges are taken into account, we arrived in accordance with Eqs. (111.104) at the same expression for the electric field as that in a free space. In particular, when there is only a volume distribution of charges, the expression for the electric field is

E(p)

1 4'7T£o

=--

f. 8 dV L

qp

(III.106)

3

V

L qp

From a mathematical point of view this equation is exactly the same as that derived in the first section for free charges only. By applying the same approach, we can write the equations of the field in integral form e

~E'dS=-

(111.107)

s

£0

where e = eo + eb is the total charge within volume V surrounded by surface S, and it includes both the free and bound charges. The first equation states that only sources create the time-invariant electric field in the presence of dielectrics; the second one demonstrates that both free and bound charges cause the electric field, and this occurs in the same manner. Now applying Stokes' and Gauss' theorems for regular points, we obtain the differential form of Eqs. (111.107). curlE

=

0

divE =

8 -

(111.108)

£0

If there is an interface of discontinuity, then making use of the same approach as that in the previous section we have E(1)

t

=

E(2)

t'

(III.I09)

That is, the tangential component of the electric field is a continuous function. While the discontinuity of the normal component across the interface is defined by a surface density of total charge, ~.

111.3 The Electric Field in the Presence of Dielectrics

245

Thus, the system of field equations in differential form can be presented in the form Coulomb's law (III. 110)

II

curlE = 0

I

II

B divE = eo

This system correctly describes the field within and outside dielectrics. However, as in the case of conductors, the right-hand side of the second equation contains an unknown quantity, namely the density of bound charges s, and k b • To eliminate these functions from the system, we will introduce a new vector related to the electric field, and for which this linkage is known for every given dielectric. With this purpose in mind, let us represent the equation B divE = eo

as

Then taking into account Eq. (11.103) we have Bo divP divE=---eo

eo

or

. ( P)

div E+ -

eo

o

=B -

(111.111)

eo

We will introduce the vector (111.112)

246

III

Electric Fields

Correspondingly, instead of Eq. (IIUll) we obtain divD = 00

(III.l13)

since EO is constant. The vector D is called the vector of electric induction, and making use of Eq, (III.92) we can express it in terms of the electric field.

D=E oE+aE=Eo(l+ :o)E=E O(1+{:l)E

(II1.1l4)

where {:l = a/Eo is the dielectric susceptibility that characterizes the dielectric. Along with this parameter we will introduce the dielectric constant E. E = 6 0 (1

+ (:l)

(III.1l5)

and correspondingly, instead of Eq. (lII.1l4) we obtain D=6E

(II1.116)

Thus, knowing the dielectric constant, it is an elementary procedure to calculate the vector of the electrical induction from the electric field and vice versa. In this relation it is appropriate to note the following. Equation (111.116) is very simple and this fact can easily produce the wrong impression that both vectors E and D have the same physical meaning, since they differ only by the scalar 6, which is often constant within a dielectric. However, the vector of the electric induction is a sum of two completely different vectors, Eq. (111.112); namely, one of them is the electric field E, and the other is the density of dipole moments, that is, the vector polarization P. Moreover, from the equality curl D = curl E E =

ECUri E

+ (grad 6 X E)

= grad EX E it follows that in general the field D is caused by vortices as well as by free charges. In particular, at the interface of media with different dielectric constants the tangential component of an electric field is a continuous function. E~l)

=E~2)

and therefore the tangential component of the vector D has a discontinuity

that indicates the presence of surface vortices (Chapter I),

III.3 The Electric Field in the Presence of Dielectrics

After these comments let us continue the

247

transformation of

Eqs, (111.110). In accordance with Eq. (I1I.103) the surface analogy of the

second equation can be represented as

I E(2) _ E(1) = n n

+

~

p(1) n

eo

I

p(2) n

eo

eo

or p(2) ] + _n_

E~2) [

_

[

E~1)

p(1) ] + _n_

eo

=

eo

'I ~ eo

or (I1I.1l7)

That is, the normal component of the vector D is a discontinuous function, but unlike the normal component of the electric field, this discontinuity is defined by the value of the density of free charges only. Now we are ready to write down the system of field equations where on the right-hand side only the known density of free charges is present. Coulomb's law

II

curlE = 0

E(2) t

I

I II

E(1) = t

0

divD = 00 I

I D=eE

(I1I.llS)

Therefore we have modified the system of field equations (111.110) in such a way that it no longer contains unknown bound charges. We were able to do this because of the assumption that there is a linear relation between the polarization and the field, and that the polarizability a is known. For a better understanding of the field behavior let us study the distribution of bound charges. In accordance with Eqs. (111.113) and (111.116) we have div s E

=

s div E + E· grad

10 =

00

248

III

Electric Fields

and since 00 + Db

divE=--Co

we obtain C

00 = -(00 + Db) Co

+ E· grad s

or (III.119) where C f = clco is the relative dielectric constant. It is convenient to distinguish two types of bound charges. Db =

0h1) + 0h2 )

(III.120)

where ,,(1) Ub -

1-c r" cr

and

Uo

The first type of bound charges arises in the vicinity of points where free charges are present; and since c f > 1, these bound charges always have a sign opposite to that of the free charges. The second type of charges can arise in the vicinity of points where the dielectric parameter e varies. If there is a projection of the electric field along the vector grad e negative charges arise, but if the vectors include an angle exceeding 7T/2, positive charges appear. In particular, in the case where the electric field is perpendicular to the direction of maximal change of parameter e, this type of bound charge does not arise. Now we shall consider surface bound charges proceeding the equation D~2)

which can be represented as D(2) n

D(1) = n

Inasmuch as

e 2 E(2) n

-

e 1 E(1) n

-

D~l)

=I o

111.3 The Electric Field in the Presence of Dielectrics

249

we have

or

(III.12l) where E av

E(l) +E(2)

=

n

n

2

n

whence

(III.122) Again, as in the case of volume distribution, we .have distinguished two types of charges. Let us illustrate these results by one very simple example. Suppose an elementary free charge with density Do is placed in a uniform medium with the dielectric constant 13. In accordance with Coulomb's law the free charge creates in the dielectric the same electric field as in free space. 1 ooL q p Eo = -----dV 41T13o L~p

(III.123)

where the point q characterizes the position of the free charge. Since the medium is uniform, there is only one type of bound charge 1-13

= _ _r Do 13 r

O~l)

(III.124)

which arises around the free charge, and its field is

E

= b

_1_ o~l)Lqp 41T13 o

dV =

L~p

_1_ ( 1 41T13 o

13 r ) 00 dV L

L~p

13 r

(III.12S) qp

Therefore, the total field caused by both charges is 1 ( 1 E = Eo + E b = - - 1 + 41T13o

-

1

) Do dV

13 r

-3--Lqp L qp

Jr

E(p)

=

1 oodV -4- - L L qp 3 1T13

qp

(III.126)

250

III

Electric Fields

Comparing Eqs. (1II.l23) and (III. 126) we see that when a free charge is placed in a uniform dielectric medium, the electric field turns out to be Sf times smaller than if this charge were in free space, and this occurs because the free charge is accompanied by bound charge having the opposite sign. Now we will derive the system of equations for the potential. As before, from the first equation curl E = 0 it follows that E = -gradU

and substituting this into the second equation of the system 011.118) we have div sgrad U =

-

00

or

V( sVU)

=

-

00

(III.127)

Certainly, this is neither Poisson's or Laplace's equation, but if in the vicinity of some point. a dielectric is uniform, then we again obtain Poisson's equation.

or

(111.128) If in addition free charge is absent, then at such points the potential becomes a harmonic function, and correspondingly we have

Performing differentiation on the left-hand side of (III.127) we obtain

or 00 VU' Vs V 2U= - - - - - s s

(111.129)

and comparing this with Eq. (111.121) we see again that the Laplacian characterizes the density of total charge located in the vicinity of a point. Inasmuch as the continuity of tangential components of the field is provided by continuity of the potential and taking into account the fact

251

IlIA Electric Current, Conductivity, and Ohm's Law

that D

n

=EE

n

=

au an

-E-

the system of equations for the potential is Coulomb's law

1 I V(EVU)

I U(1) =

U(2)

=

-8 0

(III.130)

I

I

aU2

so,

an

an

E--E-=I 2 1 0

on Si We can show that Eqs. (III.130) together with boundary conditions constitute boundary-value problems for the time-invariant electric field in the presence of dielectrics. In conclusion of this section let us make an additional comment. Various types of rocks have different dielectric constants; in particular, the parameter E f for water is 81, and oil is approximately characterized by 2-3. Such differences can, in principle, be used to distinguish formations that are saturated by oil or water, for example, in measuring the electric field in wells. However, due to the fact that rocks are both dielectrics and conductors, it is impossible to measure their dielectric constant by making use of a time-invariant electric field. To illustrate this, suppose that a charge is placed in a nonconducting borehole. Then induced charges arise on the surface of the borehole that cancel the primary electric field in the rocks (the electrostatic induction), and correspondingly the polarization inside the rocks vanishes. In the next section we will also show that in the case of constant currents, the dielectric constant of rocks does not have any influence on the behavior of the field.

IlIA Electric Current, Conductivity, and Ohm's Law In considering electrostatic induction we have established that a field caused by charges located outside of a conductor induces charges on its surface in such a way that the total field vanishes at every point within the

252

III

Electric Fields

conductor. We can also say that the field of external charges is not able to cause ordered motion of either electrons or ions, and consequently they move at random as if these charges were absent. Now we will suppose that with the help of other electric charges situated inside and on the conductor surface there arises a constant electric field, which in general can change within the conductor, but it does not vary with time. Of course, as before, this field E obeys Coulomb's law. If we knew the distribution of these charges, then the electric field at every point within the conductor could be calculated from 1 s dV E(P)=-4-f3 L qp 7TE O V L qp

1

IdS

+ -4 - fL 3- L q p 7TE O S

qp

Later we will formulate conditions that guarantee existence of constant electric field within a conducting medium. For now let us accept this fact and begin to study the movement of charges with a constant velocity through a conductor. This is the third phenomenon, after electrostatic induction and polarization of dielectrics, which we are going to consider in detail. It is obvious that if some charge is subjected to a constant electric field, it will start to move. Since the field E does not change with time, an ordered motion of charges will be observed. This phenomenon is called the electric current. In metals the current is a motion of electrons, while in sedimentary rocks, the pores of which are saturated by electrolytes, the current is composed of ions. Consequently, we will distinguish between electronic and ionic conductivity. Note, however, that in both cases rather complex, random movement of microcharges is accompanied by an ordered motion in some direction defined by the electric field. Perhaps it is proper here to notice that the motion of each microcharge is determined by the magnitude and direction of electric field in the vicinity of the point where this charge is located, and in this sense, it is independent of the field in other places in the conducting medium. The movement of microcharges in rocks is very complicated because their structure is complex, and they consist of elements having both types of conductivity with different values. To simplify the phenomenon of the electric current we will imagine that all microcharges of every sign are the same in the vicinity of some point, and they move with the same velocity. This conventional approach for performing the transition from micro to macro scale allows us to consider the current as the motion of charges distributed continuously at certain places in a conducting medium. In the same manner as we have described the distribution of sources of the gravitational and electric fields, let us introduce the current density j,

253

lIlA Electric Current, Conductivity, and Ohm's Law

which characterizes the movement of charges. This vector shows the direction of charge movement in the vicinity of the point q. It is equal to the amount of charge passing through a unit area with center at the point q, and which is perpendicular to j per unit time. Here there are three conditions, namely, (a) The area has unit value. (b) The area is perpendicular to the direction of charge movement. (c) The time has unit value. All of these conditions are essential. If one of them is not satisfied, the current density vector loses its meaning. In accordance with this definition, the vector j can be written as de

j(q)

= dSdt'io(q)

(III.l31)

where dS is the area perpendicular to the direction of the charge movement. This area is sufficiently small to assume that the same amount of charge passes through every element of this area for the same time. Also de is the charge that passes through this area during the interval dt. Finally i o is a unit vector, showing the direction of charge movement. It is obvious that the dimension of the current density is coulomb

[j] = m 2 sec Also in accordance with Eq. (111.131) we an conclude that the direction of the vector j coincides with that of movement of positive charges and is opposite to the direction of motion of negative charges, since then de < O. In general, charges of both signs can move in different directions, for instance in opposite ones, and correspondingly the total vector j is

(III.132) where j + and j - are the current densities of the positive and negative charges, respectively. As usual, the vector field j can be described with the help of the geometric models introduced in Chapter I, and in this case they are called line currents, surface currents, and tube currents. Now we will express the current density j of the positive and negative charges in terms of their densities and velocities. Suppose that dS is the cross section of an elementary tube current, shown in Fig. III.6d, and charges of both signs move along this tube. Then during the timedt charges located at distances from dS,. which are smaller than the product

254

III

Electric Fields

W dt, cross this area. Here W is the magnitude of the charge velocity. Correspondingly, the amount of charge is de = 8 dV = 8 dS W dt

and in accordance with Eq. OII.13l) the current density is j=8W

Here we see that the current density vector and the velocity have the same direction if 8 > 0, while they are opposite to each other if 8 < O. In general, both types of charge are involved in movement and then we have j+=W+8+,

(III.133)

or j = W+5+ - W-15-1

where 5+, W+ and 5-, W- are densities and velocities of the positive and negative charges, respectively, and 15-1 is the absolute value of the negative charge. Here it is appropriate to make several comments.

1. The field j does not change under a simultaneous change of the sign of both velocity and density. For instance, replacing a movement of negative charges by that of positive ones with the same magnitude but opposite direction, the field j remains the same. This shows that the movement of charges can always be reduced to the movement of positive ones only. 2. In metals, electrons form the current and j = W- 8-. In electrolytes, however, both positive and negative charges move, in opposite direction. Therefore,

where i o is a unit vector, showing the direction of movement of positive charges. 3. In accordance with Eq. (IILl33), the current density is not equal to zero if (III.134) This means that the current density j can vanish provided that both charges move in the same direction, but the equality j = 0 does not require that densities should be equal.

IlIA Electric Current, Conductivity, and Ohm's Law

255

4. The total charge density inside of every elementary volume of a conductor is

It is interesting to notice that often in the presence of current flow this total charge equals zero. In a uniform part of a metal, for instance, the sum of moving electrons and unmovable positive charges is zero. This is also true for uniform parts of electrolytes. It is clear that in such cases these elements of the volumes do not create an electric field. Now we will establish the relationship between the current density and the electric field. As the field E is applied to charges, they begin to move with some acceleration. However, the medium hampers their movement, essentially limiting the increase in velocity. The mechanism producing this phenomenon can vary. In metals, for instance, this occurs due to the collision of electrons with ions of the crystal lattice. But regardless of the mechanism, the velocity of microcharges becomes proportional to the electric field. Correspondingly, we have (111.135) Where u + and u - are positive coefficients called the mobilities of the positives and negative charges, respectively. Note in particular that in metals u + = O. Substituting these expressions into Eq. (111.133) we obtain

or j = yE

(111.136)

and (111.137) Equation (III.136) is called Ohm's law in differential form, and the coefficient of proportionality y is the conductivity of the medium. Here let us make the following comments:

1. Ohm's law establishes the relationship between two completely different fields, namely the electric field E and the current density describing the movement of charges. 2. In accordance with Eq. (111.136) the electric field causes a current density field, but not vice versa. For example, in a nonconducting medium

256

III

Electric Fields

there can be an electric field while current is absent. Let us illustrate this fact. 1

charges

I

~

~

electric field E

current density j

which clearly shows that charges create the electric field E, and in a conducting medium this field generates movement of charges, that is, the field j. It is appropriate to notice that sources of the electric field do not usually have any relationship to the charges that constitute the current density field. 3. It turns out that Ohm's law in differential form can also be applied to alternating electromagnetic fields. 4. In an anisotropic medium the conductivity depends on the direction of charge movement. In this case, y is a tensor. However, in this chapter we will consider only isotropic media and for this reason y is described by a scalar. 5. As follows from Eq. (111.137) the conductivity is directly proportional to both the charge density and mobility, and the latter characterizes the velocity of charges. For instance, the electron mobility in copper is approximately mysec

u-"'" 4.4 10- 3 -

-

Vim

This is a surprisingly small number, especially when we take into account the fact that the electric field in metals is very small. If we suppose that lmv E= - m

V

= 10- 3 -

m

then the velocity of electrons is

This shows that it takes almost 70 hours for the electron to travel 1 meter. Certainly when energy is transformed from generators to loads, it does not travel through wires, otherwise it would take years and years instead of the practically instantaneous propagation observed in reality. Thus, the high conductivity of metals is mainly caused by the high density of electrons that compensates for their low mobility.

257

llI.4 Electric Current, Conductivity, and Ohm's Law

Ions of electrolytes have an even smaller mobility, approximately 10- 9 m and this fact along with their lower density results in much smaller conductivity for electrolytes than for metals. 6. Ordered movement of charges under the action of the electric field is always accompanied by a random motion with a relatively high velocity. 2/Vsec,

m

m

10 2 < W< 106 sec sec

,

which exceeds by many orders of magnitude the velocity of charges forming the current. Due to this random motion, even in the absence of the electric field we can observe a very weak current, which is called the fluctuating current. It is obvious that the existence of such currents defines the limits of sensitivity of devices measuring the current. 7. It is convenient to mention along with conductivity 'Y the specific resistivity or simply resistivity p, which is related to 'Y by

1

p= -

(III.B8)

'Y

The dimensions of these quantities are [ 'Y ]

=

coulomb V m sec

Vm sec

and

[p]=--

and

[p]

coulomb

These units will be presented as

[y] = mhoyrn

=

ohm' m

The values of rock resistivities for metals, sedimentary, and other types of rocks are presented in the following table. The most important feature of this table is the extremely wide range of the resistivity. Rocks and sediments Limestone (marble)

Ores 12

> 10 > 10 10

Pyrrhotite

10 5_10

3

Chalcopyrite

10- 4_10- 1

Rock salt

10 6_10 7

Graphite Shales

10- 3_10 1

Granite

5000-10 6

Pyrite

10- 4-10 1

Sandstones

35-4000

Magnetite

10- 2_10 1

Moraine

8-4000

Haematite

10- 1_10 2

Limestones

120-400

Galena

10- 2-300

Clays

1-120

Zinc blende

Quartz

[After Parasnis (1979)]

> 10 4

258

III

Electric Fields

b

a

(2)

s

(1)

d

c -(2) L E =n 2E O

-(1)

E

L

=-2E

n

O

Fig. 111.7 (a) Current density flux; (b) current density flux near surface charges; (c) continuity of the normal component jn; and (d) Kirchhoff's law.

In the previous sections we have described in detail basic features of the electric field caused by charges. By analogy let us study the behavior of the current density field. First of all, knowing the vector j, it is a simple matter to calculate the amount of charge passing per unit time through an elementary surface dS, arbitrarily oriented with respect to j. As is seen from Fig. 1II.7a this is define by dI <J: dS

or

(111.139) dI

=

j dS cos(j, dS)

where dS = dS n, and n is the unit vector directed from the back to the front side of the surface dS. It is clear that this amount of charge is positive if the charges move along the normal n, and it is negative if the charges pass through the surface in the opposite direction. In particular, if the current density vector is tangential to the surface, the amount of charge dI equals zero. As follows from Eq. (111.139), the flux of the vector, j, or the amount of charges passing through an arbitrary surface S per unit

IlIA Electric Current, Conductivity, and Ohm's Law

259

time, is

(III .140) where i; is the normal component of the current density. In accordance with Eq. (111.140) the flux I is an algebraic sum of elementary fluxes through various parts of the surface, and it is called the current. As follows from the definition of current density, the dimensions of current are coulomb

[1] = - - - = ampere sec

To visualize better the movement of charges it is natural, as was already mentioned, to use the concept of vector lines. Every vector or current line shows the direction of the vector j; that is, along such lines charges move in a conducting medium. Also it is convenient to consider current tubes, whose lateral surfaces are called the current surfaces. By definition the flux of current density through current surfaces is always equal to zero since the vector j is tangential to these surfaces. In general, the cross section of a current tube can vary from point to point. Now we are prepared to formulate the basic law that characterizes the behavior of the current density field, namely the principle of charge conservation. For the time-invariant field it has the form

~j. s

dS = 0

(111.141)

That is, the flux of the current density j through any closed surface is equal to zero. Inasmuch as the current density vector describes the amount of moving charge, Eq. 011.141) allows us to make the most important conclusion about its behavior: The amount of charge arriving at any volume V is always equal to the amount of charge that leaves this volume in the same time; that is, their sum equals zero. This fact is independent of the size and shape of a surface S, surrounding an arbitrary volume V. It also does not depend on whether a uniform or nonuniform medium is considered. In particular, the surface S can intersect media with different conductivities. Moreover, Eq, (111.141) remains practically valid for a wide range of electromagnetic fields when the electric and magnetic fields, as well as the current density, change with time. As follows from Eq. (111.141) at any closed surface there are places where the normal component jn is directed outward, as well as places

260

Il1

Electric Fields

where the normal component is directed inward, and this provides flux in both directions. Suppose for a moment that Eq. (111.141) were incorrect. In this case the amount of charges arriving into the volume would not be equal to the amount leaving it. Then we can imagine two cases. 1. The amount of positive charge that arrives at the volume exceeds the amount leaving; that is, the magnitude of negative flux of the current density is greater than the positive flux magnitude. Then, during every second new positive charge arrives in the volume and accumulation of charges takes place. This means that the electric field as well as the current density would change with time. 2. The amount of positive charge leaving the volume is greater than the amount that arrives, and correspondingly the total charge within the volume would decrease. Again the electric field caused by this charge, as well as the current density, becomes a function of time. This analysis vividly demonstrates that Eq. (III.141) has to be valid, otherwise the electric field cannot be time-invariant. Equation OII.141) can be treated as the second field equation of the vector field j in integral form, but it is usually called the principle of charge conservation. There is a very good reason for this name. Indeed, if inside the volume we have a charge, then in accordance with Eq. (III.141) it remains the same at all times, as long as a time-invariant electric field exists. In particular, this charge can be equal to zero in spite of the presence of flux of current density. Let us again emphasize two essential features of the principle of charge conservation, Eq. (III.14l). 1. Because the flux of the vector j through any closed surface S equals zero, it is possible to preserve the constant electric and current density fields in a conducting medium. 2. Equation (111.141) does not mean that within a conducting medium constant charges are absent. This occurs if the equality

is valid for any surface, but the latter does not have any relation to the principle of charge conservation. Now we will derive other equations describing this principle. Applying Gauss' theorem for regular points of a medium we have

~ j . dS = s

Jv divj dV

111.4 Electric Current, Conductivity, and Ohm's Law

261

Since this is valid for any volume, we obtain divj

=

0

(III.142)

This equation also expresses the principle of charge conservation, but it can be applied only in the vicinity of points where the first derivatives of current density components exist. For instance, in the Cartesian system we have

To derive a surface analogy to Eq. (III.142) we will consider the flux of the current density through a closed cylindrical surface as shown in Fig. III.7b, which consists of three parts.

Since dS l = - n dS, dS z = n dS, and the third term disappears as the cylinder height tends to zero, we obtain

or (III.143) That is, the normal component of the current density is a continuous function across an interface between media with different conductivities. It is proper to emphasize the importance of this condition, which allows us to preserve the time-invariant electric field in a conducting medium. In fact, if the normal components of the current density were not equal at either side of the interface, then during every second we would have either an increase or a decrease of surface charge, and correspondingly the electric field would not be time-invariant. Also from Eq. (111.143) the presence of surface charge becomes evident. In other words, without the field caused by these charges it is impossible to provide continuity of the normal component of the current density and the principle of charge conservation would not be valid. To show this we will imagine that in the vicinity of some point q of the interface between media with conductivities and "z, the surface charge is absent (Fig. III.7c). Then, in accordance with Ohm's law the normal cornponents

"1

262

III

Electric Fields

of the current density are j~l)(

q) = 1'1E~l)(

q),

where E~l) and E~2) are the normal components of the electric field caused by all charges located outside of this surface as well as those situated on this surface, except at the point q, since we have assumed that in the vicinity of this point charge is absent. Inasmuch as all these charges are located at finite distances from point q, the field caused by them is a continuous function across the interface; that is, E~l)(

q) = E~2}(

q)

Then, taking into account the fact that 1'1 =F 1'2 we have to conclude that it is impossible to provide the continuity of the normal component of the current density at some point of the interface if surface charge is absent in the vicinity of this point. Thus, we have derived three forms of the second equation of the current density field, and all of them express the principle of charge conservation. divj

=0

"(2) _ -

In

·(l)

In

(111.144)

Let us also consider one special case of a current distribution: conductors are connected together at some place as shown in Fig. III.7d. This is often observed in electrotechnical practice. Then, applying Eq, (111.141) to the surface S, surrounding the point of connection of conductors and bearing in mind the fact that the flux of current density outside conductors vanishes, we have (I1I.145)

where S, is the cross section of i-conductor, and I, is the current. As is well known, Eq. (I1I.l45) expresses the first Kirchhoff law. Now we will derive the first equation of the field j. Making use of Ohm's law we have curlj = curl yE = I' curl E + VI' X E Since curl E = 0 we have at regular points of the medium, curlj = grad I' X E

(111.146)

That is, vortices of the field j are located at places where the electric field has a component in the direction perpendicular to grad y.

263

rnA Electric Current, Conductivity, and Ohm's Law

Taking into account the continuity of the tangential component of the electric field and Ohm's law for the surface analogy of Eq. (III.146), we have £(2) = £(1) = E t t t or j[Z)

jP)

Yz

Yl

(III.147) Thus (lII.148) and the right-hand side of this equation characterizes the distribution of surface vortices. In summary we will present the system of equations of the current density field j. Principle of charge conservation

Ohm's law

1 I

curlj

=

Coulomb's law

1

V'y X E

divj = 0

I

(III.149) Proceeding from Ohm's law we have j

=

-y grad U

and correspondingly the current density field is a special case of the vortex Ield, called a quasi-potential field (Chapter I). Thus in a conducting medium there are simultaneously two completely Iifferent fields, namely, the electric field caused by electric charges and he current density field arising due to the electric field. Inasmuch as in all electrical methods of applied geophysics the study of the electric field in a .onducting medium is of great practical interest, we will consider this field n detail later in the chapter.

264

III

Electric Fields

Taking into account the fact that the field E is caused by charges only, let us make use of results derived in previous sections. Therefore, the system of equations for both the field and the potential is Coulomb's law

II

on S

on S

(111.150)

U(2) -

U(I) =

0

au(2)

aU(I)

an

an

on S

on S

U(a) - U(b)

=

tE. dl' a

~

111.5 Electric Charges in a Conducting Medium

265

where 0 and 4 are the total density of volume and surface charges that include both the free and bound types of charge. In the next section we will study their distribution.

IlLS Electric Charges in a Conducting Medium

Now that we have discussed the principle of charge conservation, let us investigate the distribution of charges in a conducting medium, which always accompanies the current density field. Making use of Ohm's law we have, for regular points, divj = div yE = y divE + E· grady = 0

(111.151 )

Taking into account the second equation of the electric field

s

divE = 80

and substituting into Eq. (111.151) we obtain an expression for the volume density of charge. 0=

E . grad y (III.152)

-80----

Y

Thus volume charges arise only in places where the conductivity of the medium varies, and where the direction of its maximal rate of change is not perpendicular to the electric field. In accordance with eq. (111.152) the sign of charges depends essentially on the angle between the field E and grad y. The most typical model of a conducting earth, used in applied geophysics, is a system of uniform formations each having a different conductivity. In this case, the volume density of charges equals zero. Next we are going to study surface charges. We will make use of the surface analogy of the second equation, OIl.149).

Again applying Ohm's law we will represent this equation as

=t[(YI +Y2)(E~2)-E~1)) +(Y2-Yl)(E~2)+E~1))]

=0

(111.153)

266

III

Electric Fields

Inasmuch as

from Eq. (III.153) it follows that

~(q)

=

2s o 1'1 - 1'2 E~v (q)

=

2s/ 2 - PI E~v (q)

1'1+1'2

(III.154)

P2+PI

where E~V(q) is the average value of the normal component of the electric field in the vicinity of the point q. E av ( n

E(l)(q) q

)

=

n

+ E(2)(q) 2

n

Taking into account Eq, (111.13) we have E(1)( n

q

)=E'-q-~(q) n

2' So

where E~-q is the normal component caused by all charges except these located in the vicinity of the point q. Whence, (III.l55) Thus the density of the surfacecnal'ge is directly proportional to the contrast coefficient. .·• _PZ-PI K 12-

(III.l56)

pz +PI

and the normal component E:.-q andEq. (UI.154) can be rewritten as (III.157) This study shows that in. most :practical cases the field E, measured by electrical methods, is mainlYQIDsetl by charges arising at interfaces between media havingdIl~~~t ~si8tMties. This fact is the physical foundation for applicatioos of11k"t: "l¢¢trieal methods in geophysics. Here it is appropriate to .~ .•~ •.comments. 1. In general in tpe \1idmi~ m::lim.Y point of a conducting medium it is natural to distinguish _ ~ ·:(if ~tra1;ges, namely, (a) Positiveandn¢~;')," l~~:tQr instance, ions in electrolyteswhich constitute the ~c~ltd9Ve in opposite directions. In metals, however, onlynegathre e:~ ,fe'1'ettrons) are involved in this movement,

III.S Electric Charges in a Conducting Medium

267

while the positive ones remain at rest. It is essential to note that in both cases the sum of these charges in every elementary volume of a conductor equals zero; and correspondingly, as we already pointed out, these charges do not contribute to the electric field. (b) Volume and surface charges, which usually appear in the vicinity of points where the conducting medium is not uniform. If the electric field remains constant these charges do not also change with time, and they are the sale sources of the electric field. 2. Charges, as well as the electric and current density fields, do not arise instantly. It always requires some time for their establishment, and during this time interval the behavior of the field is governed by electromagnetic laws. Now we will consider two examples illustrating the distribution of charges.

Example 1 Current Electrode in a Uniform Medium

Suppose that an isolated wire with current I is connected with a uniform medium through a spherical electrode, as is shown in Fig III.Sa. The resistivities of the electrode and the surrounding medium are Po and p, respectively. Inasmuch as both media are uniform, volume charges are absent, but there is one interface between the electrode and the surrounding medium where surface charges arise. It is clear that due to the spherical symmetry, the current density, as well as the electric field, has only a radical component and in accordance with Eq. OII.19) we have (III.158)

where E~2) and E;1) are radial components of the electric field at the front and back sides of the electrode surface, respectively, while I is the total density of the charge at this surface. Taking into account the continuity of the normal component of the current density, Eq. (III.143), and the Ohm's law, we obtain

or (III.159)

268

III

Electric Fields

Fig. 111.8 (a) Current electrode in a uniform medium; (b) illustration of a charge density calculation; (c) distribution of charges as a function of radius;and (d) current tube.

Therefore, the total charge on the electrode surface is

or (III.160) since all current I, coming from the wire, leaves through the electrode surface except at the place where the wire is connected to the electrode. In accordance with Coulomb's law this charge creates an electric field with only a radial component, and its magnitude is

where Lop is the distance from the electrode center to the observation point p. Inasmuch as the resistivity of a formation is usually many orders

111.5 Electric Charges in a Conducting Medium

269

greater than that of the electrode, p » Po' we can write e = EopI

and

E(p) =

(III.161)

pI --2-

47TLop

Thus we have demonstrated that the charge situated on the electrode surface and surrounded by a conducting medium creates exactly the same field as if it were located in free space. It is also obvious that under the action of this field, regardless of the distance from the electrode, ions in rocks move along the field and the current density is I

j=yE=--

(III.162)

47TL~p

Of course, this result directly follows from the symmetry and the fact that all current I goes through the spherical surface with center 0, located at the middle of the electrode. In essence we have described both fields, E and j, and illustrated again the known relation ~h

~

~

the electric field ~

the current field j

It is obvious that the xitential of the electric field is

pI

U(p)

=

4L

(III.163)

Op

7T

and, correspondingly, equipotential surfaces are spherical, including the electrode surface. Thus the difference of potentials, that is, the voltage between two arbitrary points, is

[1

U( a) - U( b) = -pI 47T

LOa

- -

1]

Lab

(III.164)

Now let us resolve one paradox. As follows from Eq. (IILl60) the charge on the electrode surface is defined by the current I and resistivity of a medium p (p »Po), but it is independent of its dielectric constant. This means that if the dielectric constant of the medium is changed, the charge e, and therefore the electric field E, does not change. At the same time, the expression of the density k, Eq. (III.159), includes both the free

270

III

Electric Fields

and bound charges and, as was shown in the previous section, the bound charges depend on the dielectric constant. It seems that these two factors contradict each other. However, unlike insulators, in a conducting medium both the free and bound charges are subject to the influence of the dielectric constant, and this affects the charges in such a way that the total density of charge k becomes independent of 6. We will demonstrate this interesting phenomenon, assuming that the resistivity of the electrode is very small. Then instead of Eq. (111.159) we have (1II.l65) where k = k o + kb is the sum of the free and bound charges. Next we will express each type of charge through the current density j, and the electrical parameters P and 6. In accordance with Eq, (l1I.12l) we have, for the free charge,

where 6 I and 62 are the dielectric constants of the electrode material and the surrounding medium, respectively. Then, making use of Eq, (111.165) we obtain

ko=

6

av

pJ'r

+ (6 2 - 6 1 )Erav

or ~ av , .' ( ) P . . """'0=6 PJr + 6 Z-6 1 "2Jr=6zPJr

(III.166)

since E av r

= pjr

2

(1II.l67)

Thus the density of free charge depends on the dielectric constant of the surrounding medium only. But there is also bound charge and, as follows from Eq. (III.122),

III.S Electric Charges in a Conducting Medium

271

Or, taking into account Eqs. (III.16l), (III.167), we obtain

Thus (III.168) Therefore, the densities of the free and bound charges are subjected to the influence of the dielectric constant of the surrounding medium, but the density of the total charge, which defines the electric field, depends only on the resistivity of the medium for a given current. It is easy to see that this result is directly applicable to an arbitrary medium and any arrangement of current electrodes. The fact that the electric field in a conducting medium is independent of its dielectric constant is of great importance for the use of electrical methods in geophysics.

Example 2 Induced Charges on the Plane Interface We will assume that the current electrode A is surrounded by a conducting medium with resistivity PI' and at distance d there is a plane boundary with another medium having resistivity pz (Fig. Ill.Sb), The current of the electrode is 1. As we know, charges arise on the electrode surface and on the boundary, and both of them create an electric field at every point of the medium. Let us restrict ourselves to a study of the charge distribution only. In accordance with Eq, (111.160) the charge on the electrode surface is

272

I1I

Electric Fields

To calculate the charges on the plane interface we will make use of Eq. 011.157). (111.169) where K 12 = (P2 - PI)/(P2 + PI) and E~-q is the normal component of the field caused by all charges except the charge in the vicinity of point q. Now a great simplification takes place because the interface is planar. All induced charges on this surface that are located outside the point q do not create a normal component of the field at this point, as directly follows from Coulomb's law. In other words, E~-q is defined by the charge on the electrode surface, only.

We will choose a cylindrical system of coordinates with its origin at the center of the small spherical electrode A and with its z-axis perpendicular to the interface (Fig. III.8b). Then, in accordance with Coulomb's law the normal component of the field at any point of the boundary is (III.170) where rand d are cylindrical coordinates of the point q. Correspondingly, for the surface density k we have K 12 eA

k(q) =

~

d (r 2 + d 2 )3/ 2

(111.171)

As follows from this equation the maximal density is observed in the vicinity of the point r = 0, and then it decreases with an increase of r. At the beginning, as r < d, the density decreases slowly; but later, as the distance r exceeds d, this process occurs very rapidly, and k decreases almost as l/r 3 • Also from Eq. (111.171) it follows that the closer the electrode A approaches to the interface, the higher the density becomes in the vicinity of the point r = 0 and the more rapidly it decreases with an increase of r (Fig. III.8c). When the current electrode is placed on the interface, the density tends to infinity at the point r = 0, z = 0, while it vanishes at other points. Now we will calculate the total charge induced on the plane interface. Making use of the axial symmetry, it is convenient to first calculate the elementary charge des' induced on the ring with radius r and thickness dr.

IlLS Electric Charges in a Conducting Medium

273

Since its area is dS

=

211"rdr

we have

Integrating with respect to r from zero to infinity we obtain

Thus, the total charge arising at the interface is

(III.In) That is, its value does not depend on the distance of the current electrode from the interface, which influences only the charge distribution on the surface. As the electrode approaches the surface, induced charges become mainly concentrated near point r = O. In the limit as d tends to zero, both charges eA and es are located on the electrode, and its total charge is equal to

while the charge at the plane interface disappears. In accordance with Eq. (III.156) the coefficient K 12 varies within the region and consequently the surface charge e s cannot exceed in magnitude the electrode charge eA" If the medium where the electrode is situated is more conductive, then both charges e A and e s have the same sign but in the opposite case, where PI> P2' they have different signs. If the second medium is either an ideal conductor P2 = 0, or an insulator, pz = 00, the surface charge equals -eA and eA' respectively. Therefore, if the electrode is placed at the interface with an insulator such as air, its charge is doubled, and it equals e

A

PI!

=-

211"

(III.I74)

In all electrical methods, as current electrodes are located on the earth's surface, their charges are defined by Eq. (III. 174).

274

III

Electric Fields

111.6 Resistance

In this section we will continue to study the electric and current density fields and the relationship between them. Bearing in mind that in geophysics, unlike electronics, we mainly deal with volume conductors, let us discuss in detail the concept of resistance. We will consider an arbitrary element of tube current C, which is confined by two cross sections 51 and 52 and the lateral surface 5 f (Fig. III.8d). In accordance with Eq. (III.164) the difference of potentials, or the voltage, between two equipotential surfaces 51 and 52 is VIZ

= U(1) - U(2) = fE' d/

(III.175)

1

or VIZ

= U(l) - U(2) =

fpj· d/ I

It is obvious that since the electric field is a source field, the integral on

the right-hand side of Eq. (III.175) is path independent, and the integration can be performed along any path t having terminal points on surfaces 51 and 52' Notice that along these current lines t , which are longer and pass through more resistive parts of a medium, the magnitude of the current density is smaller than that along shorter lines passing through more conductive regions (Fig. III.9a). In other words, in general the current density is not uniformly distributed over the cross section of the current tube. Let us represent Eq. (III.175) as

U(1) - U(2) =IR I 2

(III.176)

where R 12 =

J:

2 dl' ( p--

J]

I

(III.I77)

R 12 is called the resistance of the current tube element C as the current goes from the surface 5] to 52' It is clear that the dimensions of Rare

[R] = ohm Now we will make several comments, elucidating the concept of resistance. 1. The resistance is always positive, and as the path of integration t shows the direction from 5] to 52' we assume that I> O. If the opposite

III.6 Resistance

275

Fig. III.9 (a) Distribution of current density in a conductor; (b) electric field in a uniform cylinder; (c) system of uniform cylinders; and (d) change of position of equipotential surface.

direction of t is chosen, then it is necessary to change the sign of the current. 2. The resistance R 12 , defined by Eq. (III. 177), makes physical sense only if both surfaces S 1 and S 2 are equipotential surfaces and S t is the lateral surface of the current tube. Indeed suppose that the surface S is not an equipotential surface. Then changing the position of a point "1" on this surface, the voltage and therefore the resistance of the conductor C would change, and R 12 would become meaningless. This also happens if the surface S t is not a lateral one, since in this case the current I depends on the cross section S of the conductor, while in Eq. (III.17) it is assumed that I is constant. 3. As follows from Eq. (nU77) the function R 12 characterizes the ability of the conductor C to resist the passage of the current. It is numerically equal to the difference of potentials V(l) - V(2), as the current magnitude has unit value. 4. To determine the parameters that define the resistivity R 12 , we will make use of the theorem of uniqueness, assuming that the conductor is uniform. However, later we will demonstrate that our conclusions remain valid even in the general case, where resistivity changes arbitrarily within a conductor.

276

III

Electric Fields

It is clear that the potential of the electric field satisfies the following conditions: (a) U is a solution of Laplace's equation inside the uniform conductor

V 2U = 0 (b) On the lateral surface 5 f' the normal component of the electric field is zero, and therefore

au

-=0

5f

on

an

(c) On the equipotential cross sections 5] and 52 the potential is such that 1 au - ( -d5= -I, p

1 au -f -d5=-I P an

ls, an

S2

As follows from the theorem of uniqueness, all three conditions uniquely define the electric field E inside the conductor C. In other words, there can be only one distribution of the field if the current is given. Suppose that the current equals I] and correspondingly the electric field is E], which of course satisfies all three conditions. Now let us increase the current m times, that is, 12 = ml., It is easy to see that such a change results in an increase of the magnitude of the electric field by a factor of m at every point of the conductor, but it does not change direction. The new field is E 2=mE] In fact, the potential of the field E 2 is U2 =mU] and U2 as well as U] is a solution to Laplace's equation. At the same time this field obeys boundary conditions on the lateral and cross-section surfaces since

aU2

au]

an

on

-=mThus, in accordance with the theorem of uniqueness we can say that the function E 2 describes the electric field in the conductor C if the current through its cross section equals 12 . This analysis shows that the ratios E

I

pj

or

I

III.6 Resistance

277

are independent of the current magnitude. In other words, the field of the vector j/I in the volume V is determined entirely by its boundaries 5 t , 51' 52' and the distribution of resistivity p. Thus, in accordance with Eq. (III.177) the resistance R of the conductor C is completely defined by its dimensions, shape, and resistivity. It does not, however, depend on the current or any changes in other parts of the circuit if surfaces 51 and 52 remain equipotential surfaces. This means that by defining the resistance R 12' we will imply only such ways of introducing the current into an electrical circuit that preserve constant potential on these surfaces. 5. To determine the resistance it is necessary to know the current density field, and in general it is a complicated task related to the solution of a boundary-value problem. 6. Equation (III.176) is the integral form of Ohm's law. Thus we have derived two forms of this law. j

=

yE

and

U(I) - U(2)

=

R 121

As will be shown in the next section these equations are valid only for so-called external parts of the electric circuit. 7. If the magnitude of the vector j has the same values within every section of the conductor, perpendicular to the current, then we have

J: d/

i dt

dt'

--=-=-

I

I

5

and (III.178) In particular, if the conductor is uniform,

R'2=P

f

2 dt'

1

S

(III.179)

It is appropriate to notice that Eqs. (III.178), (III.179) can be used only if

the geometry of the current density field is known, since 5 are-sections of a conductor perpendicular to the vector j. 8. Suppose that the conductor C is a uniform cylinder of arbitrary shape, confined by a lateral surface and two equipotential cross sections 51 and 52 (Fig. III.9b). In this case the electric field behavior is remarkably

278

ill

Electric Fields

simple; namely, it is constant within the conductor and tangential to its lateral surface. Indeed, the potential of such a field is a linear function and therefore satisfies Laplace's equation as well as the boundary conditions.

au

-=0

an

and

Thus, in accordance with the theorem of uniqueness, this function E is a uniform electric field. Correspondingly, we can conclude that the current density is also uniform and the expression for the resistance, Eq, (111.179), is drastically simplified.

(111.180) where t and S are the length and the cross section of the cylinder, respectively. As follows from the previous section, the electric field E within the uniform cylinder is caused by charges located on its surface. As we just proved, they are distributed in such a way that the electric and current density fields are uniform and parallel to the lateral surface. Often a conductor consists of several parts, and each one of them is a uniform cylinder with a length that significantly exceeds the dimensions of its cross section. Such elements of a circuit are called quasi-linear, and it is obvious that their resistance can be calculated from Eq. (111.180) (Fig. III.9c) 9. It is clear that if the resistivity of the cylinder, p, does not change within each cross section but varies along the conductor, then the current density field j still remains uniform and tangential to the lateral surface St. At the same time the electric field changes along the cylinder but remains constant within every cross section, and it is directly proportional to the resistivity. I

E=pj=pS

Correspondingly, instead of Eq. (111.178) we have the following expression

III.6 Resistance

279

for the resistance of a cylindrical conductor confined by equipotential surfaces 51 and 52' R 12 = -I

5

f2 pdt

(III.18I)

1

10. As was emphasized earlier, the resistance of the conductor C has meaning if surfaces 51 and 52 are equipotential surfaces, regardless of any changes in other parts of the circuit. At the same time it is very simple to perform modifications outside of the conductor C that result in a change of the potential over surfaces 51 and 52' or one of them. For instance, let us introduce an inhomogeneity T, shown in Fig. III.9d. As we know, electric charges arise on its surface, and they create a field that is certainly not perpendicular to the surface 52' It is also clear that in approaching the inhomogeneity to 52' its influence becomes stronger. However, there are cases when the surfaces 51 and 52 can be only equipotential surfaces. For instance, this occurs if they are either surfaces of ideal conductors or cross sections of linear parts of the circuit, as shown in Fig. IIUOa and b. Of course, this conclusion remains practically valid even in those cases when the volume V is confined by conductors that have great conductivity. Now we will consider three examples illustrating the concept of resistance.

Example 1 Grounding Resistance of a Spherical Electrode In the previous section we have shown that the electric field caused by charge on the electrode surface, possesses spherical symmetry, and its potential at any point p is

pI

U(p)

=

4L

(III.I82)

tr Op

Consider two spherical and equipotential surfaces with radii LOa and LOb' respectively (Fig. IIUOc). Then the difference of potential between them is

[1

I]

pI - - U(a)-U(b)=4rr

LOa

LOb

280

III

Electric Fields

Fig. 111.10 (a) Resistance of volume when its cross sections have contact with a very conductive medium; (b) resistance of volume when its cross sections have contact with quasi-linear conductors; (c) illustration to calculation of grounding resistance; and (d) potential distribution in a uniform half space.

Therefore, the resistance of this spherical layer is R

b a

= U(a) - U(b) = ~ I

47T

[_1 LOa

1_]

(III.183)

LOb

and it depends essentially on the layer thickness and the distance from the electrode, as well as, of course, the resistivity p. If we mentally represent the whole medium around the electrode as a system of spherical layers with its center at the point 0 (Fig. III.IOc), and take into account the fact that the current I goes through all of them, then we can say that they are connected in series. Correspondingly, the resistance of the whole uniform medium to the current, which moves from the electrode, can be presented in the following way:

III.6 Resistance

or

R

=_P-

e

47Ta

281

(III.184)

where a is the electrode radius. Usually R; is called the electrode or grounding resistance, and by no means does it characterize the resistance of the electrode material or that of the surface between the electrode and the medium, since the influence of both these factors is negligible. In contrast, the grounding resistance shows how all space resists the current leaving the electrode. This is a very useful concept and, in accordance with Eq. (III. 184), the electrode resistance strongly increases with a decrease of its radius. Such behavior can be expected since with a decrease of electrode surface area the current density increases in its vicinity, and correspondingly the potential decreases more rapidly near the electrode. Substituting in Eq. (III.183) LOa = a, we represent the resistance of the spherical layer bounded by the electrode surface and that with radius LOb as

R=Re(I-~)

(III.18S) LOb

The latter shows that if LOb» a, the resistance of this part of the medium differs only slightly from the grounding resistance. For instance, if LOb = lOa, this spherical layer around the electrode contributed 90% of R e , and usually in applied electrical prospecting the thickness of such a layer does not exceed 1-2 m. This consideration allows us to understand that in most cases application of two electrode arrays in electric methods is useless, and we will return to this subject in the next section. The concept of grounding resistance has been described for the case of a spherical electrode placed in a uniform medium. Of course, the grounding resistance arises if the medium is not uniform and the electrode has an arbitrary shape. In particular, if due to electrochemical processes there is a thin layer around the electrode surface with a relatively high resistivity, it can give the main contribution to the grounding resistance.

Example 2 Influence of Grounding Resistance in Measuring Voltage

Until now we have assumed that the electrode is the part of a circuit that supplies a conducting medium with a current I. Now we will consider the role of grounding resistance in measuring the voltage. Suppose we intend

282

III

Electric Fields

to determine the difference of potentials between points M and N located on the earth's surface. VMN = UM

-

UN

The distribution of the current density field as well as the equipotential surfaces are shown in Fig. III.IOd. To find the voltage in a conducting medium we will insert two electrodes in the vicinity of points M and N, connected to each other through a device (voltmeter) measuring the voltage. Here it is appropriate to make one comment; namely, the voltmeter measures the value of the integral (111.186) At the same time the voltage can be described with the help of the potential difference UM - UN' Inasmuch as the integral in Eq. (111.186) does not depend on the path of integration, it can be written as (III.187) where E is the component of the field directed along the straight line between these electrodes. Very often the distance MN is sufficiently small, and we can assume that the electric field does not change within this interval. Then Eq. (111.187) is simplified, VMN=EMN

and the measured value of the voltage of the electric field is easily calculated. Now we will continue to describe the distribution of the potential in the conducting medium in the presence of receiver electrodes M and N. As is seen from Fig. III.lOd the measuring circuit is connected in parallel with the conducting medium. Near electrodes M and N one part of the current goes into this circuit, while the other moves through the conducting medium. In the previous section we demonstrated that if the current density field j intersects the boundary between media with different resistivities, surface charges arise. Therefore, negative and positive charges appear on the surfaces of the electrodes M and N, respectively. This means that due to the current in the measuring circuit the potential of the electrode M decreases, while it becomes greater for the electrode N. In other words, the presence of receiver electrodes connected with each

111.6 Resistance

283

Fig. UI.11 (a) The influence of grounding resistance on measurements; (b) potential of the charge electrode; (c) extraneous force mechanism; and (d) electric circuit.

other, results in a distortion of equipotential surfaces (Fig. HUla); that is,

III

their vicinity (III.188)

where U/:1 and U~ are the potentials of the electrodes M and N, respectively, if some current goes through the receiver circuit. Thus in the vicinity of electrodes, when the current moves from the volume to linear conductor or vice versa, we observe a relatively strong decrease of the potential. and By definition of the resistance, Eq. (III. 176), these differences can be expressed as

(111.189) where I, is the current in the measuring circuit, while R M and R N are the grounding resistances of the electrodes M and N, respectively, which are

284

III

Electric Fields

usually unknown. Performing a summation of these equations we obtain

or (III.190) Thus, the voltage V;;N' which is the input to the measurement circuit, and the voltage V MN characterizing the field in the conducting medium, differ from each other, and this difference is proportional to the sum of their grounding resistances. For this reason it is necessary to take into account the influence of the grounding resistances, and in essence this goal constitutes the basis of all methods of measuring the voltage in a conducting medium. For illustration let us briefly discuss two such methods. The First Method Suppose that at the instant when the input V;;N is measured, the current through electrodes If equals zero. Then in accordance with Eq, (111.190) the influence of the grounding resistance vanishes and

(III.191) To achieve this compensation the receiver circuit has a source of another current, Ie' which at the instant of measurement has the same magnitude as If but the opposite direction. This is why this method is called the compensating method. The Second Method Let us represent the input V;;N as V;;N

=

IfR

where R is the total resistance of the receiver circuit, which includes the wire resistance R w and the internal resistance of the voltmeter. R =R w +R j Then Eq. (III.190) can be rewritten in the form VMN «t.cn, +R w +RM+R N)

(111.192)

where I.R, is the voltage between the terminal points of the voltmeter. If the resistance R, is chosen so that the inequality Rj»RM+RN+R w

III.6 Resistance

285

is met within a range of possible change of the grounding resistance, we have (111.193) That is, the voltage measured by the voltmeter is practically equal to the difference of potentials between the points M and N, as the influence of grounding resistances is absent. Thus using voltmeters with high internal resistance allows us, in essence, to eliminate the effect of R M and R N •

Example 3 Three-Electrode Array We will consider an array that consists of one current electrode A and two receiver electrodes M and N placed along the line (Fig. III.11b). As the current fA goes through the electrode A, a charge

eA

= COPIA

arises on its surface, which creates a primary electric field in a conducting medium.

where Lop is the distance between the current electrode and the tion point p. Also, if there are interfaces between media with resistivities, surface charges appear, which in their own turn secondary electric field E s ' Thus, the total field at any point p is the primary and secondary fields E

=

observadifferent create a a sum of

E p + Es

This vector field is described by a certain distribution of its potential, and the points M and N are located on corresponding equipotential surfaces. Correspondingly, with the help of receiver electrodes of the array, the voltage between them

V MN = U M - UN is measured. To eliminate the influence of a change of current lA' the ratio VMN

t;

UM

-

UN

fA

is calculated, and in accordance with Ohm's law it is equal to the resistance R M N of the medium, confined by equipotential surfaces passing

286

III

Electric Fields

through points M and N, respectively. RMN

VM N

(III.194)

= --

fA

Generalizing this result, we can say that any array in electrical methods measures the resistance of the medium between equipotential surfaces with potentials UM and U w We will consider two special cases where the three-electrode array plays an important role in electrical logging. First, suppose that the electrode N is located far away from the current electrode A so that we can let UN = O. Then we have a two-electrode array AM, which is usually called the normal or potential probe, and in accordance with Eq. (111.194) it measures the resistance of the conducting medium, confined by the equipotential surface, U=UM

passing through the electrode M and a spherical surface of infinitely large radius. Of course with an increase of the distance from the electrode A the contribution of different parts of the medium, confined by equipotential surfaces and having the same thickness, becomes smaller. We will consider the opposite case, when the distance between receiver electrodes is sufficiently small, and we assume that the electric field within this interval is practically constant. Then we have VMN

MN

R MN = - J = -J E A A

(III.195)

and such an array is often called a gradient probe. It is proper to notice that in a uniform medium a three-electrode array measures the resistance of the spherical layer with thickness MN and center at the point A, but in the general case of a nonuniform medium a shape of this layer can be very complicated.

III. 7 The Extraneous Field and Its Electromotive Force

Proceeding from Coulomb's law and the principle of charge conservation, we have established that at the regular points of a conducting medium, for the electric and current density fields the equations curlE

=

0

divj

=

0

(111.196)

111.7 The Extraneous Field and Its Electromotive Force

287

hold. Also it was assumed that everywhere in the medium a linear relation (Ohm's law), j

=

yE

(III.l97)

holds true. It is essential to note that in Eqs. (III.l96), (IIU97) E means the field that is caused by electric charges only, and to emphasize this fact we will often use the index "c": E = E C • However, a study of these equations shows that it is impossible to maintain a constant movement of charges in a conducting medium if the electric field is caused by charges only. To account for the constant charge movement we need to study the extraneous field. Let us discuss this subject from various points of view. First, from Ohm's law, j = 'Y grad U

it follows that

(III.198) where j t is the projection of the current density vector on some line t. Suppose t coincides with one of the current lines that are, in accordance with the principle of charge conservation, always closed. Consider the change of the potential along this line. Inasmuch as in this case the ratio j II 'Y is positive, we have to conclude that the potential continuously decreases along the current line. Therefore, from Eq. (III.198) it follows that after one complete passage along this line the potential at the initial point acquires a new value. In other words, it becomes a many-valued function that of course contradicts the definition of the potential of the Coulomb field. This result clearly indicates that something is wrong with Eq. (III.197), since it is not prudent to assume that the fundamental physical principle divj = 0 is incorrect. Now we will study this problem from a slightly different point of view and demonstrate that a nonzero electric field cannot simultaneously satisfy Eqs. (111.196) if Ohm's law, written in the form j = 'Y E, is everywhere valid. In fact, from the equation curl E = 0 it follows that the electric field does not have closed vector lines. Then, in accordance with Eq. (IIU97) we have to conclude that the current density field j also does not have closed vector lines. On the other hand, from the equation divj = 0 it follows that the field j cannot have open lines and, again in accordance with Ohm's

288

III

Electric Fields

law, Eq. (lII.197), this is also true for vector lines of the electric field. Presenting the results of this analysis in the form of a table; we have

Field E

Open line

Closed line

NO

NO NO

NO

We have to come to the conclusion that only a zero field can satisfy Eqs. (111.176), since the absence of both types of vector lines simply means that fields E and j vanish. Taking into account the fact that both equations of the system (III.l96) have been derived from physical laws, we are left with one choice only, namely, that it is necessary to modify Eq. OII.197). From the physical point of view this result leads us to the conclusion that the electric field caused by charges only cannot maintain a constant movement of charges in a conducting medium. In essence we have already observed this phenomenon in studying electrostatic induction. As we know by placing charges either inside of the conductor or on its surface, their electric field does not remain constant, but instead changes with time until it disappears altogether with the current density field. To perform modifications to Eq. (III.197) we will imagine the following experiment. Suppose that there is a tank, made from a nonconducting material with two parallel metallic plates P + and P -, placed at some distance from each other and bearing charges e+ and e", respectively (Fig. 111.11 c). It is clear that these charges create an electric field directed from plate p+ to P-. Inasmuch as the medium between them is nonconducting, both charges and their electric field remain constant. But now let us fill the tank with an ionic conductor so that part of the plates are located above the solution, and the conductivity of the ionic conductor 'Ye is much smaller than that of metallic plates. Due to the electric field E, current arises in the solution, and correspondingly positive ions move to the plate with the negative charge, while negative ions approach the plate P+, which has the positive charge. This movement of ions results in a decrease of plate charges until both the electric and current density fields vanish. Assuming that this process is sufficiently slow it can be described in a very simple manner. In accordance with Eq. (II 1.13), the electric field between plates at some instant is (III.199)

289

111.7 The Extraneous Field and Its Electromotive Force

and it is perpendicular to the plates, since the influence of their edges is neglected. Here l is the charge density at this moment. The field E creates a current density j, which equals (III.200) and correspondingly the current Ie is (III.20l) where Se is the area of each plate that is immersed into the solution, while Ie is the amount of positive and negative charge arriving every second at plates p+ and P-, respectively. It is obvious that this process leads to a discharge of the plates. To maintain a constant current, we will suppose that above the solution the plates are connected with each other through a conducting medium C j , which has a conductivity 'Yj' Thus our model, along with two plates, consists of the solution C, and the conductor C j • It is clear that in both conductors Coulomb's electric field E C has the same direction, namely, from the plate P+ to P-. Now we assume that within the conductor C j along with the Coulomb field there is another field, E?", which has an opposite direction, but it is related in the same manner with the current density, and its magnitude is greater than that of the Coulomb field; that is, (III.202) Correspondingly, within the conductor C j there are two current density fields, which are opposite to each other,

and in accordance with Eq. (III.199) the total current density in C j is (III.203) This new field, Eext, is called an extraneous field to emphasize that its generators can be of any origin (thermal, chemical, mechanical) except charges. Due to this field the positive charges move within the conductor C j against the electric field E C , and this allows us to compensate for discharge of the plate P + through the solution. At the same time negative charges move in the opposite direction in C j and increase the charge e- on the

290

III

Electric Fields

plate P-; but we will pay further attention to positive charges only. It is clear that the current inside of the conductor Cj is

t, =

l'i (

E~xt

~) s,

-

(III.204)

where S, is the area of each plate above the solution. Thus, it is natural to distinguish two parts of this model of the current flux. The first one is called external, and it corresponds to the solution C~, where only a Coulombic electric field exists. This field is directed from the plate P" to P-, creating the current, equal, by Eq. (III.20l), to

I = 'Y~-S~

I~

60

which is trying to discharge plates. In the second or internal part, conductor Cj , both the Coulomb and the extraneous forces act on charges, and in accordance with Eq, (III.202) the current in Ci is

t, = 'Yi( E~xt

-

~

)Si

which tends to restore the charges on the plates. It is obvious that if the I, are equal to each other, the charges on the plates will currents I~and not vary with time and therefore the electric and current density fields will remain constant. In other words, we have demonstrated that the presence of the extraneous field is necessary to create a time-variant current. Let us illustrate the relationship between this field and the current. becomes greater. Then, in accordance with Suppose the field E~xt Eq. (III.204) the current I, also increases, and this results in an increase in the amount of charge arriving at the area Si of the plate P+, which is spread over the whole plate. The same is true for negative charges on the plate P-. Correspondingly, the electric and current density fields increase in the solution too, until the current I~ becomes equal to Ii' A similar process is observed if the extraneous force decreases. Now we will make several comments. 1. Several features distinguish Coulomb's and the extraneous fields from each other, and they are (a) The voltage of the extraneous field E~xt, unlike the Coulomb field, is in general path dependent; that is, the voltage of this field

tE~xt.

d/

a

is a function of the contour t along which the integration is performed.

111.7 The Extraneous Field and Its Electromotive Force

291

(b) The extraneous field cannot be caused by time-invariant electric charges, and correspondingly it does not obey Coulomb's law. At the same time, as was already mentioned, generators of this field can have very different natures such as thermal, mechanical, chemical, or electromagnetic. (c) The extraneous force, unlike the Coulomb force, is usually equal to zero in those places where its generators are absent. There are exceptions, however, and one is the induction force. (d) Extraneous and Coulomb fields can create a current in a conducting medium, while it is impossible to maintain a constant movement of charges with only a Coulomb field. 3. In general every circuit consists of internal and external parts. In the external part only the Coulomb field E C acts on charges, while in the internal part both the Coulomb and extraneous fields define a movement of charges. 4. There are cases when both forces, E C and E ext , have the same direction. This occurs, for instance, when an accumulator is being charged. 5. The Coulomb and extraneous forces have only one common feature; namely, they are related to the current density vector in the same manner, and therefore we have (III.20S) The above equation is Ohm's law in differential form, which is valid within the internal and external parts of any electric circuit. It is obvious that for the external part, where E ext : ; 0, we again have j=yE

In most practical cases of electrical prospecting we are interested in the study of the behavior of the field in those places of a conducting medium where extraneous forces are absent, and correspondingly we can still make use of Eq. (HU97). 6. The voltage of the extraneous force E ext between terminal points of the internal part of a circuit

taken along a current line, is called the electromotive force :ff. (HI.206)

292

III

Electric Fields

It is clear that it' characterizes the work performed by the extraneous force in moving a unit positive charge against the Coulomb field.

Now let us consider several examples.

Example 1 Behavior of the Field and Its Potential along a Quasi-linear Circuit Suppose that the external part of the circuit, shown in Fig. lII.lld, is a uniform conductor with a constant cross section S, but within the internal part there is an electromotive force equal to it'. First consider the external part, where the current density field is uniform, since I = constant

and

S = constant

In accordance with Ohm's law, j

= '}'E

or

E=pj

we have to conclude that the electric field inside of the conductor also does not vary. E = constant This is the Coulomb field, caused by charges, and it is natural to determine the charges' location. Certainly they are not located within the conductor, since it is uniform, Eq. (I1I.152), and the sum of charges that constitute the current equals zero. Also they cannot be concentrated only near the terminal points of the internal part, because a field caused by two such charges of opposite sign by all means is nonuniform. It is impossible that the sources of this field are located outside the circuit, inasmuch as due to the electrostatic induction they do not have any influence on the field inside the conductor. Thus, there is only one place where sources of the uniform electric field E can be located, and this place is the conductor's surface. As soon as there is a current I in the circuit these charges are distributed in such a "clever" way that the electric field becomes uniform everywhere within the external part of the circuit. Schematically their distribution is shown in Figure III.lld. As usual, let us make some comments. 1. Changing the position of the external part of the circuit in space, the current I does not of course vary. This happens because the distribution of the surface charges also changes in such a manner that they maintain the same electric field in the conductor.

111.7 The Extraneous Field and Its Electromotive Force

293

2. If the cross section or a conductivity of the external part of the circuit changes, the electric field becomes in general nonuniform, and its behavior is again governed by charges. In particular, as conductivity changes both the surface and volume charges are present, and they are distributed in such a way that their electric field provides the same current through every cross section of the conductor. 3. In general there are two types of charges on a conductor surface. One of them creates the electric field inside, while another plays a completely different but also important role, since due to this charge the current in the circuit depends on the electromotive force and its resistance, only. In fact, suppose we have a charge near the circuit (Fig. III. l l d). In accordance with Coulomb's law it creates inside of the conductor a field, which can be very strong, while the current 1 does not change at all. This happens due to the charges that appear on the conductor surface, and their field along with that of the external charge, creates a zero field inside of the conductor (the electrostatic induction). As we already know, in the internal part of the circuit the current density is defined by both the extraneous and the Coulomb fields. The latter, however, is caused by all charges that arise on the whole surface of the conductor as well as inside, if it is not uniform. Now we will study the voltage between two arbitrary points a and b of the circuit (Fig. Ill.I l d). If within this interval extraneous forces are absent, then we have U(a)-U(b)= jbE.dl'= fbpj.dl' a

a

or bpj'dl' U(a) - U(b) =1 f - - =1R ab a

1

(III.207)

where R ab is the resistance of the circuit between the points a and b. In particular, when these points coincide with terminal points of the internal part (current source) we obtain U+- U_=1R e

(111.208)

where R; is the resistance of the external part of the circuit. Next suppose that the internal part with electromotive force it' is located somewhere between points a and b. Then applying Ohm's law, or

294

III

Electric Fields

we have

U(a) - U(b)

dl' f+ -- E fabE· dl'=I fbPj· a I -

=

ext

•

dl'

whence

U( a) - U( b) = IR a b

-

15'

(III.209)

where R a b consists of the resistance of the internal part R, and the resistance of that section of the external part located between points a and b. In considering the whole circuit when a = b, we obtain (III.21O) where R is the total resistance of the circuit. From Eqs. (111.208) and (III.21O) we obtain the relationship between the potential difference at the terminal points and the electromotive force. U -U +

su,

=

-

1 =15'---R, + R e 1 + (RjR e )

(III.211)

Thus this potential difference is practically equal to the electromotive force if the resistance of the external part is much greater than that of the internal one. (III.2I2) As follows from Eqs. (II1.207) and (III.209), the potential decreases along the current line within the external part while it increases in the internal part; this behavior is shown in Fig. III.12a.

Example 2 The Current Line of an Electrical Prospecting Array Now we will consider a different contour, which includes the following elements: (a) An internal part (current source) with electromotive force 15'; (b) A linear conductor (wire) with resistance R w ; (c) Two electrodes with grounding resistances R A and R B , which provide the contact with the volume conductor; and (d) An arbitrary medium with resistivity p, which in general can change from point to point.

111.7 The Extraneous Field and Its Electromotive Force

a

b

295

+

u

'----------_1 a

+

a

c

i:i-.r.J&Y··

k

2II.

_.

Fig. 1l1.12 (a) Potential distribution along a circuit; (b) the two-electrode array; and (c) double layer in a conducting medium.

Then, in accordance with Ohm's law, the current in such a circuit (Fig. III.12b) is (111.213) where R, is the resistance of the current source. As we know, the grounding resistances depend on the distribution of the medium resistivity everywhere regardless of distances from the electrodes. In other words, a change of the resistivity somewhere beneath the earth's surface has to cause a change of the current I and, respectively, measurements of the current with this simple two-electrode array can in principle give information about the electric properties of the medium. However, this is true only in theory, since the grounding resistance is mainly defined by the resistivity of that part of the medium located very close to the electrode. The size of this area seldom exceeds 1 m. Correspondingly, the current I is practically insensitive to a change of the resistivity beyond this range. For this reason the two-electrode array is not used in electric methods

296

III

Electric Fields

except for some applications in logging; instead, a standard array consists of two different parts, namely, (a) A current line AB, which includes the electromotive force. The sale function of this part is to create the electric field in a conducting medium. (b) A measuring line MN, which contains a voltmeter, aIIowing us to measure the voltage of the field between receiver electrodes caused by all charges arising in the medium. This separation of the current and receiver lines of the circuit is the distinguishing feature of most electrical prospecting arrays.

Example 3 The Extraneous Force and Electric Charges In Section 111.5 we studied the distribution of charges in a conducting medium when the extraneous force is absent. Now we will consider a more general case, proceeding from the equations

s

divE= - ,

divj = 0

80

or '(2) _

'(1)

in -in

and Ohm's law, j = y(E + E eXI ) Repeating the same procedures as in Section 111.5, we have divj = div y(E

+ E eXI ) = div yE + div yE ext = 0

or 'Y divE

+ E· grad y + div yE ext = 0

or

s

y-

+ E· grad y + div yE eXI = 0

80

Therefore the volume charges related to the extraneous force are {)ext =

div y E exl -8

0----

Y

(111.214)

111.7 The Extraneous Field and Its Electromotive Force

297

They are situated within the internal part of a circuit, and as soon as the electromotive force if; and the internal resistance R; are given, the distribution of these charges has hardly any interest for us. It is more useful to consider surface charges related to the extraneous field. Generalizing Eq. (III.153) we have

or

Inasmuch as

we have !ext =

-E

Div yE ext o yaY

(111.215)

Here Yl + yz 2

av

Y

=

Div yE ext =

Y2E~2)ext

- YIE~I)ext

and E~2)ext and E~l)ext are normal components of the extraneous force at the front and back sides of the surface, where its normal indicates the direction of the current density vector. In particular, near terminal points of the internal part we have "IC

_

Yi

..:.,-- -EO--;W

E ext

0-

(III .216)

Y

since the field E ext vanishes in the external part of a circuit. Of course, these charges have an influence on the total electric field both inside and outside of a conductor. If both components E~~ and E~~ are equal, then the charges have the same magnitude but opposite signs.

! where

+

=

EO

~Eext av Y

"IC_=

0

,

c;

Yi E 0ext Y

-EO--;W

(III.217)

298

III

Electric Fields

It is also clear that in the case when near the terminal points the conductor is uniform, Yi = Ye , we have

(III.218)

Example 4 The Contact Electromotive Force Suppose that the extraneous force E ext is distributed within a very thin layer with thickness h, and that 5 is its mean surface area, as is shown in Fig. II 1.12c. The boundary surfaces of this layer are 51 and 52' and the normal n is directed from 51 to 52. In general, the conductivities of the media inside and outside the layer are different, and the thickness h can vary. Also we will assume that the normal component of the extraneous force is positive if it has the same direction as the normal n, but it becomes negative if the normal component of the extraneous force and the normal n are opposite to each other. It is convenient to represent the field of the current density as a system of current tubes with sufficiently small cross sections. Then, every element of the layer Ci plays the role of an internal part of a corresponding circuit. It is obvious that the charges arising on surfaces 51 and 52 [Eq. (III.216) are maintained with the help of the extraneous force, while the Coulomb's field tends to discharge them. Taking into account the fact that the layer thickness is very small, it is natural to assume that within every element of the layer Ci the force E ext does not change, and therefore fti

=

f.2 E ext . do = hE~xt

(III.219)

1

Correspondingly the distribution of charges on terminal sections is defined by Eq. (III.217); that is, elementary surfaces d5 1 and d5 2 bear charges with the same magnitude but opposite signs. Now let us make one more simplification based on the assumption that the layer thickness is very small. In this case the resistance R, of every element of the layer is negligible with respect to the external resistance R; of the corresponding tube current. Then, in accordance with Eq. (III.212) we have

IlLS The Work of Coulomb and Extraneous Forces, Joule's Law

299

That is, the difference of potentials at boundary surfaces equals the electromotive force acting near the same point. It is clear that charges situated on surfaces Sl and Sz create the electric field in a conducting medium, and if the observation point is located at distances from the layer significantly exceeding h, this system of charges can be considered as a double layer. It is very easy to find a relation between the electromotive force and the density of dipole moments 1). In fact, from comparison of Eqs. 011.72) and (III.212) we have (III.220)

Inasmuch as the field of the double layer is defined by the distribution of the function 1), let us consider the limiting case,

but the dipole moment 1) remains the same. This means that we have arrived at the mathematical model of the double layer, when the positive and negative charges are situated on the front and back sides of the surface S, respectively. Then, making use of the results derived in Section III.2, we have

au + at

au at

-----

a?;c at

(111.221)

and

eu; an

au an

where z" = ?;c = 1) / Co is called the contact electromotive force. Also, applying the concept of the solid angle it is not difficult to calculate the field of the double layer at any point. This approach is widely used in the theory of methods, based on a study of natural electric fields.

III.S The Work of Coulomb and Extraneous Forces, Joule's Law As is well known the movement of charges in a conducting medium (current) is always accompanied by heat production. The relationship between this heat and the fields E C , Eext,j is discussed in this section. First,

300

III

Electric Fields

we will suppose that the conductor is a quasi-linear circuit surrounded by free space, as shown in Fig. III. 11d; later, a more general case will be studied. In the previous section we demonstrated that a constant current can exist in a conducting medium only if there is an extraneous force, which can have any origin except Coulombic. Induction or electromagnetic fields, for instance, are a very important class of extraneous forces widely used to generate constant currents. Extraneous forces also arise due to chemical or physical inhomogeneities in conductors. Such forces appear near contacts having different chemical contents (galvanic element, accumulator), or different temperatures (thermoelement), or in the presence of a concentration gradient of electrolyte ions (concentration galvanic element). Mechanical forces can also provide transportation of charges (electrostatic generators). Inasmuch as all of these systems are vital for the existence of constant current, they are usually called current sources. Now we will consider the work produced by a current source and the distribution of its energy along a circuit. In general, a moving charge is subjected to the influence of both the Coulomb and extraneous forces (Ohm's law), and correspondingly they perform work, which can be easily calculated. In fact, during the time interval dt the amount of charge passing through every cross section of the circuit element C l Z is the same, and it is equal to de

=

(III.222)

I dt

where I is the current. In particular, the amount of charge that enters and leaves the element C\2 is the same. It may be proper to notice that opposite charges with an equal magnitude pass through various sections, and their velocity is extremely small. Thus the elementary work of forces along the displacement dl' can be written dA

=

I dt(E

+ E ext ) • dl'

(III.223)

because E = I dt(E + E ext ) is the force acting on charge de. Respectively, the work related to the translation of the charge between section S 1 and Sz during one second is

A= If\E + E

ext

) •

1

where dl' is directed along the current line.

dl'

(III.224)

111.8 The Work of Coulomb and Extraneous Forces, Joule's Law

301

It is clear that A has the dimension of power, and in SI units it is measured in watts.

1 watt

=

1 volt

X

1 ampere

In accordance with the principle of conservation of energy this work has to be transformed into some form of energy W, and the amount of this energy is the same as A.

W=Ij,\E+E eXI ) 'd/

(III.225)

1

At the same time experiments demonstrate that this energy appears as heat Q, within the element C 12 , per unit time. Q =1 j,\E + E e XI )

.

d/

(III.226)

1

In other words, the total work of Coulomb and extraneous forces, performed within some element of the conductor, is converted into heat within the same element. Equation (III.226) is in essence a formulation of Joule's law, and it is valid provided that neither chemical reactions nor conductor motion consume the energy of a current source. In particular, the amount of heat that appears in the whole circuit is (III.227) since the circulation of the electric field vanishes, but

P=If?

(III.228)

is the work performed by the current source during one second. Thus, all energy of the source of the constant current in a conductor is transformed into heat. In addition, let us make several comments. 1. The equality ¢E' d/= 0 means that the work done by the Coulomb force along the whole circuit is zero. This result can be expected, since in the opposite case, due to a decrease of the energy of this field, it would not be possible to preserve the time-invariant current. 2. In the external part of the circuit the extraneous force is absent and in accordance with Eq. (III.226) we have

(!II.229)

302

III

Electric Fields

This shows that every second the work of the electric field is converted into heat, and this fact can serve as another argument illustrating the necessity of a continuous extraneous force. 3. Joule's law, Eq, (111.226), can be described in terms of the current I, resistance R, and voltages of the electric and extraneous forces. In fact, making use of Ohm's law we have (III.230) where R 12 is the resistance of the conductor element en' If the extraneous force is absent within this element we have (111.231) That is, the amount of heat that appears every second is directly proportional to the resistance and square of the current magnitude. In particular, for the external part with resistance R; we obtain Qe =/(U + -U - )=J 2R e

(111.232)

4. In the internal part of the circuit the amount of heat that appears every second is

or

{+ (E ext -

Qi =J

E) dt= te- I(U_- U +)

(III.233)

since the fields E ext and E have opposite directions, but E'" > E. Equation (111.233) can be written as (111.234) It is clear that the first term on the right-hand side of the equation describes the work of the Coulomb electric field E related to the movement of charge J dt per unit time along the external part of the circuit.

Ae=J(U+-U_) =IJ- E·d/ +

In other words, this term characterizes the amount of current source energy required to maintain the constant electric field E. Certainly, the current source must perform at least this work, A e , but in reality it also produces additional work, Ai' For instance, in an electrostatic generator of the current a significant part of the work is related to heat losses due to

111.8 The Work of Coulomb and Extraneous Forces, Joule's Law

303

friction between its elements. This additional work A j can be described by introducing the internal resistance R, of the current source, which absorbs the heat energy so that

Also we can imagine a current source that in fact has an internal resistance; and then, in accordance with Joule's law, the amount of heat that appears within the source is

5. Considering alternating fields we will demonstrate that some part of the current source energy travels through the surrounding medium as electromagnetic energy, and within the external part of the circuit it is converted into energy of the electric field. Correspondingly the transformation of the source energy into heat can be approximately presented in the following way: Current source energy, P

additional work: Ai = Q i

electromagnetic energy

1 electric field energy

6. In general, the total amount of heat that appears due to the constant current does not coincide with Joule's heat Q. This is related with the fact that near a contact between different conductors Peltier heat arises; in addition, Thomson heat appears in the presence of a temperature gradient. However, unlike Joule's heat, these two forms of heat are linear functions of the current and usually their contribution is very small.

304

III

Electric Fields

7. This analysis remains valid if instead of a quasi-linear conductor a current tube is considered. Let us take an elementary volume dV = (dS . dl), where dS is a section of the current tube. Then, the amount of charge passing through this surface during the time interval dt, is de = (j . dS) dt and the work performed by the Coulomb and extraneous forces can be presented as (j . dS){(E

+E

eX1

)

•

dl'} dt

Therefore, the amount of heat that appears every second is

{j . (E + E

dQ =

eX1

) } (

dS . dl')

{j '(E+EeX1)}dV

=

At the same time, the work done by the extraneous force is

Thus, in general in a unit volume during one second two processes occur, namely, (a) The extraneous force performs work. (b) Heat appears, and its amount is given by

(III.235) where p is the resistivity of the medium. In particular, for an elementary volume of the external part of a current tube we have dQ

dP

-=0 dV

_

'

£2

=j .E= _

dV

p

=pj2

(III.236)

1II.9 Determination of the Electric Field in a Conducting Medium In the previous sections we have attempted to describe the most essential features of the electric and current density fields in a conducting medium, and in particular the following questions have been studied: (a) The behavior of the electric field at regular points and near interfaces of media with different resistivities.

IlI.9 Determination of the Electric Field in a Conducting Medium

(b) (c) law. (d) (e)

305

The distribution of charges in a conducting medium. The current density field and its relation to the electric field, Ohm's The role of extraneous forces. Electromotive force and resistance, and so on.

Now we will discuss basic questions related to solving the forward problem, namely how to determine the electric field in a conducting medium. The importance of this subject is obvious since the calculation of the electric field is a cornerstone of the quantitative interpretation of all electrical methods. As was shown in Chapter II the solution of the forward problem of the gravitational field is based on direct use of Newton's law, Eq. (I1.5),

and therefore it consists only of numerical integration. The density of mass (5(q) is a physical parameter of the medium, and of course it is indepen-

dent of the field g and the geometry of the model. In other words, as soon as a model of the medium is chosen, the sources of the gravitational field are simultaneously specified; that is, we know the sources before their field is calculated. This fact allows us to make use of Newton's law and to determine the field g in the most simple way. In contrast, Coulomb's law 1 --f 47Tco v L

(5(q)

E(p) =

--3- L qp qp

dV +

1 --I. 47Tco S L

l(q)

-3- L qp

dS

qp

cannot be used to determine the electric field in a conducting medium because the density of electric charges, unlike the density of mass, is not a physical parameter of the medium that can be specified. Instead, it depends on such factors as resistivity and electric field. In fact, in accordance with Eqs. (III.152) and (III.154), (5 =

1 -copE' grad -, p

and in particular, this means that we can expect an infinite number of charge distributions given a single model of a conducting medium. Thus, to calculate the electric field E, we have to know the density of charges (5 and l. These quantities, however, can be defined only if the field is already known. So again, as in the cases of electrostatic induction and the polarization of dielectrics, we are faced with "the closed circle"

306

III

Electric Fields

problem, which does not allow us to use Coulomb's law for the field calculation. This fact alone requires us to formulate boundary-value problems and explains the fundamental difference in how we approach solutions of forward problems in the theory of the gravitational and electrical methods. In electrical methods we are forced to solve boundary-value problems, and certainly this approach is much more complicated than it would be if we knew the charge distribution and could directly apply Coulomb's law. Before we formulate the boundary-value problems it is appropriate to present the system of field equations for both fields E and j. Summarizing the results derived in previous sections, we have [Eq, (III.237)] Coulomb's law

(A)

Ohm's law

Principle of charge conservation

divE =

curlE = 0 curl] = 'VI' X E

-

s

EO

divj

E~2)

(B)

j?) - j~l)

=

0

=E[1)

= (1'2 - I'1)E t

(III.237) Let us make several comments concerning this system. 1. Since the dielectric constant E does not have any influence on the electric field in a conducting medium, we assume that all parts of the medium have the same dielectric constant, equal to that of free space, and therefore 0 and k are volume and surface densities of free charges only. 2. The system (III.237) describes both fields E and j at the external part of the current tubes, where extraneous forces E ex! are absent. This is hardly a shortcoming, since in electrical prospecting practice the internal part of a circuit is not usually considered. 3. The system is written for two fields E and j, but the latter can be determined, provided that the electric field is already known. Moreover,

III.9 Determination of the Electric Field in a Conducting Medium

307

taking into account the fact that only the voltage of the electric field is measured in electrical methods, we will concentrate our attention entirely on the field E. As follows from Eq. (III.237) and as was demonstrated earlier, the system of equations for the electric field is divE

curlE = 0

s

=-

EO

(III.238)

It is clear that this system is the same as that in free space. This is not surprising since the origin of the constant electric field remains the same regardless of the type of medium. However, there is one fundamental difference. Unlike free space, the charge density 0 and 1, cannot be specified prior to calculation. Since 0 and 1, are unknown in the above equations, it is natural to replace them by two equivalent equations that describe the principle of charge conservation, Eqs. (III. 142), (III. 143). These equations also contain information about charges, though they are not present in explicit form. Indeed, the equation div j = 0 can be rewritten as

div j = div y E

=

y div E

+ E . grad y = 0

or divE

Egrad 'Y = ----

y

That is, we again derive the second equation of the system (III.238). Also the continuity of the normal component of the current density,

holds only if a certain amount of charge appears near this point; that is, it establishes a relation between the field and charges, as well as the equation 1,

E(2) n

E(l) n

=-

EO

Then, after this replacement we obtain the system of field equations of the constant electric field in a conducting medium, which is used to solve the

308

III

Electric Fields

forward problem in the theory of electrical methods [Eq. (III.239)]. Ohm's law

Coulomb's law

Principle of charge conservation

t I

t

curlE = 0

II

£(2) _ £(1) = t

t

0I

'V

12

(III.239)

div yE = 0

£(2) = n

'V

11

£(1) n

I

Let us make several comments here. 1. This system describes the behavior of the electric field at the regular points of a conducting medium as well as at interfaces of media with different resistivities. 2. The field equations (III.239) have been derived from three physical laws, namely Coulomb's law, Ohm's law, and the principle of charge conservation. For this reason they contain exactly the same information as these laws. If we imagine, for instance, that the first equation is invalid,

curlE "1= 0

or

then this would mean that the field E does not obey Coulomb's law, and that the field is caused by vortices as well as sources. If the field does not satisfy the second equation, divj "1= 0, this means that charges continuously accumulate and therefore the field cannot be time-invariant. 3. This system does not describe the field behavior due to a double layer. This special case was considered in detail earlier. 4. It is clear that any constant electric field, regardless of the distribution of resistivity of the medium and the position of primary sources, is a solution of the system (III.239). This happens because every time-invariant electric field must obey Coulomb's law, Ohm's law, and the principle of

309

111.9 Determination of the Electric Field in a Conducting Medium

charge conservation. Respectively, we can say that the origin of the system is such that it has an unlimited number of solutions. Now we will use the analogy between the field in free space and in the presence of dielectrics, and introduce a system of equations describing the potential U in a conducting medium. First of all, from the first equation curl E = 0, it follows that E = -gradU Substituting this into the second equation div j points of a medium

=

0, we obtain for regular

V(yVU)=O

(III.240)

Then making use of the equality

au

E t- at -

conditions for the electric field at interfaces can be represented as and where U1 and Uz are values of the potential at the back and front sides of the interface, respectively. Inasmuch as both components of the electric field have finite values, the potential U is a continuous function at an interface with some charge density 2,. In fact, from the equality

it follows that as the points p] and F:» which are located on opposite sides of the interface, approach each other, the potentials at these points become equal. U] = Uz It is clear that if the potential does not change through an interface then its tangential derivatives at both sides are equal to each other and therefore the conditions

and are equivalent.

au]

auz

-=-

at

at

310

TIl

Electric Fields

Summarizing these results we arrive at the following system of equations describing the behavior of the potential in a conducting medium: Coulomb's law

Ohm's law

Principle of charge conservation

(III.241)

aUI 1'1

auz

an = I'z an

As usual, let us make several comments. 1. Since the potential U is a scalar function, very often it is more convenient to solve the forward problem making use of Eq. (111.241) rather than Eq. (111.239). 2. In essence Eq, (111.240) is Poisson's equation. In fact, differentiating we obtain

or

and as was shown earlier, the right-hand side characterizes the density of charges at regular points. 3. In most practical cases in electrical prospecting and electric logging we can assume that the conducting medium is a piecewise uniform one, and that it is composed of different media each having a constant resistivity. Because of these assumptions, the equation for the potential is drastically simplified and we again arrive at Laplace's equation. (III.242)

From the physical point this result is obvious since, in a piecewise uniform medium, volume charges are absent. 4. As in the case of the system for the electric field, Eqs. (11.241) do not take into account the behavior of the potential caused by a double layer.

111.9 Determination of the Electric Field in a Conducting Medium

311

Fig. 111.13 (a) Illustration for theorem of uniqueness; (b) the first type of medium model; (c) the second type of medium model; and (d) illustration in deriving the integral equation.

5. The potential U of any constant electric field is a solution of the system (III.241), and correspondingly the latter does not uniquely define U. Moreover the equation E = - grad U determines the potential up to a constant. Thus, the system (III.241) has an infinite number of solutions. Our next step is to outline an approach that will allow us, in principle, to determine the constant electric field in any conducting medium. Since Coulomb's law cannot be used for this purpose, it is natural to refer either to the system of field equations or that for the potential U. However, these systems alone do not uniquely define the field and therefore it is necessary to bring additional information. As is known, this task is solved with the help of the theorem of uniqueness; the importance of this theorem is discussed in detail in Chapter I. With some insignificant modifications that take into account an arbitrary change of the medium conductivity, we will again describe this theorem and formulate boundary-value problems. Suppose that the potential U is considered in a volume V, which is surrounded by boundary surfaces 51 and 52' Also there is an interface 5 12, where the conductivity can have different values at the front and back sides (Fig. III.l3a). Of course, the potential U within this volume is a solution of the system (III.24l). It is appropriate to notice that the surfaces 51 and 52 are usually

312

III

Electric Fields

different surfaces. They can be a surface of the current electrode, or the earth's surface, or a spherical surface of infinitely large radius, etc. To determine conditions at boundary surfaces that uniquely define the field, let us assume that there are two different solutions of the system (III.24l): U(1) and U(2); that is,

V('Y VU(I)) = 0,

V( 'Y VU(2») =

°

and

UI

(I)

=

U(1) 2'

aup) an -

aup) an

auF)

aUF)

'Y I - - -'Y 2 - -

(111.243)

and U I(2) =

U(2)

'Y I

2'

---a;;- = 'Y2---a;;-

where 'YI' UI and 'Y2' U2 are the conductivity and potential at the back and front sides of the interface S12' respectively. We will consider the difference of these solutions. U(3) =

U(2) - U(1)

which in accordance with Eqs. (III.243) satisfies the following conditions:

and (3) - U(3) U1-2

aup) an- --

'Y I -

aUp) 2 an

'Y -

(III.244)

Then we will introduce the vector X (III.245) and make use of Gauss' theorem

1v* divX dV

=

~ X . dS + ~ X . dS + ~ X . dS Sj

Sz

(111.246)

So

where So is a "safety" surface surrounding the interface S12' where the vector X is a discontinuous function, because the conductivities 'YI and 'Y2 are not equal to each other. Correspondingly Gauss' theorem is applied to the volume V*, which is confined by the surfaces Sl' S2' and So'

111.9 Determination of the Electric Field in a Condncting Medium

313

Substituting Eq. (III.245) into Eq, (I1I.246) and differentiating, we obtain

As So approaches S12 the integration over So is reduced to that over both sides of the interface S12' Taking into account the fact that at the back and front sides of this surface and

"2 =

-no

respectively, we have

and V* ~ V. Now making use of Eqs. (III.244), the equality (111.247) is strongly simplified and we obtain

The latter relates values of the potential and the field on the boundary surfaces S1 and 52 with those inside of the volume. Suppose that the surface integrals in Eq. 011.249) vanish. Then the right-hand side is also zero.

Inasmuch as the integrand y(VV(3))Z is positive (y > 0), we have to conclude that grad

V(3)

=0

(111.250)

That is, the function V3 does not change within the volume V. Bearing in mind the fact that V3 is the difference of two solutions of the system (111.241) we can say that if the surface integrals of Eq, (III.249) are equal to zero, then two arbitrary solutions of the system (111.241), V 1 and

314

III

Electric Fields

U2 , can differ from each other by a constant. However, the field E is uniquely defined since

E=

~

grad U =

-

grade U + C)

Now we will formulate boundary conditions, such that the surface integrals in Eq. (I1I.249) become equal to zero. At this point it is natural to make use of results derived in Chapter I. Then we obtain three forms of the boundary conditions on the surfaces 51 and 52'

1.

U = cp( q)

2.

au an = t/J(q)

3.

au ¢.Yas =1 an

(I1I.251)

5

and 5 is an equipotential surface. Here cp(q), t/J(q), and 1 are given functions. It is essential that every boundary condition of (III.251) together with the system (I1I.241) uniquely defines the electric field in the volume V. This is the main result derived from the theorem of uniqueness. Correspondingly, we can formulate three boundary-value problems.

1.

Dirichlet's Problem at usual points

\7(y\7U) =0

( a)

(b)

YI

aUI an

=

Y2

aU2 an

on

-812

(c) Neumann's Problem

2.

(b)

( c)

at usual points

\7(y\7U) =0

(a) 1=

U2

,

aUI YI an

=

aU2 Y2 an on

52

111.9 Determination of the Electric Field in a Conducting Medium

3.

315

The Third Boundary Problem (a)

'V( y'VU) = 0

au!

(b)

(c)

at regular points

1'1

au

an

~ y-d5 =/, 51

an

aU2 = 1'2

an

au ~ l'-d5=-/ 52

an

where 51 and 52 are equipotential surfaces. Now we will make several comments that perhaps will help to clarify the role of the theorem of uniqueness. 1. The theorem of uniqueness does not provide an algorithm for determining the field, but instead it formulates conditions that uniquely define the field. 2. Every boundary problem consists of two parts, namely, (a) The system of equations for the potential (111.241). (b) One of the boundary conditions. 3. All three boundary conditions always have clear physical meaning, and it is usually very simple to formulate them. 4. It is essential to understand that types of boundary conditions can vary from point to point, and of course they can be different at different boundary surfaces. At the same time it is necessary to note that at all points of boundary surfaces without exception one of these boundary conditions must be specified. 5. In deriving the system of equations for the potential (11.241) ·and formulating the boundary conditions, we have assumed that the model of the conducting medium has one interface. However, it is obvious that our results remain valid in the general case, when there are several interfaces between media with different resistivities. In accordance with Eq. (III.24l), at every interface both the potential and the normal component of the current density have to be continuous. 6. In spite of the fact that the theorem of uniqueness does not suggest an algorithm for calculating the field, it formulates the main steps of solution that have to be accomplished in order to find the electric field.

As examples of the process of formulating boundary-value problems, we will consider two models of a conducting medium often used in the theory of logging and electrical prospecting.

316

III

Electric Fields

Example 1 The First Model (Fig. IlI.13b)

Suppose that a current / goes into a medium through the electrode A, located near the point O. The medium is everywhere uniform except at the interface S12' Now let us start to extract information about the behavior of the field that is essential to formulating a boundary-value problem. First, at regular points the potential U satisfies Laplace's equation since y is constant. Then it is obvious that in accordance with Coulomb's law the potential caused by all surface charges tends to zero as the distance Lop from the electrode increases unlimitedly. U--70 In other words, we can say that on a spherical surface S2 with infinitely large radius, the potential U is equal to zero. In other words, we have formulated the boundary condition at infinity. Concerning the behavior of the potential near the interface S12' it is clear that both the potential and the normal component of the current density have to be continuous.

au! Yl

an

aU2 =Y2

an

The latter can be considered to be the surface analogy of Eq. (111.240). We have described the behavior of the potential everywhere except at the electrode surface SI' which along with S2 confines the volume V. Since as the electrode conductivity is many orders of magnitude greater than that of the surrounding medium, it is possible to treat S 1 as an equipotential surface.

U( SI) = constant Also it is natural to assume that the current I through the electrode surface is known, and in accordance with Ohm's law it is related to the potential U by

au

~ y-dS =/ Sl

an

where n is normal directed into the electrode.

III.9 Determination of the Electric Field in a Conducting Medium

317

Thus, the boundary-value problem in this case can be described by the potential satisfying the following conditions: 1. At the equipotential boundary surface of the electrode

au

~"h-dS=! 51

(111.252)

an

2. At regular points of the medium the potential is a solution of Laplace's equation

3. The potential and the normal component of the current density at either side of the interface are related by aU2

aUI 'YI

an

= 'Yl

an

4. At the boundary surface 8 2 , which has an infinitely large radius, the potential tends to zero. U-+O

As follows from the theorem of uniqueness, these four conditions uniquely define the electric field, as well as the potential, and therefore we can say that the boundary-value problem is well formulated. At the same time, it is proper to notice that the boundary condition near the source can very often be simplified. Suppose that the current electrode A is a small sphere with radius a; then we can express the potential U as a sum.

where Uo is the potential due to the charge on the electrode surface, while Ul is the potential caused by charges that appear at the interface Sl2' Then making use of Eq. (1II.163) we have

U=

PI! +Ul 41TL op

where Pl is the resistivity of the medium that directly surrounds the electrode. If we assume that the electrode radius is sufficiently small, then in approaching its surface the potential Ul , caused by remote charges, tends to some finite value, while the potential Uo becomes very large.

318

III

Electric Fields

Therefore, the boundary condition near the source can be rewritten as V

-7

Va =

PI! ---

41T'L op

if

Lop> a

Perhaps it is appropriate to make two comments. (a) In the limit we can replace the surface charge by a point charge. This replacement does not change the field outside the electrode. (b) From the physical point of view it is clear that this simplification is valid even in cases where instead of a spherical electrode there is a small electrode with an arbitrary shape, and the surface SI is located at some distance from this electrode. Thus the boundary-value problem can be represented as 1.

VI

-7

Va

PI! = ---

41T'L op

2.

V2 V = O

3.

Y;a;; = Yi+ 1

au;

as

Lop -7 0 (III.253)

aU;+1

----a;;-

on

S;

4. Here we have made one obvious generalization, by assuming that a medium contains several interfaces S;. Certainly the boundary condition V -7 Va is much simpler than Eq, (III.252), but the latter is more general, and in particular it has to be used if the electrode is located sufficiently close to charges arising at interfaces.

Example 2 The Second Model (Fig. III.13d This model consists of an upper nonconducting half space and a piecewise uniform conducting medium located beneath the earth's surface. Now we will formulate the boundary problem for the conducting half space, surrounded by the electrode surface SI' and the boundary surface S2' which includes itself, the earth's surface So, and a half-spherical surface of infinitely large radius Ss' S2 = So + S, Taking into account the fact that the normal component of the current density equals zero at the boundary with the nonconducting medium, it is

I1I.9 Determination of the Electric Field in a Conducting Medium

319

very simple to formulate the boundary condition at the earth's surface. In fact, according to Ohm's law, the normal component of the electric field also vanishes at the conducting side of this surface. Thus, the boundary-value problem is formulated in the following way: \l 2 u = 0

1.

2.

au; Yia;;

aU;+1

v. + I ----a;;-

=

on

s,

3.

4.

5.

(III.254)

au

-.-70

on

an

So

U.-70

Let us make two comments. 1. Because the normal component of the electric field is known at all points of the earth's surface (En == 0), we are able to determine the field in a conducting medium without considering free space. 2. In accordance with Eq. (III.174) the potential due to the charge of the current electrode located at the earth's surface is

Uo =

-

PI! -

-

2'77" Lop

and correspondingly the boundary condition on the surface S I is

U.-7 Uo =

PI!

---

2'77"Lop

After we have formulated boundary-value problems for the electric field in a conducting medium, it is natural to take the next step and consider methods to solve these problems. There are at least three such methods. 1. The method of separation of variables 2. The method of finite differences 3. The method of integral equations

320

m

ElectrieFltl1d$

But their. study is thesUbjeet of applied mathematics, and far beyond the scope of tIns .monogra,t)h. Neve~less It1t~(} next section we will demonstrate several times the appn~tlon thtm~thoo of separation of variables. The use of this method . ,~er. is .limited to models of the medium with relatively simple ~~ nfooterfaces between medi? w~th different con?uctivities. ~*. tonslderationof the field beha.vlOr In such models IS of great l?~ 'itrt~rest, and in fact these studle.s helped to develop the basic ~,~f electrical methods. At the same time more complex models of a ~tbJotlng medium require that we apply much more complicated ap:proaches than the method of variable separation, such as the methods of lntegra!equation and finite differences. Both of these methods vividly demonstrate the degree to which solution of the forward problem in electrical methods is more complicated than in gravity methods. For illustration we will derive here the integral equation with respect to the potential of the electric field, and with this purpose in mind, we will consider a two-layered medium with an arbitrary inhomogeneity within the upper layer, as is shown in Fig. III.13d. We will use the following notations: 11 and 12 are the conductivities of the upper and second layer, respectively; 'Yi and Sj are the conductivities and the surface of inhomogeneity. So and SI are the earth's surface and the interface between layers, respectively. To derive the integral equation we will proceed from Green's formula.

·.iii'

fv(G'V u - UV 2

2G)

au

aG)

dV =,{,. (G - - U- dS ~ an an

(III.255)

where n is the normal directed outward from the surface S, which surrounds the conducting medium, and G is an arbitrary Green's function. At the same time U is the potential of the electric field and, in accordance with Eqs. (111.254), satisfies the following conditions: 1. At regular points it is a solution of Laplace's equation,

2. Near the current electrode it tends to Uo,

where the electrode is located in the vicinity of the point O.

111.9 Determination of the Electric Field in a Conducting Medium

321

3. At infinity the potential tends to zero as if the source were a point charge,

c V~--~O

Lop

where C is some unknown constant. 4. At the interface between the upper layer and the formation

eo, avz /" an = /,z an

V, = Vz,

5. At the surface of the inhomogeneity Sj

eu,

/"an

Vj=Vj ,

eu, =/'ia;;

6. At the earth's surface the normal component of the current density is zero; that is,

eo,

-=0

an

Now we will choose a function G such that the volume integral in Eq. (111.255) vanishes, but the surface integral is reduced to that over the surface inhomogeneity only. With this purpose in mind, let us suppose that the Green function G, which depends on two points q and p, is, up to a constant, the potential at the point q, caused by a unit charge situated at the point p in a two-layered medium when the inhomogeneity is absent. Correspondingly, the function G satisfies the following conditions: 1. Within the upper layer and the formation, except at the point p,

VZG =0 2. Near the point p, which is an observation point, 1 G(q,p)~-

i:

3. At the interface between the upper layer and the formation and

aG

j

/' 1 ---;;;; = /'

aG z z---;;;;

4. At the inhomogeneity surface and

aG an

j

eo, an

--=-

322

III

Electric Fields

5. At the earth's surface

aG

j

-=0

an

6. At infinity function G decreases as

L qp ~oo In applying Eq. 011.255) we have to take into account the fact that it is only valid provided that the first and second derivatives of functions V and G exist. For this reason we will surround the current electrode and the point p by "safety" surfaces SA and Sp, respectively, and apply Green's formula to each uniform part of the conducting medium. Within the second layer we have

avz

aGz] dS=O

f [G z-sn: -Vz-an_

(111.256)

51

since VZG z = VZV = 0 and both functions G z and V z decrease at least as fast as 1/L q p at infinity. Here n., is directed into the upper layer. Applying Green's formula to the volume occupied by the inhomogeneity, we have (III.257) because VZGi = VZVj = 0, and 11+ is directed outward from the volume. Finally, outside the inhomogeneity and within the upper layer we obtain

(III .258) The first three integrals can be drastically simplified. We will consider the first integral where the integration is performed over the sphere around the current electrode, and in the limit its radius tends to zero. In approaching the current electrode V, ~ p,I/41TR, where R is the radius

11l.9 Determination of the Electric Field in a Conducting Medium

323

of the spherical surface, we have au] aR

PII - 41TR 2

au] an

or

PII 41TR 2

since the directions of Rand n are opposite to each other. Taking into account the fact that the Green function and its derivative have finite values in the vicinity of the electrode, and making use of the mean value theorem, we obtain

~

G

SA [

l

au ] aG I] --u-_· dS an I an

=

[G ~ - !!L aG ]41TR2 47TR 2 41TR an I

as

=PIIGI(O,p)

R-40

(III.259)

It is easy to see that the right-hand side of Eq. (III.259) equals 41TU*, where U* is the potential at the point p in the horizontal layered medium when the inhomogeneity is absent; that is,

~

aUI GI [ SA an

-

aG I] U ] - dS = 47TU*(p) an

(III.260)

The second integral equals zero, because both derivatives aG jan vanish at the earth's surface; that is,

~

So[

aUI aG I] G I - - UI - dS=O an an

(III.261)

Finally, we consider the third integral around the observation point p, where function G tends to infinity at a rate proportional to IjR in approaching point p; that is, G

1 R

-4 -

I

aG I

and

-an

1 R2

= -

since nand R have opposite directions. Applying again the mean value theorem, we obtain

~

S

[ p

aG]] aUI G - - U - dS I an 1 an =

1 aUI [R an

-

1 ] 41TR 2 R2

U I

as

R

-4

°

(III.262)

324

III

Electric Fields

where UI(p) is the potential of the electric field in the presence of the inhomogeneity. Thus, instead of Eq. (III.258) we can write 47TB* (p) - 47TUI( p)

+

f [G Bn; aUt 51

I

[G imaU:I _ UaGn : I ] d5

+~

I

~i

U I aG Bn;

j ]

I

d5 = 0

(III.263)

Now multiplying Eqs. (III.256) and (III.263) by 'Yz and 1'1' respectively, and adding them we have

(III.264) Taking into account the fact that at the interface 51

eo, 1'1

an

eu, =

'Yz

an

and aG I

aG z

1'1--;;;; = l' z--;;;;

but since n l and n z have opposite directions, the sum of the last two integrals in Eq. 011.264) equals zero, and therefore 47T'YP*(p) - 47T'YPt(p)

Suppose the point p approaches the surface 5 i . Then applying the method used for the study of the double-layer field we have

III.9 Determination of the Electric Field in a Conducting Medium

325

Whence

(III.265) Next, considering the point p inside the inhomogeneity we can obtain

In the limit, when the point p approaches Sj we have (III.266) Adding Eqs. (III.265), (III.266) we obtain

47TYP*(p) - 27T{YP\(P) + YP2(P)}

Since and

au! Y\

an

aU2 =Yi

an

while we finally obtain

u(p)

=

~u*(p) Yj +Y1

(111.267) Equation 011.267) is an integral equation with respect to the potential U, since both points p and q are located on the surface Sj'

326

ill

Electric Fields

Here it is appropriate to make the following comments: 1. The potential U*(p) in a layered medium is assumed to be known, and it can easily be derived by applying the method of separation of variables. 2. The integral equation (111.267) remains valid if the inhomogeneity is situated in a horizontally layered medium with n layers. In this case, the function G has to describe the potential in this layered medium. 3. In particular, an inhomogeneity can be some structure at one of the interfaces of the layered medium. 4. Considering electrostatic induction, we demonstrated that the integral equation with respect to charges can be treated as a system of linear equations. Of course, this is also true for Eq. (111.267). 5. As soon as the potential U is known at the surface Sj, it can be calculated at any point outside the inhomogeneity. 6. The integration on the right-hand side of Eq. (III.267) is performed at the surface Sj, except the point p where the potential U(p) is calculated.

III.10 Behavior of the Electric Field in a Conducting Medium

Now we will consider several examples that illustrate field behavior in different models of a conducting medium. They are chosen in a way to demonstrate the application of electrical methods in different areas of geophysics. Also this study will include a discussion of the resolution of electrical methods, their depth of investigation, the influence of geological noise, etc.

Example 1 Influence of an Inhomogeneity on the Electric Field

Suppose that an inhomogeneity with conductivity Yj is surrounded by a uniform medium having conductivity Ye (Fig. III.l4a), and Eo describes the field behavior in the absence of the inhomogeneity. This field is often called the primary field. Due to this field, charges arise at the surface of the inhomogeneity, and in accordance with Eq. (III.154) their density is defined by (III.268)

III.10 Behavior of the Electric Field in a Conducting Medium

327

Fig. III.14 (a) Behavior of the field if Yi > Ye; (b) behavior of the field if Yi < Ye; (c) electric and current density lines at an interface; and (d) field behavior in the presence of a resistive inhomogeneity.

where Pi and Pe are the resistivities of the inhomogeneity and the surrounding medium, respectively, while E~v is the average value of the normal component, and the normal n is directed toward the inside of the inhomogeneity. Proceeding from the principle of charge conservation, we have to conclude that the magnitude and sign of ~ vary in such a way that the total amount of surface charge equals zero. In most practical cases this means that the positive charges arise at one side of the inhomogeneity surface, while negative ones appear at the opposite side. Therefore, there is a closed line of points along which the density ~ equals zero. Of course in general the distribution of charges is much more complicated. These charges are sources of the secondary field E s ' and, correspondingly, the total field outside and inside of the inhomogeneity is (III.269)

328

III

Electric Fields

Here, it is proper to make several comments. 1. In accordance with Coulomb's law, the secondary field E, is related to the charge density 4 by

Es{p)

=

_1_~ 47TEO

4{q~Lqp s

dS

L qp

where 4(q) is unknown. At the same time this equation allows us to derive some useful features of the behavior of the field. 2. In general both fields, Eo and E, have different magnitudes and directions. 3. It is obvious that only the secondary field contains information about the resistivity, shape, dimensions, and location of the inhomogeneity. All of these quantities, as well as Pe' are usually called geoelectric parameters. 4. The potential and current density fields can be represented as the sum of two fields, while

for points inside and outside the inhomogeneity, respectively. Let us assume that a body with a relatively simple shape is more conductive than the surrounding medium ('Yi > 'Ye), and the primary electric field is directed from the back to the front side of the inhomogeneity (Fig. III. 14a). Then in accordance with Eq, (IIL268) negative charges appear at the back side, while positive ones arise at the front side. In fact, the normal component E~v is positive at the back side, but the contrast coefficient Pi -Pe Pi +Pe

is negative, and correspondingly 4 < O. In contrast, at the front side positive charges arise, since E~v < O. Now we will consider the behavior of the component of the total field, which is directed along the primary field. It is convenient to present the secondary field as a sum,

E, = E:+ E; where E: and E; are the fields caused by the positive and negative charges, respectively. First we will take some point PI' located outside and in front of the inhomogeneity. In this case the field E: produces a positive

111.10 Behavior of the Electric Field in a Conducting Medium

329

component along the field Eo, while the field of the negative charges, E;, has the opposite direction. Taking into account the fact that the amount of positive and negative charge is the same, but the latter is located at greater distances, we can say that along the primary field

IE,"o1 > IE;:oI and therefore, due to the presence of the inhomogeneity, the field increases at the point PI; that is, (1II.270) where E,o is the vector component of the secondary field in the direction of the primary field. Next consider the field behavior at the point Pz, also located outside of the body and opposite the point PI' In this case the field of negative charges, E;, is directed toward the body, opposite to the field of positive charges. Since the negative charges are located closer, we can again conclude that in this area the field increases in the direction of the primary field. Now we will take the point P3' also located in the surrounding medium but near the equatorial plane of this model. As is seen from Fig. III.14a both fields E: and E; produce vector components that are opposite to the primary field; that is (1II.271) Summarizing these results we can distinguish three areas, namely, in front of and behind the conductive inhomogeneity where the field increases in the direction of the primary field, and the area around the equatorial plane where the field becomes smaller. At these boundaries the component of the secondary field along the primary field equals zero. Now we will consider the behavior of the field inside the inhomogeneity if 'Yi > 'Ye. Inasmuch as the positive and negative charges are located at the front and back sides of the inhomogeneity, their field produces a component opposite to the primary field. Correspondingly, the total field in the direction of the primary field decreases. With an increase in the ratio of conductivities 'YJ'Ye' the secondary field inside the inhomogeneity also increases and in the limit becomes equal in magnitude to the primary field. Therefore, in this case of the ideal conductor the field within the inhomogeneity vanishes. As follows from this analysis, with an increase of conductivity 'Yi' the voltage between two arbitrary points of the inhomogeneity decreases. In other words the better the conductor the smaller the electric field is inside it, and to some extent this linkage can serve as a characteristic of a conductor.

330

~.

III

Electric Fields

Next we will study the behavior of the potential of the secondary field From the equation Us(p)

1 ,(..~(q)dS

=

-'Y"

4r.Eo s

L qp

it follows that the potential is positive if the point p is located near the front side of the inhomogeneity, and it is negative if the observation points are near the back side. . Correspondingly, there is a surface passing through the inhomogeneity where the potential Us equals zero. In the case of the ideal conductor, the potential of the total field remains constant at all points of this body, even if the potential of the primary field changes. Finally, let us discuss the current density field. Outside the inhomogeneity the behavior of the current field is similar to that of the electric field, since j = 'YeE. Inside the body the current density increases in spite of a decrease of the component of the electric field in the direction of Eo. Since the amount of charge passing through any elementary surface cannot be infinitely large, with an increase of the conductivity 'Yi the magnitude of the current density tends to some finite limit that depends on the shape and size of the inhomogeneity. The vector j is directed along the primary field Eo. Until now we have studied the behavior of the field in the presence of a more conductive inhomogeneity. Next suppose that the body is more resistive, 'Yi < 'Ye • As follows from Eq, (III.268), positive and negative charges arise at the back and front sides of the body, respectively (Fig. III.14b), and therefore the distribution of the field is opposite to that considered in the previous case. For instance, the field increases inside the inhomogeneity as well as in the area around the equatorial plane, while it decreases in front of and behind the body. With an increase of the resistivity, Pi' the electric field inside gradually increases and in the limiting case of an insulator it reaches some finite value, which depends on the shape and size of the inhomogeneity. As concerns the potential Us, its behavior is similar to that in the previous case, but the positions of zones with the positive and negative values U, change. It is obvious that the behavior of the current density outside the inhomogeneity is similar to that of the electric field, but inside the current density tends to zero as the resistivity increases. In addition let us consider the behavior of the vector lines of both fields E and j near the inhomogeneity surface. As follows from Eqs, OII.237), and

j~ =j~

(III.272)

III.lO Behavior of the Electric Field in a Conducting Medium

331

where the indices "i" and "e" correspond to points of the inhomogeneity and the surrounding medium, respectively. Then making use of Ohm's law we also have and

(III.273)

Taking into account the continuity of the tangential component of the electric field and the normal component of the current density, Eq. (III.272), we obtain .e

-j

it

it

Yij~

and

Yej~

(III.274)

As is seen from Fig. III.l4c the direction of the field near the interface can be characterized by the angle Q', formed by the field E with the normal n. It is clear that

Then, Eqs. 011.274) can be rewritten as 'Y;

tan

'Ye Q';

tan Q'e

or

tan

Q'j

'Y;

tan

Q'e

'Ye

(III.275)

Of course, the same result is obtained if we proceed from the equation

In accordance with Eqs. (III.275) the vector lines of the fields E and j refract near the boundary, and this occurs in such a way that at either side the value of tan Q' is proportional to the conductivity of the medium. These vector lines approach the normal in the medium with higher resistivity as if they were trying to reduce their path length in this medium. Also it is clear that if one of the angles equals 0 or 1T/2, then the other angle has the same value, provided that both conductivities have finite values. In other words, in these cases vector lines are not refracted, although one of the field components has a discontinuity. This study shows that the vector lines of the field j concentrate inside of the inhomogeneity as well as in front and behind it, if 'Yi > 'Ye . This is

332

ID

Electric Fields

accompanied by rarefaction of these lines near the lateral part of the body surface. In the opposite case of a more resistive inhomogeneity, the vector lines of j are more rarefied in front of and behind the body as well as inside it. At the same time they are concentrated outside near the lateral surface. This behavior of current lines creates the impression that current tends to concentrate in the more conductive medium. If the inhomogeneity is an ideal conductor, then its surface becomes an equipotential surface and therefore the tangential component of the field E" equals zero. This means that the vector lines of both fields E and j are perpendicular to the interface. As concerns the behavior of the field inside the body the electric field vanishes, since Pi = 0, but the current density lines pass through continuously. In the opposite case, when the body is an insulator, 'Yi = 0, the normal component of the current density equals zero, j~ = 0, near the interface and therefore E~ also vanishes. Correspondingly the vector lines of both fields are tangential to the interface of the insulator. Let us consider one more feature of the behavior of the field related to the influence of an arbitrary inhomogeneity. Inasmuch as the sum of induced charges on the inhomogeneity surface equals zero, their field at distances essentially exceeding dimensions of the body approaches that of an electric dipole and, in accordance with Eq, (111.59), we have for the potential U. 1 M' L q p U ---+ ------;;-~ L qp ---+00 (III.276) 5

47TE O

L~p

where L q p is the distance between the observation point and any point within the inhomogeneity; M is the dipole moment, which is proportional to the amount of charges having one sign, and it depends on the primary field Eo as well as the shape and dimensions of the inhomogeneity. In general the field inside the body is not uniform, but there is an exception: When the inhomogeneity has a relatively simple shape and the primary field Eo is uniform within its vicinity. In this case the dipole moment M is proportional to the field Eo, and it has either the same direction as r. > 'Ye , or the opposite one if 'Yi < 'Ye . It is also obvious that Eq. 011.276) can be useful in evaluating the secondary field when observation points are located at distances sufficiently exceeding inhomogeneity dimensions. In contrast, often lateral changes of resistivity occur near the earth's surface, and in such cases measurements are usually performed outside and inside the inhomogeneity. For illustration the behavior of the potential and the tangential component of the electric field, caused by surface charges, is shown in Fig. III. 14d.

III.IO Behavior of the Electric Field in a Conducting Medium

333

Unlike the potential, the normal component of the electric field is a discontinuous function at the surface of the inhomogeneity, and this discontinuity is defined from Eq. OII.272). j~ = j~

or

Such inhomogeneities can be objects of investigation when electric methods are used for mapping; but at the same time they can present geological noise, which is often a very serious obstacle for the application of electrical prospecting. Until now we have considered the influence of an arbitrary inhomogeneity on the electric and current density fields. Now let us briefly discuss a case when the electric field is not disturbed by the presence of lateral resistivity changes. As follows from Eq. (111.268), charges are absent on the inhomogeneity surface if at every point the normal component of the primary electric field E nO equals zero. In fact, in such cases, continuity of the normal component of the current density is automatically satisfied without accumulation of surface charges. For instance, suppose that the primary field Eo is directed along the strike of a two-dimensional dike, shown in Fig. III.15a. Then the density of surface charges I is equal to zero, and therefore, the secondary field E, is absent. This means that everywhere the electric field coincides with the primary one and in this case it is impossible to discover the dike. At the same time the current density can be extremely large within the conducting body, since if

r.> Ye

Of course, if there is a component of the primary field perpendicular to the dike surface, then charges appear and they create a secondary field, which contains information about the geoelectric parameters of the body.

Example 2 Distribution of Charges in a Layered Medium

Suppose that a medium is a system of uniform regions with different resistivities and the current electrode A is located in its internal part enclosed by surfaces S;, as is shown in Fig. III.15b. Let us calculate the amount of charge arising at each interface. First of all, the electrode

334

III

Electric Fields

Fig. 111.15 (a) Electric field along a two-dimensional model; (b) the current electrode inside of a piecewise uniform medium; (c) model of a layered medium; and (d) a sphere in a uniform field.

charge eA equals

(III.277) where PI is the resistivity of the medium that surrounds the electrode, whose resistivity is neglected. In accordance with Eq, (111.19), the density of charge at the interface S, between media with resistivities Pi and Pi+ I is

I

(E ni + l -Eni )

=E o

or making use of Ohm's law and the principle of charge conservation we have

(111.278) where q is an arbitrary point of the surface Si' Taking into account the fact that the surfaces S, are closed around the electrode, the same amount of charge I passes through every one of them.

III.10 Behavior of the Electric Field in a Conducting Medium

335

Correspondingly, the total charge on the surface S, can be calculated as

or (III.279) Certainly, this is a very simple expression that shows that the amount of charge distributed on the interface S, is directly proportional to the difference of resistivities and the current. Before we continue, it is appropriate to make several comments. 1. Equations (111.278), (111.279) do not apply at the interface with an insulator since Pi+ I ~ 00. 2. The sign of the charge e i is defined by the resistivity difference. It is positive if Pi + I > Pi' but negative if Pi + I < Pi' At the same time the sign of the density I can vary from point to point. 3. Equation 011.279) determines the total amount of charge, but its distribution still remains unknown.

Now we will write down expressions for the charges arising at each surface Si' including the current electrode. Then we have eA = lOoPI] e l = loo(pz - PI)! e z = lOO(P3 - pz)! e 3 = lO O(P4 - P3)]

eN = lOO(PN - PN-I)]

where PN is the resistivity of the external region. Performing a summation over all the charges that appear in the medium, we obtain N

e

=

Lei = lOOPN] i~

(111.280)

1

This is an interesting result which demonstrates that the total charge

336

III

Electric Fields

arising at all interfaces coincides with the charge on the electrode surface, as it if were located in the external uniform medium with resistivity PN' As in the case of gravitational masses, any distribution of the volume or surface charges confined within some volume V creates practically the same field as that of a point charge, if the observation points are located far away and e O. Therefore, the asymptotic behavior of the potential of the total field in the medium, shown in Fig. IIUSb, is

*

U(p)~-

PN1

41T"L q p

(III.281)

This result is of great practical importance since it shows that if the current and receiver electrodes are located on the surface of the layered medium (Fig. III.ISc), and the distance between electrodes increases, the depth of investigation of such an array also increases. In other words, the voltage, measured between receiver electrodes located in the medium with resistivity PI' approaches that of a uniform half-space with resistivity PN' In essence, Eq. (III.281) explains one of the most important features of geometrical soundings used in electrical prospecting as well as in logging. In addition let us notice the following: 1. Strictly speaking each region of the layered medium is not enclosed by its two interfaces (Fig. III.1Sc). However, our results remain valid, since we can mentally imagine two surfaces at infinity, which make every layer closed, but the amount of charge passing through them equals zero. 2. Charges induced at each interface of the layered medium are not confined with an area of finite dimensions, but are spread over the entire interface. However, as was shown in Section III.S, their density ~ decreases relatively quickly, and due to this fact the field of these charges tends to that of a point source as the distance from the current electrode becomes sufficiently great. 3. Let us note that the amount of charge in the medium does not change if there is some region surrounded by a surface S * (Fig. III.1Sb) and located outside of the current electrode. In fact, the total charge arising on this surface is

and in accordance with the principle of charge conservation, the integral on the right-hand side equals zero. Here Po is the resistivity of this region. This means that an equal amount of positive and negative charge appears on the surface S *, and correspondingly, with an increase of distance their

111.10 Behavior of the Electric Field in a Conducting Medium

337

field tends to that of an electric dipole. In other words, in the presence of such an inhomogeneity the asymptotic behavior of the field is still described by Eq. (III.28l).

Example 3 A Conducting Sphere in a Uniform Electric Field (Fig. m.isn Suppose a sphere with radius a and conductivity 1'z is situated in a uniform electric field Eo. The surrounding medium has conductivity 1'1' Before we discuss the solution of the boundary-value problem, let us describe several obvious features of the primary and secondary fields. First of all we have assumed that the primary field is uniform. Certainly such a field cannot exist everywhere, since that would require an infinite power source. Nevertheless this approximation is very useful, if we restrict ourselves to a study of the field within some finite region around the sphere and assume that sources of the primary field are located at great distances from observation points. Inasmuch as the field Eo intersects the surface of the sphere, both positive and negative charges arise and their sum equals zero. These charges are sources of the secondary field and due to their presence we are able in principle to detect a conductor. It is natural to assume a distribution of charges possessing axial symmetry with respect to the axis, which is directed along the primary field and passes through the sphere center. For this reason we can expect that the potential, as well as both the fields E and j, have the same symmetry. Of course it is an assumption only, but very soon with the help of the theorem of uniqueness we will prove that this assumption, as well as others, is correct. Assuming the axial symmetry of the field, we will introduce a spherical system of coordinates R, 8, ep with its origin located at the center of the sphere and z-axis directed along the primary field. Then in accordance with Eqs. (III.253) the boundary-value problem is formulated in the following way: at regular points

1. 2.

3.

aU1

auz

1'1 aR

= 'Yz aR as

R -)

where Uo is the potential of the primary field.

if 00

R=a

(III.282)

338

m

Electric Fields

The boundary condition at infinity is obvious since the secondary field of induced charges decreases with an increase of the distance from the sphere. Taking into account the relative simplicity of the problem let us attempt to find a solution proceeding from our understanding of the behavior of the field. The potential of the primary field can be determined easily since the field Eo has only a component along the z-axis and therefore E

su;

o

(III.283)

=--

az

Whence

Inasmuch as the constant is not essential for determination of the field, we will put it equal zero. In other words, it is assumed that the potential of the primary field vanishes in the plane z = O. Then we have (III.284) Now let us consider the secondary field caused by surface charges. As we know the total charge equals zero and correspondingly far away from the sphere their field is equivalent to that of an electric dipole. Now we will assume that this behavior is observed everywhere outside of the sphere, regardless of the distance from the origin. Also let us suppose that an unknown dipole moment M is directed along the field Eo. Then the potential of the secondary field outside of the conductor is

U

M·R

(111.285)

= ------.,,s 47TS R 3

o

Thus, the potential outside of the sphere is UI =

~EoR

cos e +

Mcose 47TS oR

2

if

n».«

(III.286)

Finally, we will assume that charges are distributed in such a way that inside the sphere they create a uniform field, directed along the z-axis. Then the total field within the conductor can be represented as and if where C is also an unknown constant.

n s. a

(III.287)

rn.lO Behavior of the Electric Field in a Conducting Medium

339

With the help of several assumptions we have described a potential by Eqs. (III.286) and (III.287). Now it is time to check whether all of these assumptions are correct or wrong and, if they are valid, to determine the unknowns M and C. Of course, this task will be performed with the help of the theorem of uniqueness, and this means that using Eqs. (II1.286), (III.287) we will attempt to satisfy conditions of the boundary-value problem, Eq. (111.282). First, it is clear that the potentials of all electric fields caused by electric charges and considered outside of these charges, satisfy Laplace's equation; and, in particular, the potential of the uniform field and that of the electric dipole are its solutions. This fact can also be proved by substituting these functions into Laplace's equation. As follows from Eq. 011.286), the condition at infinity is satisfied too. Now let us find out whether we can provide continuity of the potential and the radial component of the current density at the sphere surface, using Eqs. (III.286), (III.287). Then we have for the potential,

and for the normal component of the current density, Yl{ - Eo -

2M 3 }cos 8 = -YzEoC cos 8 47TE oa

Thus, we have obtained two equations with two unknowns.

(III.288)

As is well known, a linear system of two equations with two unknowns does not always have a solution, and if the system (111.288) cannot be solved, this means that our assumptions or part of them were incorrect. However, Eqs. 011.288) have a solution, and we obtain M=47TEo

Yz - Yl Yz + 2Yl

3

a Eo

(111.289) 3Yl

C=--Yz + 2Yl

340

III

Electric Fields

Thus instead of Eqs. (III.286), 011.287) we have if

R

~a

(III.290)

and if

R
Then in accordance with the theorem of uniqueness we can conclude that Eqs, (111.290) describe the potential of the electric field caused by charges on the surface of the sphere, if the primary field is uniform. In fact, the functions UI and U2 satisfy all three conditions of the boundary-value problem, Eq. 011.282). Moreover, as follows from this theorem, there is no other solution except that given by Eqs. (111.290). This example vividly illustrates that regardless of the approach used to find a field, as well as the assumptions which have been made, only the theorem of uniqueness decides whether this function is a solution of the given problem. Now we will describe some features of the secondary field inside and outside of the sphere. First, consider the distribution of surface charges. Making use of Eq. (III.19) we have (III.291) since nand R are directed inside and outside of the sphere, respectively. Taking derivatives of the potential with respect to R we obtain

aUI aR

=

-Eocos8-2

Y2 - YI a

Y2 + 2YI R

and

Since E

au

R

3

- 3 Eo cos

=--

aR

8

341

111.10 Behavior of the Electric Field in a Conducting Medium

we have

or ( III.292) Thus, the density of charges I is distributed symmetrically with respect to the z-axis, and it decreases as cos 8 with an increase of the angle 8. The maximal magnitude of charge is observed at () = 0, where the primary field is perpendicular to the surface, and it is equal to zero in the equatorial plane, 8 = 'TT12. The dependence of the charge density on conductivities is defined by the coefficient K iz(III.293)

°

which varies between -1/2 and 1, as the ratio YZ/'Yl varies from to 00. lt is clear that the same relatively weak dependence on conductivity is inherent for the electric field, too. This means that the resolving capabilities of electrical methods, applied in mining prospecting, are usually very poor. As follows from Eq. (III.293), even for relatively small values of Yz/Yl the coefficient K 12 is almost equal to unity. That is, conductors having completely different conductivities can create practically the same field. In other words, the high sensitivity of electrical methods allows us on the one hand to detect relatively small changes of resistivity, and on the other hand it does not permit us to determine the resistivity of these conductors. Now let us consider the behavior of both fields E and j inside of the sphere. In accordance with Eq. (III.290), the electric field E z is uniform and directed, as is the primary one, along the z-axis.

auz E = -z

Ez =

aR

3Yl Yz+2YI

or

Eo =

3

2+(Yz/Yl)

(III.294)

Eo

Thus, with an increase of the conductivity of the sphere (yz > Y 1)' the field decreases and in the case of the ideal conductor tends to zero, while the

342

III

Electric Fields

current density

increases and approaches its limit, which is independent of the conductivity of the sphere and is equal to

jz

=

3y 1Eo

In the opposite case of a more resistive sphere, the electric field E z becomes greater with an increase of resistivity pz; but this change is relatively small. In fact, for an insulating sphere we have

£z = I.5E o In conclusion it is appropriate to notice that this analysis of the field behavior is useful for understanding different aspects of electrical methods in mining and engineering geophysics and also for studying the influence of geological noise in electromagnetic methods such as magnetotelluric soundings.

Example 4 Elliptical Cylinder in a Uniform Electric Field

Now we will study a field in a slightly more complicated case. Consider an elliptical cylinder located in a uniform medium and a primary electric field Eo, which is uniform and perpendicular to the cylinder axis (Fig. III.16a). In solving the boundary-value problem, Eq. (III.253), we have to provide continuity of the potential and the normal component of the current density at the cylinder surface. To simplify this procedure we will introduce an elliptical system of coordinates g, TJ. y Z

= a cosh

g cos TJ,

Z

= a sinh g sin TJ

Z

(III.295)

where a = {a - b ) l / Z is the eccentricity of the cylinder, and a and bare the major and minor semiaxes of the cylinder, respectively. This system is defined by two families of elliptical and hyperbolic cylinders that are orthogonal to each other and have the same focus at the points y=±a,

z=O

Both coordinates change in the following way:

os g <

00,

0

~ TJ

< 27T

and the coordinate TJ is measured from the y-axis.

III.I0 Behavior of the Electric Field in a Conducting Medium

343

Fig. III.16 (a) Elliptical cylinder in a uniform field; (b) current electrode at the borehole axis; (c) behavior of Bessel functions; and (d) behavior of the integrand in Eq. 011.336).

It is essential to note that the cylinder surface coincides with one of the coordinate surfaces, g = go. In this system the metric coefficients are

hl=h 2=a(cosh

2

g-COS

2

1) )1/2

and correspondingly Laplace's equation is written in the very simple form (III.296) In the same manner as in the previous case the secondary field arises due to induced charges that appear on the cylinder surface, and correspondingly we represent the potential of the total field in the form (III.297) Inasmuch as the primary electric field is directed along axis y, its potential

344

III

Electric Fields

Ua is Ua = -Eay = -Eaa cosh g cos

7]

or

(III.298) Substituting Eq. (III.298) into Eq. (III.296) it is easy to see that functions of the type or satisfy Laplace's equation. Now we will make two assumptions about the field behavior, namely, (a) Inside the cylinder the field remains uniform, and it is still directed along the y-axis, (b) Outside the cylinder the secondary field decays with an increase of the distance from the cylinder as a function e-e and depends on the angle 7] in the same manner as the primary field. Correspondingly, we can write

(III.299) Uz = -aEaB cosh

g cos 7]

Here A and B are unknowns that are independent of the coordinates g and 7]. It is obvious that both functions U1 and Uz satisfy Laplace's equation, and U1 obeys the boundary condition at infinity. as From the two conditions at the interface where

1'1

aU1 ag =

I'z

g = ga,

eo, ag

we obtain a system of equations determining the unknown coefficients A

lII.lO Behavior of the Electric Field in a Conducting Medium

345

and B. Solving this system we have

~

(1 -

A= 1+

) sinh

goe~o

'Y2

- tanh go 'YI

(III.300)

1 + tanh go B

= -~'Y""'2--

1+ -

tanh

go

'YI

Therefore, the functions UI and U2 , given by Eqs. (III.299), (111.300), satisfy all the requirements of the boundary-value problem, and correspondingly they describe the potential of the electric field. For the components of the field E we have 1

E

au

=--'7

h 2 aTJ

or

(III.301)

To describe some features of the behavior of the field, we will take into account the fact that tanh

b

go = - , a

1 + b/a 1 -b/a

First let us consider the distribution of charge as a function of the parameters of the elliptical cylinder and conductivity of the surrounding

346

III

Electric Fields

medium. As was shown earlier, the surface density I is defined by the discontinuity of the normal component of the electric field.

and in accordance with Eqs. (111.301) we have

(

~:

- 1) ego tanh go cos 7] (III.302)

The latter shows that the charge density has finite values at all points on the cylinder surface and at 7] = ±1T/2 it is zero; but it increases toward points 7] = 0, 1T, where under the conditions 1'2 b b --«1'-«1 1'1 a

a

it is

Earlier it was shown that the relationship between surface charge density and field strength can be written as

where E~ and E~-q are the normal components of the primary field and the field of the surface charges, respectively, except for the charge located at the point q. As follows from Eq. (111.301), the components E~ and E~-q have the same direction if 1'2> 1'1' and they are opposite to each other if 1'2 < 1'1' For this reason the charge density and the secondary field in the presence of a highly conductive cylinder, if (a> b), are greater than in the case of the highly resistive inhomogeneity. Proceeding from Eqs. (III.301) we will briefly describe the field behavior outside the cylinder. First of all it is clear that if the cylinder is more resistive than the surrounding medium 1'2 < 1'1' the electric field depends weakly on the conductivities and is mainly defined by the value of b/ a. With an increase of b/ a the electric field ETJ increases as well, because the observation point becomes closer to the charges. When the cylinder is characterized by higher conductivity than the surrounding medium

1II.10 Behavior of the Electric Field in a Conducting Medium (1'2/1'1» 1),

347

and the ratio of axes is not small, or more precisely when 1'2 b

--» 1 1'1 a the secondary field is controlled by the geometric parameters only. However, with a sufficiently elongated cylinder in the direction of the field Eo, 1'2 b - - «1 1'1

a

b

'

-« 1 a

(III.303)

the field of charges is relatively small, E~ «Eo, but it depends on the conductivity of the cylinder in both cases, whether 1'2/1'1 > 1 or 1'2/1'1 < 1. In essence, inequalities (111.303) are conditions, as the influence of an elongated cylinder on the electric field practically vanishes, and it behaves as an infinite layer with thickness 2b. It is interesting also to notice that the ratio of the field magnitudes contributed by charges on the surface of an ideally conducting cylinder and those on an insulating one is equal to the ratio of semiaxes of the cylinder, alb. Finally, consider the field inside of the elliptical cylinder, which can be rewritten as E 2 -

1 +bla 1 + ('Y2 bl'Y1 a )

E

°

or if

a rb

>1

(III.304)

where b ro L=--1 +bla

is often called the depolarization factor, since it characterizes the field decrease inside of the conductor as 1'2/1'1 > 1. Let us point out the following features of the field E 2 : 1. Surface charges are distributed in such a way that both the secondary field inside of the elliptical cylinder and the primary field are uniform and have the same direction.

348

III

Electric Fields

2. In the case of the circular cylinder we have

1

L=-

and

2

2Eo £2=----1 + (Y2IYl)

3. With an increase of the ratio alb the influence of charges decreases and we have if

Y2 b --~O

Yl a

This shows that with an increase of conductivity the cylinder has to be more elongated in order to us to neglect the field due to surface charges. 4. With a decrease of the conductivity of the surrounding medium Y 1 , the field inside of the cylinder decreases as well. As YI approaches zero the surrounding medium becomes an insulator, and the surface charges are so strong that their field completely compensates the primary one. Thus, in this case the elliptical cylinder behaves as an ideal conductor regardless of its resistivity. Of course, this conclusion is valid for an arbitrary conductor and any type of primary electric field (electrostatic induction). 5. If the elliptical cylinder is an insulator we have

E

Eo l-L'

if

=-2

a~ b

and correspondingly the maximal increase of the field E 2 is observed for the circular cylinder when L = 1/2.

E 2 = 2Eo 6. In accordance with Ohm's law the current density in the conducting cylinder is Y2 E o

1+(~~-I)L Here it is appropriate to distinguish two cases. (a) A very elongated cylinder when the field E 2 is practically equal to Eo- Then the current density becomes directly proportional to the cylinder conductivity.

III.lO Behavior of the Electric Field in a Conducting Medium

349

(b) A very conductive cylinder for which the product

is much greater than unity, even for a relatively elongated conductor. In such cases the current density is independent of the cylinder conductivity.

but it can essentially exceed the value of the normal field jo as a » b.

We have considered two examples, when a sphere or an elliptical cylinder is placed in a uniform field. In both cases the field inside of conductors also remains uniform. Such behavior is not a simple coincidence. In fact, we can prove that the field inside an ellipsoid with any ratios of axes a, b, c and arbitrary orientation with respect to the primary field Eo also remains uniform; but in general it does not have the same direction as Eo. In particular, if the primary field is directed along the major axis of the spheroid then the field inside is

and 2

1- e ( 1+e ) L = -3- In---2e 2e 1- e Here and

b=c


For instance, for elongated spheroids the depolarization factor L is

350

m

Electric Fields

simplified and we have

b? (2a

)

L z -Z In--1 «1 a b

if

a/b»1

In the next example we will consider a problem that plays a fundamental role in electrical logging.

Example 5 The Electric Field of the Point Source at the Borehole Axis Suppose that a very small current electrode A is placed at the borehole axis (Fig. III.16b). The borehole radius is a, and the conductivities of the borehole and surrounding media are 'Yl and 'Yz, respectively. ! is the current through the electrode. Our goal is to determine the electric field within the borehole. Before we formulate the boundary value problem, let us make use of the results derived in Example 2 and discuss the distribution of charges and some general features of the behavior of the field. First of all the charge arising on the electrode surface is

(III.305) and it creates a primary electric field

PI!

E O = -L4 3 Lop 7T"

(111.306)

Op

where Lop is the distance from the electrode to an observation point p, In the presence of this field electric charges also appear at the interface between the borehole and the formation, and as is well known their density is

where E~v is the normal component of the field, caused by charge eA and surface charges except in the vicinity of the point q. From this equation we can conclude that the surface density I decreases very rapidly with an increase of the distance from the electrode. In accordance with Eq. (111.278) we can represent the density I as

1= co(pz - Pl)jn

III.10 Behavior of the Electric Field in a Conducting Medium

351

since the normal component of the current density is a continuous function. Then, integrating over the entire borehole surface we obtain

Therefore, there are two charges. One of them is located on the surface of the small current electrode, the other is the charge distributed with a different density on the borehole surface, and their sum is e = eA + e s = EOPII + EO(P2 - PI)I = EOP21

That is, the total charge coincides with the charge on the current electrode as if it were located in a uniform medium with resistivity P2' The potential of the primary field is PII Uo(P)=-4 L 1T

(III.30?)

Op

while the potential caused by surface charges is defined by 1

Us(p) = 41TEO

f :2:(q) dS S

L qp

(III.308) It is obvious that the latter cannot be used for calculation of the field since i; is not known, but it is useful for our study. Let us consider the influence of the resistivity of the borehole and the formation near and far away from the electrode. As follows from Eqs. (III.30?), (III.308), in approaching the electrode the primary field becomes very large if the electrode radius is sufficiently small, while the secondary field tends to some constant, since the distance Lop is always greater or equal to the borehole radius. Thus near the electrode the primary field prevails, and therefore only information about the borehole resistivity can be obtained. In the opposite case, when the observation point is very far away from the electrode, charges spread over the borehole surface create practically the same field as that of an elementary charge es ' located at any point near the current electrode. As was mentioned earlier, the same equivalence is observed for the gravitational field, since as soon as the field is considered at sufficiently great distances from either masses or charges

352

III

Electric Fields

their distribution inside some volume is not important, and we can place all the mass or charge at any point of this volume. In our case, considering the electric field far away, one can mentally replace the surface distribution of charges by the elementary charge e s ' situated at the electrode, along with the charge eA. Thus the potential and the electric field at large distances from the current electrode can be represented as

(III.309) as

Lop» a

These equations show that as the distance between the current electrode and an observation point p increases, the field approaches that of a uniform medium with resistivity P2' in spite of the fact that the point p can be located within the borehole having resistivity Pl. As follows from the analysis performed in Example 2, the same asymptotical behavior is observed in the case in which there is an invaded zone with resistivity PI:>. between the borehole and the formation. After this qualitative analysis of the behavior of the field we will take the next step and derive exact formulas. These will establish relations between the field and its potential on the one hand and medium resistivities, the borehole radius, and the distance from the observation point to the current electrode on the other hand. First we will choose a cylindrical system of coordinates r, ip, z with its z-axis directed along the borehole axis and with its origin located at the center of the current electrode (Fig. III.16b). Then due to the symmetry of the model with respect to the z-axis and the plane z = 0, where the current electrode is located, it is natural to assume that the potential is independent of the coordinate cp and that it is an even function of z; that is, U=U(r,z)=U(r,-z)

(III.31O)

Now we are ready to formulate the boundary-value problem, which in accordance with Eq. (III.253), describes the behavior of the potential in the following way: 1. Inside of the borehole and within the formation, the potential U satisfies Laplace's equation.

111.10 Behavior of the Electric Field in a Conducting Medium

353

2. Near the current electrode, potential UI tends to that of the electrode charge.

as

R~O

where

3. With an increase of the distance R from the current electrode the potential in the borehole, U I , as well as in the formation, U2 , approach zero. as

R

~

00

4. At the borehole surface both the potential and the normal component of the current density are continuous functions.

au!

1'!

aU2

a; = 1'2a;

if

r =a

We will look for the potential as a function satisfying all four requirements. First of all we will find a solution to Laplace's equation. Taking into account the axial symmetry, it is proper to present this equation in the cylindrical system of coordinates.

a2u 1 au a2u -+ --+-= 0 ar 2 r ar az 2

(HI.311 )

since

au

-=0 alp

This is a differential equation of second order with partial derivatives, since the potential U depends on two coordinates, rand z. To solve this equation we will suppose that its solution can be represented as the product of two functions, so that each function depends on one argument only, and consequently we have

U=T(r)cP(z)

(III.312)

Substituting Eq. (111.312) into Eq. (111.311) we obtain

d 2T cP dT d2cP cP dr 2 + -; dr + T dz 2

=

0

354

III

Electric Fields

and dividing both sides of the equation by 1>T, 1 d 2T

1 dT

1 d 21>

--+ ---+--=0 2 T dr

rT dr

1>

dz?

(III.3l3)

On the left-hand side of Eq. (III.313) it is natural to distinguish two terms. Term 1 =

1 d 2T -

-2

T dr

1 dT

+ -rT dr

and

At first glance it seems that they depend on the arguments rand z, respectively, and Eq. (III.313) can be represented as Term 1( r) + Term 2( z)

=a

However, such equality is impossible, since changing one of the arguments, for example r, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of rand z. Therefore, we have to conclude that every term does not depend on the coordinates and is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the potential as a product of two functions. For convenience let us represent this constant in the form ±m2 , where m is called a constant of separation. Thus instead of Laplace's equation we obtain two ordinary differential equations of the second order. 1 d 2T

1 dT

T dr?

rT dr

---+--=+m 2 -

(III.314)

Let us emphasize that replacement of the differential equation with partial derivatives by two ordinary differential equations is the main purpose of the method of separation of variables, since the solution of the latter equations is known. To choose the proper sign on the right-hand side of Eq, (III.314) we will take into account the fact that the potential U in the borehole and in

355

III.10 Behavior of the Electric Field in a Conducting Medium

the formation is a symmetrical function with respect to the coordinate z, Eq, (III.31O). For this reason we will choose the minus sign on the

right-hand side of the equation for 4>, and correspondingly we have (III.315) where 2

4>"(z)

d 4> dz?

=-

As is well known the latter has two independent solutions, sin mz and cos mz; but we will use only cos mz, since it is an even function of the z-coordinate. Thus, the function 4> can be written as cfJ(z, m) = Cm cos mz

(III.316)

where Cm is an arbitrary constant independent of z. As follows from Eqs. (III.314), on the right-hand side of the equation for function T(r) we have to take the sign" +" and therefore, 1 T" ( r) + - T' ( r) - m 2 T = 0 r

( III.317)

where dT T'=-

dr '

Introducing the variable x = mr we have dT dT dx dT -=--=mdr

dx dr

dx

and

Substituting these equalities into Eq, OII.317) we obtain d 2T 1 dT -+---T=O dx? x dx

(III.318)

This equation is also very well known and is often used in various boundary-value problems with cylindrical interfaces. Its solution is modified Bessel functions of the first and second type but zero order, [o(x) and Ko(x), respectively. Their behavior is shown in Fig. III.16c, and they have been studied in detail along with other modified Bessel functions. Also we will use modified Bessel functions of the first order, [I(X) and KI(x), which describe derivatives of functions [o(x) and Ko(x); the relations

356

III

Electric Fields

between them are (111.319) Graphs of these functions are also given in Fig. 1II.16c. It is useful to show the asymptotic behavior of these functions.

Ko(x)

fo(x)~l

fj(x)

x

~

2"

~

-In x 1

Kj(x)~-

x

(111.320) as

x~o

and 1 fo(x)~exJ

27TX 1

f j ( x) ~ e' fj;;X

27TX

Ko(x)

~e-xv

Kj(x)

~e-xv

7T

2x 7T

2x

as

x~oo

Let us notice that modified Bessel functions are described in numerous monographs, there are many tables of their values, different representations of these functions, relations between them, polynomial approximations, etc. Certainly, application of these functions is as convenient as that of elementary functions. Thus, a solution of Eq. (111.318) can be represented as

T(x) = Dfo(x) + FKo(x) or (1II.321) where D m and Fm are arbitrary constants that are independent of r. Now making use of Eq. (III.312), for each value of m we have (1II.322) where Am and Em are unknown coefficients that depend on m. It is clear that the function VCr, Z, m) satisfies Laplace's equation and we might think that the first step of solving the boundary-value problem is accomplished. However, this assumption is incorrect, since the function VCr, Z, m) depends on m, which appears as a result of the transformation of Laplace's equation into two ordinary differential equations, while the potential V describing the electric field in the medium is independent of

111.10 Behavior of the Electric Field in a Conducting Medium

357

m, Inasmuch as the function Ut;r, z, m) given by Eq, (111.322) satisfies

Laplace's equation for any m, we will represent U as the definite integral.

which is independent of m, Thus we have arrived at the general solution of Laplace's equation, which includes an infinite number of solutions corresponding to different coefficients Am and B m. Now we are ready to perform the second step in solving the boundary-value problem: to choose among the functions Am and B m solutions, which obey the boundary conditions near the current electrode and at infinity. With this purpose in mind, we will take into account the asymptotic behavior of the functions Io(mr) and Ko(mr). As was shown earlier, Ko(mr) tends to infinity as its argument approaches zero, and therefore this function cannot describe the potential of the secondary field within the borehole. At the same time the function IaCmr) increases unlimitedly with an increase of r and correspondingly it cannot describe the field outside of the borehole. Thus, instead of Eq. (111.323), we can write if

r -s a

(111.324) if r ~ a

It is clear that these functions satisfy both Laplace's equation and the boundary conditions. In fact, in approaching the current electrode the function U1 tends to the potential caused by the charge on its surface, while with an increase of r the function Uz, due to the presence of Ko(mr), decreases. Also both integrands in Eqs. 011.324) contain the oscillating factor cos mz, and therefore the functions U1 and Uz tend ~o zero as the distance along the z-axis increases. This asymptotic behavior will be considered later in detail. Next we will satisfy the last requirement of the boundary-value problem and find coefficients Am and Bm such that they provide continuity of the potential and the normal component of the current density on the borehole surface. To simplify our transformations, it is important to represent the potential of the primary field in terms of the same functions as the

358

III

Electric Fields

secondary field. Such a representation is very well known, and it is called the Sommerfeld integral.

1

R

2 -1 Ko(mr) cos mzdm 00

- =

(II1.325)

17" 0

where and

r =1= 0

Then an expression for the potential inside the borehole is if

r'*O (II1.326)

where PI 1 C=217"2

Now the conditions at the interface r = a can be written as

(III.327) and

1'21o BI1IK~(ma) 00

=

cos mzdm

where

100x) =

dl~X)

,

dK (x)

Ko(x) = __ 0_ dx

Both equations contain an infinite number of unknowns Am and B m , and they can be considered as integral equations with respect to Am and Bm • Fortunately, there is one remarkable feature of these integrals that allows us to drastically simplify them. In fact, they are Fourier cosine transforms, and from the theory of these integrals it follows that their equality results in the equality of their integrands. Therefore we have CKo(ma) +Amlo(ma) =BmKo(ma)

1'1{ -CK1(ma) +AmI1(ma)} =

-1'2

Bm K l( ma )

(III.328)

359

111.10 Behavior of the Electric Field in a Conducting Medium

since

This is a dramatic simplification, inasmuch as instead of integral equations we obtain for every value of m a system of two linear equations with two unknowns, Am and Em' whose solution is

and

Let us notice that in deriving the expression for Em the equality

has been used. Thus the functions VJ(r, z ) and Vir, z ), given by Eqs. (III.324) and (111.329), satisfy all requirements of the theorem of uniqueness; correspondingly we can say that these functions describe the potential of the electric field caused by the electrode charge and charges distributed on the borehole surface. Inasmuch as measurements of the potential difference as well as the electric field are of a great practical importance in electrical logging, let us write their expressions in the borehole.

PJI[

V(r,z)=417

1

2

/r 2+z 2 00

X

E,

=

PJI[

-4 17

2

fa z

(r +z

+-(YJ-Y2) 17

Ko(ma)KJ(ma)lo(mr) cos mz dm ] Y2 Io(ma)K J(ma) + y11](ma)Ko(ma) 2 2)3/2 + - ( Y] - Y2) 17

] xlo Y21o(mKo(ma)KJ(ma)lo(mr)sinmz dm ma) K]( ma) + Y]1]( ma) K o( ma) 00

(III.330)

360

III

Electric Fields

since

au

E

=--

az

Z

In particular, if measurements are performed at the borehole axis r = 0, these expressions are slightly simplified, since r = and we have

°

PI I [ 1 2 U(O,L) = 47T L + 7T(Y]-Y2)

Ko(ma)K](ma) cos mLdm + y]I](ma)K o(ma)

00

X

fa

1

Y2 10( ma ) K I( ma )

(III .331)

and PI!

[1 2

EAO,L)= 47T L 2 + 7T(Y]-Y2) 00

X

fa

mKo(ma)K](ma) sin mLdm yJo(ma)KJCma) + y]I J ( m a ) K o(ma)

1

where L is the distance between the current and receiver electrodes of the normal and lateral probes, which, as was described earlier, measure the potential and the electric field, respectively. In essence in both cases a difference of potentials or voltage is measured. However, with the normal probe the second receiver electrode is located far away and its potential is practically zero, while with the lateral probe both receiver electrodes are close to each other, and we can say that the voltage is equal to the product of the electric field and the distance between these electrodes. Here it is appropriate to make to comments. 1. Algorithms for integration and calculation of modified Bessel functions are very well elaborated, and for this reason determination of the potential and the electric field by Eqs. (III.33l) is a relatively simple task. 2. Applying Eqs, (III.33l) and making use of the principle of superposition, we can solve the forward problem for more complicated probes formed by several current and receiver electrodes, such as the seven-electrode laterolog.

Now let us consider Eqs. OII.33l) in detail. For simplification let us introduce new variables

x=ma,

L

a=-, a

YI

P2

Y2

PI

s=-=-

111.10 Behavior of the Electric Field in a Conducting Medium

361

and introducing them into Eq. OII.33!) we obtain U(O, L)

=

Ua[1 + (5 - l)a

~ {.oA~

cos axdx] (III.332)

Ez(O, L)

=

E az [1 + (s -1)a 2

~ fa"'xA: sin axdx]

where Ull and E az are the potential and electric field at the z-axis, caused by the charge eA if it were located in a uniform medium with resistivity Pl' PI!

U,=--

a

4rr L '

(III.333)

and

Thus the electric field and its potential can be represented as the product (III.334) and the functions FE and Fu depend on two parameters, namely, the probe length, expressed in units of the borehole radius a, and the ratio of conductivities s. It is obvious that these functions characterize the influence of the medium and the probe length, since they show how the field and its potential at the borehole axis differ from the corresponding functions in a uniform medium with the borehole resistivity PI' Let us rewrite Eqs. (III.334) as and

F

U =u

Ua

Very often these ratios are presented in the following form: and

p~ U -=-=F PI Ua u

(III.335)

where p; and p~ are called the apparent resistivity of the lateral and normal probes, respectively. Of course, we can introduce similar expressions for the apparent resistivity of more complicated probes. As an example, we will consider the relationship between the potential U and the geoelectric parameters of the medium. With this purpose in

362

III

Electric Fields

mind, we will study the behavior of the function F u ' In accordance with Eqs. (III.332), 2a -1 A~ cos a x dx 00

Fu

=

1 + (s - 1)

7T

(III.336)

0

where A~ is given by Eq. OII.333) and it is independent of the parameter a. With a decrease of the probe length, the ratio a tends to zero and according to Eq. (III.336) we obtain

Fu~l+(s-l)-lA~dx~l

2a 7T

00

if

0

a~O

That is, this potential approaches the potential caused by the charge eA only, which is located on the surface of the current electrode. Certainly, this is a known result, but it is derived in a different way. Now we will investigate the opposite case in which the probe length increases, and correspondingly the parameter a tends to infinity. To explain the asymptotic behavior of the function Fu ' let us pay attention to the integrand in Eq. (III.336). This is the product of two functions. A~ cos ax One of these functions, A~(x), gradually decreases without a change of sign, while cos aX is an oscillating function. The interval ~x, within which it does not change sign, is defined by the condition 7T

Lix= a With an increase of the parameter a this interval decreases and, correspondingly, A~ becomes practically constant within every interval ~x. Taking into account the fact that A~ is a continuous function of x, we can say that with a decrease of ~x the integrals over neighboring intervals are almost equal in magnitude, but have opposite sign. In other words, they cancel each other, and with an increase of a this behavior manifests itself for smaller x. This means that in the limit as a tends to infinity the integral in Eq. (III.336) is defined by very small values of x, and this is illustrated in Fig. III. 16d. Taking this fact into account, we will simplify the expression for A~(x), replacing the functions Io(x) and I/x) by their asymptotical formulas, Eq. (III.320). Then, instead of Eq. (III.333) we obtain if

x~O

363

111.10 Behavior of the Electric Field in a Conducting Medium

Correspondingly, the asymptotic expression for the potential is

V(O, L) =

o;[1 + (s -

2a 1) --;;:-

fa Ka(x) cos ax dx 00

]

if

a

~

00

and in accordance with Eq. (111.325) we have

V(O, L)

=

Va 1 + (s - 1)

[

a]

~

vl +a z

=

pz!

Vas = 47TL

as

a

~

00

Again we have demonstrated that with an increase in probe length the potential in the borehole approaches that of a uniform medium with the resistivity of the formation. Let us note that by applying the same approach we can derive similar expression for the electric field. Now we will discuss the results of calculation of the ratio

o,

V

Pi

ti;

presented in log-log scale in Fig. III.17a. The index of the curves is s = Pz!Pi' and the parameter a is plotted along the horizontal axis. The function V is given by Eq. (III.336). We will distinguish several features of these curves, which reflect the behavior of the potential at the borehole axis. 1. All curves have left and right asymptotes, corresponding to the resistivity of the borehole and formation, respectively. 2. With an increase of the formation resistivity, or more precisely s, the approach to the right asymptote takes place at greater distances from the current electrode. Similar behavior is observed in the case when the formation is more conductive. 3. With an increase of the separation all curves intersect the right asymptote, have a maximum, and then decrease, gradually approaching their asymptotes. This indicates that there is a range of separations where the potential exceeds that in a uniform media with resistivity P:» if s> 1. 4. In the case of a more conductive formation, apparent resistivity curves also intersect the right asymptote. 5. The more the parameter s differs from unity, the smaller the distance should be between the current electrode and the observation point in order to neglect the influence of charges at the borehole axis. 6. If the formation is relatively resistive, s » 1, there is an intermediate zone where the apparent resistivity curve has a slope that is approximately equal to 45°. This zone becomes wider with an increase of s. Such behavior of the apparent resistivity means that the potential remains

364

III

Electric Fields

practically constant. In fact, from Eqs. (III.307) and (111.335) it follows that Pa = 47TUL

log Pa = log 47TU + log L

or

and this equation describes a straight line with a slope of 4SO if U = constant. Now we will study the behavior of the field within this zone. In the simplest case, when the formation is an insulator, P2 = 00, the distribution of currents in the borehole can be represented in the following way. Very near the electrode A the current lines are almost radial in direction. With an increase in distance they tend to be parallel to the borehole axis. Correspondingly, at sufficiently large distances from the electrode A we can expect a uniform distribution of current density, which has only a z-component, equal to I jz= - 22

7Ta

Here I is the current arriving at the electrode from the wire, while a is the borehole radius. The coefficient 1/2 is introduced since the current I is symmetrically distributed with respect to the electrode A. Applying Ohm's law,

.

Ez

J-= " PI

we find that the electric field inside of a borehole with resistivity PI is E

PII 27Ta 2

I 25

=--=z

(III.337)

where 5 is the borehole conductance (III.338) Equation (111.337) describes an electric field, which is uniform within a cross section of the borehole, and it does not change along the z-axis. Therefore, the voltage V, measured between two arbitrary points M and N, V= U(M) - U(N) =EzMN

(III.339)

remains constant. It is obvious that the current is accompanied by the appearance of surface charges that create completely different fields inside and outside of the borehole. In fact, inside the borehole the electric field is uniform if

365

111.10 Behavior of the Electric Field in a Conducting Medium

a »1 and it is directed along the z-axis, Eq. (111.337), while outside, r > a, charges create only a radial field. A different behavior of the field and currents occurs when the surrounding medium is conductive (pz =1= (0) since the radial component of the current density outside the borehole is not equal to zero. In other words, a leakage of current from the borehole into the formation occurs. Correspondingly, with an increase in the distance from the electrode A the current through the borehole cross section decreases. It is obvious that this behavior of the current along the borehole is observed because both vertical components of the current density and the electric field decrease with the distance z. It is clear that with an increase of the formation conductivity, the electric field E; would decrease more rapidly. This description allows us to assume that within the intermediate zone the borehole behaves as a transmission line, and now we will demonstrate that this assumption is correct. In accordance with Ohm's law the change of the potential dU along an arbitrary element of the borehole, dz (Fig. m.rn», is

I( z) dz dU= - - - S

(III.340)

where dzjS is the resistance of the borehole element dz, U(z) is the potential at the point p, and It z) is the current through the borehole cross section at point p. The negative sign in Eq. (III.340) is introduced since

dU= U(z + .lz) - U(z)

U(z) > U(z + .lz)

and

A change in current along the borehole occurs due to leakage into the formation. In other words, we can consider that the surrounding medium with resistivity pz is connected in parallel with the borehole. Consequently, within the interval dz this leakage current dl, is equal to
T = -

dz

T

dI

= r

-

-

dz

dI( z)

(111.341)

where T is the resistance per unit length of the formation to radial current. Thus, Eqs. (1ll.340), 011.341) can be represented as dI 1 -=--U(z) dz T

(III.342)

366

III

Electric Fields

After differentiation of this system we obtain (111.343 )

and where 1

n=--

(III.344)

ISf

Thus, the distribution of potential and current is defined by the parameter n, provided that our assumption about the field behavior within the intermediate zone is correct. Let us note that U(z) is the potential at some point of the borehole with respect to infinity, where U equals zero. In particular, any cylindrical surface with a sufficiently large radius that is coaxial with the borehole has practically zero potential. As is well known, any solution of Eqs. (III.343) has the form

where A and B are unknown constants. Due to leakage, the current in the borehole tends to zero as z ~ ± 00, and we have

I(z) =Be- n z

Z> 0

if

(III.345)

Because of the symmetry with respect to the electrode A, half of the current goes in one direction, while the other half goes in the opposite direction. Therefore, the initial condition is t

I( z) =

"2

z=0

as

since the effect of the leakage is negligible near the source. Correspondingly, we have f fez) = <e ? " (III.346) 2 where I is the current near the electrode. Making use of Ohm's law we have, for the electric field at any point of the borehole,

. E;

=

I(z)

Pj}z = PI lra 2

I(z) =

-S-

or I E = -e- z / P z 2S

(111.347)

III.I0 Behavior of the Electric Field in a Conducting Medium

367

Taking into account the fact that

au az

E

=-Z

we obtain for the potential

Iff

-e- z / P S

U(z)=2

(111.348)

To compare this result with the curves given in Fig. III.17a we will again introduce the apparent resistivity, corresponding to Eq. (III.348). Then making use of Eq. 011.335) we have Pa = 27T"L PI

!!.e-

VS

PI

L/

P

(III.349)

where L is the probe length. . It is easy to see that the results of calculations by Eq, (111.349) coincide with the data given in Fig. III.17a, provided that the transverse resistance T equals P2 T=P2

(IIl.350)

and the formation is much more resistive than the borehole. This fact strongly confirms that within the intermediate zone, where P2 »> Pl' the behavior of the current and the potential is governed by the transmission line equation when its conductance is equal to that of the borehole. The transverse resistance is equal to the formation resistivity if the surrounding medium is uniform. Let us write expressions for the potential, the electric field, and the second derivative of the potential.

Ilt2

U(z)=2

I

.~ _ e - Z/ySP2 S

E(z) = _e-z/YS;;;

25

(111.351)

As follows from their comparison, the second derivative of the potential is more sensitive to the formation resistivity. In fact, in accordance with

368

ill

Electric Fields

Fig. III.I7 (a) Apparent resistivity curves; (b) electric field in a borehole surrounded by an insulator; (c) two-layered medium; and (d) Bessel functions Jo, Jo.

IIl".lO Behavier of the Electric Field in a Conducting Medium

369

Eqs. (III.343) we can conclude that

d 2U(z) U(z) dz 2 S

12

=

(III.352)

and since these equations are differential equations, the latter can also be used in the case when the formation is a horizontally layered medium. In conclusion we will make two comments. 1. Equation (III.352) shows that in principle we can measure the formation conductivity through the casing as well as during drilling. 2. Comparison of the results of calculations by Eqs. (III.349), (III.350) with the apparent resistivity curves presented in Fig. III.17a also demonstrates that even for a relatively conductive formation, 10 < s < 1000, the leakage effect described by Eq. (111.343) plays a very important role.

Example 6 The Electric Field on the Surface of a Medium with Two Horizontal Interfaces (Fig. III.17c) Suppose that a current electrode A is placed on the surface of a two-layered medium. The upper layer has thickness h, and 11 and 12 are conductivities of the layer and the basement, respectively. First let us consider the distribution of charges, which appear at interfaces only. In accordance with Eq. (III.173) the charge on the electrode surface is (III.353) and correspondingly the potential and the electric field of the primary field are and

(III.354)

Due to the presence of the primary electric field Eo, surface charges arise at the interface between the layer and the basement; and in accordance with Eq, (III.178) their density is

Taking into account the fact that the current through this interface equals

370

III

Electric Fields

!, we obtain for the total charge at the surface

(111.355)

or es

=

EO(PZ - PI)!

The charges distributed over the interface then induce charges at the earth's surface. As was demonstrated earlier, Eq. (111.172), each elementary charge located in a conducting medium creates at the boundary with an insulator a surface charge with the same sign and magnitude. For this reason, the induced charge on the earth's surface, eo, coincides with that at the interface, e s , but it is distributed in a different manner; that is,

(III.356) Thus the total surface charge in a medium is (111.357) This means that the charge, which arises at all surfaces, coincides with the electrode charge, as if the electrode were located at the surface of a uniform half space with the resistivity of the basement, pz. As in the previous example, this analysis allows us to establish the asymptotic behavior of the field as a function of the distance between the current and receiver electrodes. In fact, let us represent the potential as the sum PI!

(III.358)

U=-+U 21fT

s

where U. is the potential caused by charges at the bottom of the layer and at the earth's surface. Inasmuch as the potential U. has a finite value everywhere, in approaching the current electrode both the potential and electric field are mainly defined by the charge at the electrode surface; that is, and

PI!

Er(r) ~ - 2z

(111.359)

1fr

In other words, with a decrease of the separation between the current and receiver electrodes, r, the depth of investigation also decreases, since the field is practically defined by the resistivity of the upper layer, PI' only. In the opposite case, when the observation point is far away from the current electrode, the influence of surface charges, as in the previous example, is the same as if the total charge were placed at the current

ill.to Behavior of the Electric Field in a Conducting Medium

371

electrode. Then, in accordance with Eq. CIII.357) we obtain pz!

U(r)--7

pz! Er--721Trz

and

(1II.360) 21T r, This means that with an increase of the separation r, the depth of investigation increases as well, in spite of the fact that the current and receiver electrodes are placed in the upper layer with resistivity Pl' This happens because the electric field and its potential become functions of the basement resistivity pz only, and this result does not depend on the thickness of the upper layer. Moreover, as follows from the study of the charge distribution in a layered medium, this asymptotic behavior remains valid, regardless of the number of layers and the presence of structures having finite dimensions. In essence we have explained the main concept of geometric soundings based on measuring the voltage at different separations from the current electrode. Now we will derive equations for the potential and the electric field at any separation from the current electrode, and with this purpose in mind, we will solve the boundary-value problem. First of all taking into account the axial symmetry with respect to the vertical axis, perpendicular to interfaces and passing through the current electrode, we will choose a cylindrical system of coordinates r, q;, Z as shown in Fig. III.17c. In accordance with Eqs, (111.254) the boundary-value problem is formulated in the following way: 1. Within the layer and the basement, the potential satisfies Laplace's equation, if

0

~z

~

h

and if

z:2:: h

2. In approaching the current electrode, the potential UI tends to that caused by the charge at the current electrode. PI!

U

--7 - -

21TR

I

if

R

--7

V

0

where R = r z + z z . 3. At the earth's surface the normal component of the current density is zero, and therefore

aUI -=0

az

as

z

=

0

372

III

Electric Fields

4. With an increase of the distance R from the current electrode, the field and its potential tend to zero. as

R

-+ 00

5. At the interface z == h the potential and the vertical component of the current density are continuous functions.

aU1 1'1

az

aU2 =1'2

if

az

z= h

Let us notice that, since the conducting medium is surrounded by surfaces at every point of which boundary conditions are defined, we do not need to consider the field above the earth's surface. Now we will determine the potential that satisfies all these conditions, beginning with Laplace's equation. In accordance with Eq. CIII.31l) we have

a2u 1 au a2u -+ --+-=0 ar2 r ar az 2 since, due to the axial symmetry, the potential U is independent of the coordinate cpo Applying the method of separation of variables and representing the potential as

U(r,z) == T(r)q,(cp) we again obtain two ordinary differential equations. 1 d 2T T dr?

1 dT rT dr

---+--=+m

2

-

If we choose the positive sign on the right-hand side of the equation for TCr), as was done in the previous example, then the solutions would be

the modified Bessel functions IoCmr) and KoCmr), which have singularities either at infinity or at points of the z-axis, respectively. Inasmuch as all these points are located within the upper layer and in the basement functions, IoCmr) and Kimr) cannot describe the potential, which everywhere has a finite value except at the origin of coordinates. For this reason we will take the negative sign on the right-hand side of the equation for T

111.10 Behavior of the Electric Field in a Conducting Medium

373

and then obtain (III .361)

(111.362)

The solution of the second equation consists of exponential functions. (III.363)

where em and D m are unknown coefficients, which are independent of z. Introducing a new variable x = mr we can represent the first equation in the form d 2T 1 dT -+--+T=O dx? x dx

(111.364)

The solutions of this equation are Bessel functions of the first and second type but zero order, Jo(x) and Yo(x), respectively, which are thoroughly studied and widely used in numerous theoretical and engineering problems. Inasmuch as the function Yo(mr) has a logarithmic singularity at points on the z-axis, it cannot be used to describe the field. The behavior of the function Jo(x) is shown in Fig. III.l7d. In particular, we have if

x« 1

and

(III.365)

if

x» 1

Thus for every value of the separation constant m we obtain

and correspondingly the general solution of Laplace's equation, which does not depend on m, is (111.366)

Having accomplished the first step in solving the boundary-value problem, let us satisfy the other conditions.

374

III

Electric Fields

Representing the function Uir, z ) in the upper layer by

we see that Uj(r, z ) satisfies the boundary condition near the electrode as well as Laplace's equation. To satisfy the condition at the earth's surface, auj/az = 0, we will take the first derivative of the potential with respect to z. Then we obtain

eu, az

Pj[z 21T(r 2 +z 2 ) 3/ 2 (III.368)

Letting z

=

0 we have

(111.369) This is a very complicated integral equation with respect to unknowns and D m , but fortunately for us integrals of the type

em

have one remarkable feature similar to that of Fourier integrals; namely, from the equality

or (111.370) it follows that (111.371 ) Thus, instead of Eq. (111.369) we have

Ill.I0 Behavior of the Electric Field in a Conducting Medium

375

and correspondingly the expression for the function U, is slightly simplified. (I1I.372) In the basement, where z increases unlimitedly, we will represent the solution to Laplace's equation in the form

Uz(r,z) = ['Bme-mZJa(mr)dm o

(I1I.373)

which of course satisfies the condition at infinity, as z ~ 00. Also, due to the oscillating character of Ja(mr), both functions U1 and Uz tend to zero as the coordinate r increases unlimitedly. To satisfy the conditions at the bottom of the upper layer it is necessary, as in Example 5, to represent the potential of the primary field in the form corresponding to that of the secondary field. With this purpose in mind, we will make use of the Lipschitz integral.

or

(111.374) if

z> 0

Substituting Eq. (111.374) into Eq, (I1I.372) we have

Ul ( r, z)

=

{Or Ce- mz + Cm( e mz + e- mz)] Ja( mr) dm a

where

pJ

C=27T Respectively, continuity of the potential and the normal component of the current density occurs at the interface z = h, if

(111.375)

376

III

Electric Fields

and

= -"/2[OBme- mhmJo(mr) dm o Now, again making use of Eqs, (III.370), (III.371), a drastic simplification occurs and we obtain for every m two equations with two unknowns. Ce- mh + Cm( e mh + e- mh) = Bme- mh (III.376) "/1[ _Ce- mh + Cm( e mh - e- mh) 1= -"/2 Bme- mh Let us note here that in both the case of the borehole and that of the horizontally layered medium we, in essence, observe one of the most important features of the so-called special functions, namely, their orthogonality. This feature makes them extremely useful for solving numerous boundary-value problems. Solving the system (III.376) we have K e- 2mh P 1 C = 12 1 m 1 - Kl2e-2mh 2'77" (III.377)

where _ P2 -P1 K 12 P2+PI

Therefore, the functions UI and U2 , given by Eqs. (III. 372), (III.373), satisfy all the conditions of the boundary-value problem, provided that the coefficients Cm and B m are defined by Eq, (IIL377). Correspondingly, U1 and U2 describe the potential of the electric field when the current electrode is located at the origin of coordinates. In particular, if the observation points are located at the earth's surface we have 2mh ] PII[l e- + 2K 12 _2mhJO(mr) dm UI(r) = 2'77" r 0 l-K e

1

00

and

12

(III.378)

377

111.10 Behavior of the Electric Field in a Conducting Medium

where E, = J1(x)

-aular

is the radial component of the electric field and

= -dJO<x)/dx is the Bessel function of first order (Fig. III.l7d).

We have derived formulas (IIl.378) for a two-electrode array, but by applying the principle of superposition it is a simple matter to obtain an expression for the voltage of an arbitrary system of electrodes. Also, by making use of the same approach in solving the boundary-value problem it is easy to generalize Eq. OII.378) for an n-layered medium. Algorithms for calculating integrals that describe the field and its potential at the surface of a horizontally layered medium are very well developed and have standard procedures. Introducing a new variable, x = mr , we will write Eq. (III.378) for the electric field as (III.379) or

where E Or is the primary electric field and FE is a function that depends on two parameters only. Pz

s=-

and

h a=r

PI

Before we demonstrate the results of calculation of the electric field, let us study its asymptotic behavior. As follows from Eq, (III.378), when approaching the current electrode the integral tends to zero, since J1(x) ~ 0 as x ~ 0, and correspondingly the field is defined by the electrode charge. Er~EOr=

PII 27rr

(1II.380)

--2

In the opposite case, when the distance r increases and r/h >>>- 1, we will take into account that, due to the oscillating character of the Bessel function J1(x), the integral is mainly defined by values of the integrand at the initial part of the integration. Then, letting the exponents approach a value of one, we have as

m~O

378

III

Electric Fields

and therefore,

1 00

1+2 K12 E; = -PJ [2" mJ 1( m r ) dm 27T r l-K I2 a

if

]

r

~ 00

Inasmuch as mJ 1( m r )

=

a ar

-Ja(mr)

-

then, in accordance with Eq. (111.374), 1 -

r

1a Ja(mr) dm 00

=

and we have

r~

if

00

or if

r~

00

(111.381 )

since _ P2-Pl K 12 -

P2 +Pl

Thus, regardless of the ratio of resistivities, with an increase of the separation r, the electric field approaches the value of the field in a uniform half space with the resistivity of the basement. Of course, both asymptotics, as either r ~ 0 or r ~ 00, have been derived earlier proceeding from the charge distribution. Now we will consider the special case when the basement is an insulator and the separation r is much greater than the upper layer thickness h. Then, letting K 12 = 1 and assuming that m ~ 0, we have me- 2m h -----;;:---;1 - e- 2m h

since e- 2m h

:::: 1

- 2mh.

m ~

1

-- = -

2mh

2h

as

m

~O

111.10 Behavior of the Electric Field in a Conducting Medium

379

Therefore we obtain

1 Jo( mr) dm ar

= -PI I [ -12 - -1 -a 27T r

h

00

]

a

if

r

~

00

Finally, by neglecting the first term we have I

E=-r 27TSr

if

r

-» 1 h

(III.382)

where S = Y1h is the conductance of the upper layer. Equation 011.382) shows that with an increase of the distance the electric field becomes inversely proportional to the conductance S, and it does not depend separately on the thickness and resistivity of the upper layer. In other words, by performing measurements of the field far away from the current electrode we can determine the conductance S of the layer only, if P2 = 00. It is helpful to arrive at Eq. (III.382) in a different way. As is seen from Fig. III.18a, with an increase of r the vector of the current density becomes practically horizontal and independent of both coordinates rand z. Then the total current through any lateral surface of the cylinder with radius r and height h is

and taking into account Ohm's law, j, = YIE" we again obtain Eq. (111.382). It is always useful to derive the same equation from the mathematical and physical points of view. For instance in our case this approach allowed us to understand that the asymptotic behavior of the field described by Eq. (111.382) corresponds to a uniform distribution of the current density in the upper layer. Moreover, now we are able to generalize Eq. (111.382) for an n-layered medium, provided that basement is an insulator (Fig. III.18b). Since the electric field is horizontal far away from the current electrode, the equipotential surfaces become themselves lateral surfaces of cylinders with axis z. We will consider two arbitrary equipotential surfaces located

380

III Electric Fields

Fig. HI.1S (a) Current density distribution in a conductive layer; (b) horizontally layered medium and distribution of currents in the far zone; (c) apparent resistivity curves; and (d) model of a medium with a vertical contact.

at distance

/).r.

The voltage between them can be written as (1II.383)

where I, is the current in ith layer, R, is the resistance of the cylindrical layer with thickness Sr and height hi. The latter is the thickness of ith layer. It is obvious that

v=

Er/).r

and

Pi/).r

/).r

R·=--=-I 27Trh i 27TrS j

where S, is the conductance of the ith layer. Then instead of Eq. (111.383) we obtain

t,

=-=

Sj

(111.384)

III. to Behavior of the Electric Field in a Conducting Medium

381

Also taking into account the fact that far away from the electrode we can assume that the layers are connected in parallel, we have (III.385) or S2 S3 I=I1 + - I1 + - Ii S1 S1

+ ...

Sn +-1 S 1 1

Thus (III.386) where n

is called the total conductance of a system of layers, and this parameter often plays an important role in the interpretation of an electrical sounding. Finally, from Eqs. (III.384) and OII.386) we have II I E = -- = -r

27TrS I

27TrS

if

r

- » 1, h

Pn+1

=

00

(III.387)

that, in fact, is the general version of Eq. (III.382). Let us note that the range of separations, where Eq. OII.387) describes the field, is usually called the S-zone. To complete the study of the asymptotic behavior of the field, it would be natural to investigate the case when the basement is an ideal conductor. Such analysis, however, would require an appendix about some remarkable features of functions of a complex variables. For this reason let us restrict ourselves to only one comment, namely, that the field far away from the current electrode decays exponentially, if the basement is an ideal conductor, and the power of the exponent is proportional to the ratio r/h, Now we will consider the apparent resistivity curves, Pa/Pl> calculated by Eq. (III.379), provided that

and presented on a log-log scale in Fig. III. 18c. The index of the curves is the parameter s, equal to PZ/PI'

382

III

Electric Fields

In accordance with the behavior of the electric field, the left asymptote of all curves is equal to unity; that is, as

r -

h

--7

0

Then, with an increase in separation the influence of the basement gradually increases and in the limit the curves approach their right asymptote, equal to Pzlpl; that is, if

r h

--700

As follows from a study of the distribution of surface charges, it is seen from the curves that with an increase in resistivity difference, approach to the right asymptote takes place at greater separations. In the case of a more resistive basement, S » 1, there is an intermediate range of separations, when all curves approach that which corresponds to a nonconducting basement. Therefore, within this range, the apparent resistivity P« depends on only one parameter of the medium, namely the conductance of the upper layer S. The behavior of the curves of PalPI' given in Fig. III. 18c, clearly demonstrates the main concept of the geometrical or Schlumberger soundings, which are mainly performed with the symmetrical four-electrode array, shown also in Fig. III.l8c. This method is widely used in groundwater and engineering geophysics, and the results of measurement are usually presented in the form of the apparent resistivity P« as a function of the distance between the middle point 0 and the current electrode, AB 12. Concluding this example, let us make several comments related to the solution of the inverse problem, that is, to interpretation of Schlumberger soundings. 1. In accordance with Eq. (III.379), the apparent resistivity Pa can be represented as

Applying the same approach for a solution of the forward problem in an n-layered medium, we can show that the apparent resistivity has a similar expression.

111.10 Behavior of the Electric Field in a Conducting Medium

383

where Pi and hi are the resistivity and thickness of the ith layer, respectively. 2. Presenting the apparent resistivity curves on a log-log scale, we have log o, = log PI + log F( A,log

:1 )

(III. 389)

where A is the set of parameters of the medium pJpl' hJh 1 • Equation OII.389) shows that a change of the resistivity of the first layer, Pi> as well as its thickness hI' do not alter the shape of the curve Pa but results in a parallel shift only. This fact essentially simplifies the interpretation. 3. In the theory of inverse problems of geophysics it has been proved that the inverse problem for geometric soundings performed at the surface of a horizontally layered medium is unique. In other words, only one set of geoelectric parameters generates a given curve of the apparent resistivity or the field. Certainly, this is a very important result, representing in essence the theoretical foundation of the interpretation of geometrical soundings. However, uniqueness of this inverse problem only holds provided that the field is measured with absolute accuracy, which of course does not correspond to real conditions. In fact, several factors are always present and introduce some error into values of the apparent resistivity. These factors are (a) Errors in measuring the voltage between receiver electrodes. (b) Errors in determining distances between electrodes. (c) The presence of a geological noise, which includes lateral changes of resistivity, topography effect, etc.; in other words, everything that produces deviation of the real model of the medium from a horizontally layered one. Thus, there is always a difference between the measured curve of the apparent resistivity and that calculated for a horizontally layered medium. Correspondingly we can say that the interpretation of geometrical soundings is an ill-posed problem. This means that by matching the measured and theoretical curves we are only able to establish limited ranges within which every parameter can vary, instead of determining their exact values. As a rule, the width of these ranges is different for different parameters, and this can be explained in the following way. The electric field measured at the earth's surface is caused by all charges distributed at interfaces, and it is obvious that the relative contribution of charges at the top and bottom of some layer strongly depends on its resistivity, thickness, and position. For this reason it is natural to expect that some parameters of a given medium can

384

III

Electric Fields

be determined with a great error only, while others are defined with an accuracy sufficient for practical applications. Correspondingly, we can say that interpretation of sounding data consists of determining parameters that characterize with relatively high accuracy certain features of a horizontally layered medium. In particular, these parameters can be either the longitudinal conductance of some layers, S; = h;/p;, or the transverse resistance, 1'; = p;h;, of others. This consideration clearly shows some similarity between the interpretation of gravitational data and electric soundings.

Example 7 The Electric Field at the Surface of a Medium with a Vertical Contact We will consider the behavior of the electric field in the presence of a vertical contact, shown in Fig. III.18d. Let us introduce a Cartesian system of coordinates x, y, Z, with its origin at the center of the current electrode A and x-axis directed perpendicular to the contact between media with conductivities 'Yl and 'Yz. Suppose that the electrode A is located in the medium with conductivity 'Yl at a distance d from the contact. It is clear that the primary electric field, caused by the electrode charge, gives rise to an appearance of charges at the contact. In turn these charges create an electric field that generates charges at the earth's surface. Certainly, the distribution of charges at both surfaces is established as the result of their interaction. Therefore, the potential at every point of a medium is a sum of the potentials, caused by the electrode charge Uo and surface charges

u..

U= Uo + U. By analogy with previous examples, it is convenient to write the potential in the conducting medium as x~d

(III.390)

x~d

Let us note that the potential Uz is the potential of the total field, caused by all charges. Now in accordance with Eq. (111.254) we will formulate the boundaryvalue problem for the points of a conducting medium in the following way: 1. At regular points the potential satisfies Laplace's equation.

385

111.10 Behavior of the Electric Field in a Conducting Medium

2. At the earth's surface,

Z

= 0,

au az

-=0

since the normal component of the current density vanishes. 3. The potential and the normal component of the current density are continuous functions near the vertical contact.

aUI Yj

ax

aU2 =

Y2

ax

4. Near the current electrode the potential tends to that of the primary

field. as

R ----) 0

(111.391 )

In particular, if the current electrode is located at the earth's surface, pjI

U----)Uo = - 2-rrR 5. With an increase of the distance from the current electrode the potential decreases.

U----)O

as

R ----)

00

where R= ( x 2 +y 2 +z 2)

1/ 2

All these conditions uniquely define the potential in the conducting medium, and it is not necessary to look for a solution above the earth's surface. Unlike Example 6, application of the method of separation of variables for a solution of this boundary-value problem does not allow us to derive simple expressions for the potential. This is related to the fact that the vertical contact does not coincide with any coordinate surface of the Cartesian system. However, there is an elegant approach allowing us to reduce this problem to another one whose solution is much simpler. With this purpose in mind, we will mentally transform the mirror image of the conducting medium with respect to the earth's surface to the upper half space. After this transformation we obtain a new model of a conducting medium with only one planar surface x = d and two current electrodes having equal charges, p j Is o/ 4-rr , and symmetrically situated with respect to the plane corresponding to the earth's surface (Fig. III.l9a). Next

386

III

Electric Fields

a

b "(,

A

"(,

"(2

2A

"(2

d

0 A

"(,

c Pa

P,"

"(2

"(1

"(2

d Pa

A...--.1.-. M

p;-

A"

L

MN

P2

P2

X

X

P1