Geophysical Field Theory and Method, Part B, Volume 49: Electromagnetic Fields I (International Geophysics)

j(t,z)=~ y

2i z

(T) 1/2 etrt

r

/

t

(1.288)

where 'YJ.LZ

2

T=--

4

(1.289)

We call this parameter the time constant. It is directly proportional to the conductivity, the magnetic permeability, and the square of the distance from the conductor surface. Thus, the parameter T depends strongly on the position of the observation point with respect to the surface, Z = O. It is proper to distinguish three stages or intervals of the transient response of the current density jyo 1. The early stage, t « T 2. The intermediate stage when the time of measurement is comparable with T, and finally 3. The late stage, when the time t is several times greater than T (t» T). It is clear that these three stages are observed at any point inside the medium, regardless of the distance from the surface z = O. But with an increase of this distance, each stage manifests itself at greater times.

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As follows from Eq. 0.288) at the early stage the current density i, grows exponentially with time. In the intermediate stage, the current density reaches a maximum and in accordance with Eq. 0.288) the maximum occurs at

YMZ 2

t max =2T=-2

(1.290)

Let us make several remarks: (a) At different distances from the conductor surface the maximal value of i, is observed at different times and these times become greater with increasing z-coordinate. (b) With a decrease of conductivity and magnetic permeability, /max also decreases. (c) From Eqs, (1.288) and (1.290) it follows that the maximal magnitude of the current density is: . J

max

_ ( 2 ) 1/2 i oy - -

ire

Z

(1.291)

that is, it is inversely proportional to the distance z. Next we consider the late stage when T

- « 1 t

(1.292)

In accordance with Eq, 0.288) at relatively large times, the current density IS

. _. (YM) 1/2

i,

-lOy

trt

(1.293)

that is, uniformly distributed along the z-axis and decreases with time as /-1/2.

Representing the exponent e- t / T as

we have for the current density

.

J

y

2i oy =-

~zn=o

L 00

(-1((T)n+l/2 n!

-

/

(1.294)

This form is very convenient to describe the late stage behavior, since the series converges very quickly when t ts» T.

1.9 Diffusion of a Quasistationary Field

115

It is appropriate to note that a similar representation, which contains integer and fractional powers of the parameter Tit, describes the late stage of a transient response of the current and the field in much more complicated models of a layered medium. Now we consider transient responses of the current density as a function of time when J.L = J.Lo' These responses are shown in Figure I.9c and the index of each curve is the distance z from the conductor surface. It is clearly seen that with an increase of the distance from the conductor surface, the transient responses of current density change. For instance, their maximum becomes smaller and as follows from Eq. (1.288), for every given parameter T, the magnitude of jy is inversely proportional to the coordinate z. It is also useful to discuss the dependence of the current density on the distance z. With this purpose in mind, we distinguish three ranges of distances which correspond to the small, intermediate, and large values of the parameter Tit. For instance, at relatively small distances from the surface z = 0, when t » T, the current is almost uniformly distributed and decreases at a rate inversely proportional to the square of time. Then, with an increase of z the density j y starts to decrease. Examples illustrating the behavior of the current density j y as a function of z are shown in Figure I.9d. The index of each curve is time t, These curves vividly demonstrate the process of penetration of currents into the conducting medium. At the beginning they are located in the vicinity of the conductor surface when the skin effect manifests itself. Then, with an increase of time they appear at greater distances. It is essential that the character of the current distribution varies with time. In particular, at the first instant, the volume density j y is equal to zero, while in the opposite case of relatively large times, the currents are almost uniformly distributed in a large part of the conducting medium. We have described the behavior of currents in a conducting medium as a function of the time and of the distance from the surface where they appear at the initial moment. This process of penetration of currents into the conducting medium is called diffusion. From a mathematical point of view, it is related to the fact that Eqs. (1.265), describing the quasistationary field, are diffusion equations. As was mentioned earlier, these equations also describe the penetration of the heat and movement of ions in solutions where the concentration of ions changes. Certainly, there is some similarity in the behavior of functions describing these completely different phenomena. For instance, the magnitude of currents, the amount of heat, as well as number of ions change in such a way that their maximum appears at different places at different times. However, it is appropriate to make one

116

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comment. In our case the current density has only one component jy. This means that charges located at some depth z = Zo remain during diffusion always at the same distance from the conductor surface. In other words, the charges which form the current are not involved in the movement along the z-axis. Of course, in more complicated models of the conducting medium and generators of the primary field, charges can move in all directions at different times, but this fact is not an essential feature of current diffusion. In accordance with Eq. (1.287), the magnitude of the current density differs from zero at any distance from the surface of a conducting medium as soon as time t is not zero. This feature of the behavior of current clearly demonstrates the fundamental difference between propagation and diffusion, since it implies that induced currents appear instantly at all points of the conducting medium. This paradox only emphasizes the approximate character of the quasistationary field. Next we study the diffusion of the electromagnetic field. Inasmuch as the behavior of the electric field,

- . _. (Pf.L) 1/2 e Ey-Ply-1oy 'TT't

_y!"z2/(41)

(1.295)

as a function of the distance z and time t is the same as that of the current density, we only pay attention to the magnetic field. With this purpose in mind, let us proceed from Eq, 0.273):

In order to determine the field B; from this differential equation, it is necessary to know the field at some point of the medium. For this reason it is useful to study the field behavior at infinitely large distances from the conducting surface. First of all, consider again the initial moment (t = 0) when the primary field Box vanishes. Correspondingly, induced currents which arise at this moment at the plane z = 0 create the same magnetic field Box everywhere in the conducting medium, including infinity. Moreover, taking into account the antisymmetry of the field B x ' caused by these currents with respect to the plane z = 0, we have to conclude that above the conducting medium the magnetic field is if t = 0

and

z<0

(1.296)

A natural question now arises. What happens to the field B x at infinitely (z ~ (0) and above the conductor when the currents penetrate into the

1.9 Diffusion of a Quasistationary Field

117

medium? To solve this problem, let us recall that the total current I passing through an elementary vertical strip, shown in Figure 1.9b, is independent of time [Eq. (1.285)]. Also, it is useful to represent mentally a current distribution as a system of elementary horizontal layers with density j(z, r), Then, taking into account the fact that the magnetic field on either side of each current layer is uniform and the total current of all layers is constant, we can conclude that at infinity (z --7 00) the field B x ' caused by induced currents, remains equal to the primary field. At the same time, above the conducting medium the magnetic field B, has an opposite direction. Thus, regardless of time, we obtain if z --7 00 if z < 0

(1.297)

In addition, let us make several comments: 1. In deriving Eq. (1.297) we used the fact that the field depends only on one coordinate (one-dimensional model). 2. This remarkable boundary condition also remains valid in the medium when conductivity changes only along the z-axis, provided that the primary magnetic field is uniform and

a

3. It is simple matter to see that in the case when the primary field Box arises at some instant t = 0, we have

if z --7 00 if z < 0

(1.298)

Thus, the magnetic field above the conductor surface, in particular in its vicinity, is two times greater than the primary field Box. 4. The independence of the quasistationary magnetic field of the conductivity and time, when the field is considered above the medium (z < 0), is a fundamental feature of a one-dimensional electromagnetic field. Applying the principle of superposition, it is easy to show that this result is still valid when the primary field is an arbitrary function of time. Now consider again Eq. (1.273). Integrating this equation from z to infinity and taking into account Eq. 0.297), we obtain

BOx - Biz, t) = JL f""jy dz z

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or

BAz,t) =BOx-JL [XJjy dz

(1.299)

z

Let us introduce a new variable

where

'YJL 4t

p=-

Then, we can represent Eq. (1.299) as

Inasmuch as

and

F(u)

=

2 t" 2 -1, e- X dx

{;;o

(1.300)

is the probability integral, we have

BxC z, t) =BoxC1- 2{1- F(u)}) or

BAz,t) =BoJ2F(u) -1]

(1.301)

where (1.302) Thus, we have expressed the field B in terms of the probability integral, which is a very well-known function. Its behavior is extremely simple and is shown in Figure I.lOa. In particular, when the argument u is small, the function Ftu) can be represented as a power series:

2

F(u) = {;;

00

(_1)k U2k + 1

Eo kl(2k + 1)

(1.303)

1.9 Diffusion of a Quasistationary Field

119

a F(u) 1.0

+----:::;;::::-----

0.5

b

.5

u

0 + - - - - + - . , - - - - - - - . - - - - -... 1 2 3

-.5

-1

o

Fig. 1.10 (a) Behavior of function F(u); (b) transient responses of magnetic field; (c) impulse as difference of step functions; (d) system of impulses of primary field. (Figure continues.)

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Electromagnetic Fields

c Be(t}

Be(t)

o

o

d Be(t)

Fig.1.10 (Continued)

In the opposite case of large values of u, the probability integral is described by the asymptotic series

F(u)::.:d-

e-

('7T)

U2 [

co

1/21+:E(-1) u

k=l

k

1 ' 3 ' 5 ' 7 ... (2k-l)] 2 k 2k (1.304) U

Let us notice that the probability integral is also a solution of the diffusion equation (1.269), since the functions Ex and Fiu) differ from each other by only some constants. Now we are prepared to study the behavior of the magnetic field. Taking into account Eqs. 0.301) and (1.304), the field Ex at the early stage

1.9 Diffusion of a Quasistationary Field

121

is if tiT « 1

(1.305)

As is seen from this equation, the behavior of the field B clearly demonstrates the principle of inertia of magnetic flux. In fact, we have if t «T and at greater distances from the conductor surface this approximate equality holds for greater times. It is interesting to notice that within the early stage the difference between the secondary and primary field is defined by a very small exponential term. In other words, at the early stage the field B is practically independent of time, as well as of distance, and it is almost equal to Box. However, this conclusion does not apply to its derivative with respect to time, 8B x18t. Next, consider the opposite case, that is, the late stage. Since the parameter u is small, it is appropriate to make use of the series (1.303). Then we obtain

(

t)

Box -;

00

=

(_l)k

-Box + 2Box kL:o k!(2k + 1)

(T)k+l/2

t

(1.306)

Thus, at the late stage the magnetic field also consists of two parts, namely: 1. A constant which describes the field everywhere above the conductor surface and is equal to - Box. 2. A second part which is a series containing fractional powers of the ratio Tit. Discarding all terms of this series except the first three, we have

n,(; ) ~ - Box[1 - 2( ~

f/2 + ~ ( ~ f/2 - ~ (~

r/2

+ ... ],

if t»

T

(1.307)

The behavior of the transient response of the magnetic field is shown in Figure I.10b. Here it is proper to note that in spite of the absence of the primary magnetic field when t > 0, the leading term of the series describing the late stage is a constant and it differs from the primary field Box only in sign. As will be shown later, this behavior is an exception and is

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related to the fact that the field depends on one coordinate and does not vanish at infinity, Now let us discuss the relationship between the field B, and induced currents which generate this field. In other words, we will try to explain the transient response of the magnetic field proceeding from the function i; Suppose that an observation point is located at some distance Zo from the conductor surface. At the beginning of diffusion, induced currents appear mainly at smaller distances (z < zo), and they have a relatively high density since the total current remains the same. Inasmuch as such a change of the current distribution within the interval 0 < z < Zo does not have an influence on the field at the point z = zo, it is almost equal to the primary field BoX" Also, it is clear that with an increase of the distance Zo this feature of the field behavior will be observed at greater times. With further increase of time, currents with an appreciable magnitude begin to appear at distances exceeding zoo Inasmuch as their magnetic field at the point z = Zo has the direction which is opposite to that of the field generated by currents located above this point, the total field B; becomes smaller. In particular, there is a moment when the magnetic field vanishes. At sufficiently great times the influence of currents located beneath the point z = z 0 becomes dominant, and the magnetic field gradually approaches - Box. 1.10 Diffusion and Periodic Quasistationary Fields

Now we study diffusion when the primary field Box is a more complicated function of time than a step function. First, suppose that the field Box is described by an impulse with width I1t shown in Figure I.lOc. In accordance with the principle of superposition, induced currents behave as if they were generated by two step functions of the primary field, having the same magnitude but opposite sign and arising at different times, as shown in Figure I.lOc. Therefore, we can treat the current distribution caused by the impulse as the difference of two current systems which arise on the conductor surface at the moments t = 0 and t = I1t, respectively. It is clear that if we observe the currents and the field within the time interval 0 < t < 11 t, then their behavior is determined by the first step function only. Before we consider the distribution of currents induced in a conducting medium due to an impulse of the primary field, let us make one comment. It concerns currents arising at the conductor surface and in its vicinity when Bo/t) is a step function. As we know, the surface current exists only at the first instant and then disappears. In accordance with Eq, (I.288), the current very quickly arises within an extremely thin layer beneath the

123

1.10 Diffusion and Periodic Quasistationary Fields

conductor surface, and its distribution is characterized by the volume density jy. Since we consider the same model of the medium and the primary field as in the previous section, the current density caused by an impulse of Box, is if t >

~t

(1.308) Of course, this equation does not describe the behavior of the surface density of currents i y' The latter appears at the instant t = 0 and is equal to zero during the time interval ~t. Then, at the moment t = At is arises again and has the same magnitude but the opposite direction. At time exceeding ~t we have only a volume density of currents. Now consider the behavior of these currents at different distances and times. As follows from Eq. (1.308) near the surface T « 1 and the transient process occurs relatively quickly. In other words, the time interval during which we observe the early, intermediate, and the beginning of the late stage is either smaller or comparable to the impulse width. Therefore, there is an essential difference between the magnitudes of the currents caused by each step function. In particular, in approaching the conductor surface we can always find sufficiently small distances z where the behavior of induced currents, generated by the first and second step functions, corresponds-to the late and early stages, respectively. Thus, in spite of the opposite directions of these currents, the magnitude of the total current density can be practically the same as that caused by either step function of the primary field. This means that near the conductor surface a compensation effect due to currents induced by the second step function is usually very small, except for sufficiently large times when currents are negligible. At the same time, it is clear that with an increase of impulse width ~t as well as the medium resistivity, this behavior holds at greater distances from the surface z = O. Next, suppose that the observation point is located at a greater depth inside the medium. Taking into account the fact that with an increase of the distance z the transient response of currents occurs later, we assume that the time t, which corresponds to the intermediate stage, is much greater than ~t:

t

»~t

and

t

~ T

Also, we will not pay any attention to the early stages where currents are

124

I

Electromagnetic Fields

usually very small and specifically so with an increase of the distance z. Then, making use of the expansion li.t) (t _li.t)I/2 "'" t i / 2( 1- 1 -t

2"

and e(l1t/tX'r/t) "'"

MT 1+ - t

t

we can represent Eq. (1.308) as j "'" j y

Iy

(~_t

~)2

li.t t

(1.309)

where jly is the current density caused by the first step function of the primary field. As follows from Eq, (1.309), the compensation effect manifests itself when Tli.t

- -«1 t t

(1.310)

This inequality can be interpreted in the following way. Near the surface (z = 0) we see this effect only at the very late stage. But with an increase

of the distance z it can be observed at the intermediate stage and at greater depths even at the early stage of the transient response. Of course, with an increase of time the influence of the compensation effect becomes stronger. In other words, comparing the same stages, we can say that with an increase of the distance, the current density jy due to the impulse becomes much smaller than that generated by the single step function. It is obvious that with a decrease of the impulse width and the resistivity, the compensation effect manifests itself at smaller distances from the conductor surface. For instance, in the case of an ideal conductor, the surface current caused by the impulse of the primary field is equal to zero, if t > sr. Thus, we demonstrated that induced currents caused by the impulse decay more rapidly with time and distance, z provided that inequality (1.310) is met. In particular, in accordance with Eq, (1.309) the behavior of

1.10 Diffusion and Periodic Quasistationary Fields

125

currents at the late stage is .

i Oy

7

1/2

.:lt

J :::::: - - --:c-=-3 2

y

..;:;z

t /

'

if t ts-

7

(1.311)

In contrast, with an increase of the impulse width or the resistivity, the compensation effect becomes noticeable only at greater distances from the conductor surface. Let us consider one more feature of the current distribution generated by the impulse of the primary field. As follows from Eq. (1.309), the current density as a function of either the distance or time changes sign unlike the case of the step function excitation. This happens approximately at the instant when the currents generated by the first step function reach their maximum, Eq. (1.290): t:::::: 27,

.:It if - « 1 t

It is easy to see that with an increase of .:lt, the diffusion of currents is described by an almost antisymmetric function of time and distance. Such behavior is observed if the time interval which includes the early, intermediate, and an initial part of the late stage of the response, caused by the step function, is much smaller than .:It; that is,

.:It > k-r ,

if k» 1

(1.312)

However, with an increase of the distance from the conducting surface, the parameter 7 increases rapidly, too, and correspondingly the inequality becomes incorrect. One more time this fact shows that the transient response of induced currents changes with depth z. Next, we assume that the primary magnetic field is described by a system of impulses. These impulses are characterized by the same magnitude and width, as well as by equal intervals between them (Fig. I.10d). At any point of the conducting medium, located at some distance from the surface z = 0, the induced current is caused by all impulses of the primary field which occurred earlier. If the time of observation corresponds to the interval inside some impulse, then the effect of the last step function should also be added. Of course, impulses which occurred much earlier do not have a practical influence on the current distribution at the moment t. As in the case of a single impulse of the primary field, we observe the transient response of currents, but there is one fundamental difference between them. In fact, now the primary field is a periodic function and its period includes the interval between neighboring impulses and the width

126

I

Electromagnetic Fields

of the impulse. Respectively, the transient response of currents is also a periodic function of time. For instance, suppose that the time intervals between impulses and their width are T[ and T z , correspondingly. Then, we observe the transient response caused by all previous impulses if the time t satisfies the condition

where to is the time which characterizes the front of the nearest impulse. In the next part of the period,

currents are caused by all previous impulses and the step function which appears at the instance t = to + T[. If we study the transient response when the time obeys the condition

then its behavior is exactly the same as in the previous time interval. Therefore, we can say the function describing the current at any point of a conductor is the periodic function of time, and its period,

coincides with the period of the impulses of the primary field. •Inasmuch as the current generated by each impulse of the primary field depends on both time t and distance z, the distribution of currents due to the system of impulses is not a periodic function of z, unlike the dependence on time (Fig. I.10d). Of course, this consideration also applies to the electric and magnetic fields. It is interesting to discuss one special case, when the system of impulses, having the same magnitude and sign, does not induce currents in a conducting medium. As we know, every impulse is formed by two step functions of opposite sign arising with time delay T z. Therefore, a decrease of the interval between impulses T[ results in a cancellation of the effect caused by neighboring step functions. And in the limit when the primary field Box becomes constant, the induced currents vanish. Now we consider a primary field which is described by a system of alternating impulses as shown in Figure I.11a. It is clear that, adding the constant primary field, this system of impulses coincides with the first one (Fig. I.10d). Thus, both of them produce the same distribution of currents in the conducting medium.

1.10 Diffusion and Periodic Quasistationary Fields

127

a

.. t

b 0.5 0.4

0.2

0.0

-0.2

-0.4 -0.5 -+--------.--------r-------~p .01 .1 Fig. 1.11 (a) Comparison of sinusoids and system of impulses; (b) spectrum of complex amplitude of current density; (c) flux of Poynting vector; (d) Poynting vector within the external and internal parts of current circuit. (Figure continues.)

It is also obvious that with a decrease of the period T, the step functions describing the primary field approach each other; therefore, due to the compensation effect caused by them, induced currents are located relatively closer to the surface. In contrast, with an increase of the oscillation period T, the separation between neighboring step functions also increases, and consequently the compensation effect manifests itself

128

I

Electromagnetic Fields

c ""'-

Ie'

n

.:

.

y

.'"" •. . •7\

,.0>

,.. :'.':;,;:

Fig.I.ll

dS

'/;"8BW}L,) ..<"(j

:".:

\

+

....

.'.

-

>{

...........

(Continued)

at greater depths. Thus, changing the frequency of oscillations f CJ = liT), we change the current distribution in the medium. In particular, when the frequency is relatively high, the currents are mainly located near the surface. In other words, we again observe the tendency of currents to concentrate near the conductive surface. This is the reason why this phenomenon is also called the skin effect, even if instead of surface currents there are only volume currents concentrated near the surface. Certainly, it is appropriate to distinguish the two types of skin effect. In the first case, the effect is due to the inertia of the flux of the magnetic field, and induced currents

1.10 Diffusion and Periodic Quasistationary Fields

129

appear only on the conductor surface when the primary field B changes instantaneously. However, in the second case, when the primary field is a periodic function, the volume currents are always present. Their concentration near the surface is explained by the fact that the compensation effect manifests itself more strongly at greater distances from the surface. Next, let us establish an approximate relationship between the period of oscillation T and the depth of penetration of currents. As we know, the compensation effect is strong if the separation between the two neighboring step functions of the primary field, having opposite sign, is smaller than the time constant 7'. In other words, the induced currents are relatively small at such distances from the surface that satisfy the condition 7'

>T

or T )1/2 2 & ( z> 2 'YJ.L = ('YJ.Lf) 1/2 = 2 ('YJ.LW) 1/2

(1.313)

where w = 27T f is the angular frequency. Thus, induced currents are mainly located within the range 2 O~z<

1/2

(1.314)

( 'YJ.L f)

and this interval decreases with an increase of the frequency of oscillations of the primary field as well as with an increase of the conductivity and magnetic permeability of the medium. Before we consider the quasistationary field in a conducting medium, when Box is a sinusoidal function of time, it is appropriate to make one comment. Earlier we demonstrated that transient responses of induced currents caused by the step function BoX
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I

Electromagnetic Fields

Suppose that the primary field is (1.315) Box = B o sin cot Then, as was demonstrated in Section lA, the behavior of the electromagnetic field and currents is described by sinusoidal functions, provided that the parameters of the medium Y, IL, and E do not depend on frequency. Therefore, for a one-dimensional model in which the electric and magnetic field are described by one component, we have

i, = jsin( cot + 'P e) E y = E sine tot + 'Pe) B; = B sine tot + 'Pb)

(1.316)

where B, E, and j are amplitudes of oscillations, and 'Pb and 'P e are their phases. It is essential that the amplitudes and phases are independent of time, but they can be functions of several parameters-namely, Box, w, y, IL, and z. The fact that the secondary and primary fields have the same time dependence is a very remarkable feature of the sinusoidal oscillations. In particular, it means that regardless of the distance from the conductor surface, the field dependence on time remains the same, which in general is not correct. We start the study of a sinusoidal electromagnetic field in a conducting medium from the magnetic field and represent it as ,

B;

= -

Im B* e- iw t

(1.317)

where B * is the complex amplitude of the magnetic field and

B* =Be-i'Pb In accordance with Eq. 0.116), the complex amplitude B * satisfies the equation (1.318)

where k2

=

iYILW

(1.319)

and k is the wave number which corresponds to the quasistationary field. It is clear that the electric field and the current density obey the same equation: (1.320)

1.10 Diffusion and Periodic Quasistationary Fields

131

where E * and j * are the complex amplitudes of the electric field and the current density, respectively. Now we introduce a parameter which is very useful to characterize a distribution of sinusoidal currents and fields. With this purpose in mind, let us represent the wave number as YMW )1/ 2 1+i k = ViYMW = ( -2(1 +i) = -8-

(1.321 )

8= (_2 )1 /2

(1.322)

where

YMw

is called the skin depth. In 51 units it has the dimension of meters and can be written in the form 10 3 8 = -(1OpT) 1/2 27T

(1.323)

It is natural to expect that the skin depth 8 and parameter T characterizing the diffusion of currents are related to each other. In fact, from Eqs. (1.289) and (1.322) we have ort

>

~(~

r ~p2 =

(1.324)

Next, we consider Eq. 0.318), which is an ordinary differential equation of the second order with constant coefficients. As is well known, its solution is (1.325)

From the physical point of view it is impossible that the field increases without limit when the distance z tends to infinity. Then, taking into account the fact that z z z ikz=i(1+i)-=i--8 8 8

and

we have to assume that the coefficient C is equal to zero. Thus, we obtain if z

~

00

(1.326)

132

I

Electromagnetic Fields

or if z --)

00

where B~ is the complex amplitude of the secondary magnetic field caused by induced currents only. Correspondingly, the solution of Eq. (1.318), satisfying the condition at infinity, is B* =Ae i k z or

(1.327)

where A is unknown. In order to determine this constant, we have to consider again the behavior of the field at infinity. As follows from Eq. (1.326), the secondary magnetic field, caused by induced currents, is equal to - B o at infinity, regardless of the frequency. Then, making use of the fact that a one-dimensional distribution of currents generates the same magnetic field as a system of elementary and uniform current layers (Section 1.9), we arrive at the conclusion that on the conductor surface and above if

z:'O: 0

lpus, the total magnetic field on the surface is if z

=

(1.328)

0

Comparison of Eqs. (1.327) and 0.328) shows that A

=

2B o and

if z > 0

(1.329)

Z

(1.330)

where p =

i

= (

y~W

f/2

It is proper to note that the boundary condition for the sinusoidal

magnetic field and its initial condition, in the case when the primary field Box is a step function arising at some moment, coincides with each other. Often the parameter p is called the induction number, and it characterizes the distance from an observation point to the conductor surface expressed in units of the skin depth. It is clear that in a one-dimensional and uniform medium the parameter p along defines the field behavior. In particular, at the distance z

1.10 Diffusion and Periodic Quasistationary Fields

133

which is equal to the skin depth 8, the field magnitude decreases by a factor e, and this fact is usually used to introduce the concept of the skin depth. We can say that induction number p describes the diffusion of sinusoidal fields in the same manner as the parameter (7 /t)1/2 characterizes the transient field. In essence, the ratio: (7 /t)1/2 can also be called the induction number. In accordance with Eqs. (1.317) and (1.329), the magnetic field can be represented as (1.331 )

Therefore, if the distance between two points .:lz is such that .:lz = 27ro, the phase of the field changes by 27r. For this reason, the distance A = 27ro

(1.332)

is sometimes called the wavelength, even though the propagation effect is neglected. Let us determine the current density from the second of Maxwell's equations: JB*

----a;- =

JL j *

.

Then, taking into account Eq. (1.329), we have ik

ik z J. * -- -2B0 e

JL

or

(1.333)

and 37r

cp=-p-4

Inasmuch as the parameter p defines the behavior of currents as well as the field, it is natural to distinguish three ranges. 1. The range of small parameters: p < 1. 2. The intermediate range. 3. The range of large parameters: p » 1. These ranges can also be considered as the low-, intermediate-, and

134

I

Electromagnetic Fields

high-frequency part of the spectrum, respectively, and each of them is observed regardless of the distance from the conductor surface. As follows from Eq, (1.330), near the surface the low-frequency spectrum can be observed at relatively high frequencies. At the same time, with an increase of the distance z, this range of the spectrum holds at lower frequencies. This tendency is also seen within the intermediate- and high-frequency parts of the spectrum. For instance, with an increase of the distance z, the high-frequency part of the spectrum begins to manifest itself at lower frequency. This analysis clearly shows that the field behavior is controlled by the value of parameter p. Before we study the distribution of currents, let us note the following. In accordance with Eqs. (1.329) and (1.333) and taking into account that E* = pj ; we have

or 1 E* -=-z B

*

p,

xy

where (1.334) This ratio is often called the impedance of the one-dimensional field in a uniform conducting medium, and it plays a very important role in the theory of magnetotelluric soundings. As is seen from Eq. (1.334), the magnitude and phase of the impedance are independent of the distance from the surface of a uniform half space. In a later chapter we generalize this result and with some modifications apply it for a horizontally layered medium. Presenting the complex amplitude of the current density as the sum of the in-phase and quadrature components, we obtain from Eq (1.333) 2BO In j* = - --pe- P(cos p p,z

+ sin p)

2B Qj* =--ope-P(cosp-sinp)

(1.335)

p,z

Now we are ready to study the current distribution. In the range of small

135

1.10 Diffusion and Periodic Quasistationary Fields

parameter (p« 1), the exponent and trigonometric functions (1.335) can be represented as p3

sinp =::.p -

III

Eqs.

p2

6'

cosp

=::.1--

2

Then, for the current density we have

(1.336) if p < 1 or, making use of Eq. (1.324), we express both components in terms of the time constant and the frequency: In i;

=::. -

2fiB o

- - - {T 1/2W 1/2 -

2T 3 / 2w 3 / 2 + ... }

J-LZ

Qj*

=::.

2fiB o J-LZ

(1.337) {T 1/2W 1/2 -

2fiwT +

2T 3 / 2w 3 / 2 -

..• },

if

WT«

1

Preserving all terms in the expansion of the functions e- P , sin p, and cos p, it is Fasy to see that the low-frequency spectrum of both components of the current density contains both integer and fractional powers of os, It is useful to note that in a layered medium the low-frequency spectrum of the field and currents, caused by arbitrary generators of the primary field, has a similar representation but also contains logarithmic terms in ca. As follows from Eqs. (1.337), in the range of small parameter p, both components of the current are almost uniformly distributed along the z-axis, and with a decrease of the frequency such behavior occurs at greater distances z. Next, we discuss the opposite case, when the parameter p is sufficiently large. In accordance with Eqs. (1.335), both components of the current oscillate and rapidly decrease with an increase of either the distance z or the frequency w. It is clear the regardless of how low the frequency which is used, there are always distances z where the current behavior corresponds to the range of large parameters (p » 1). At the same time, with an increase of the frequency this behavior of currents manifests itself relatively close to the conductor surface, and at sufficiently high frequen-

136

I

Electromagnetic Fields

des, when w» l/T, currents are mainly concentrated near the surface. This phenomenon can hardly surprise us since we demonstrated earlier that with an increase of frequency of impulses of the primary field, the compensation effect becomes stronger. Correspondingly, induced currents decrease more rapidly with the distance z. Thus, there is a clear analogy between the behavior of currents at the early stage of the transient response and the high-frequency spectrum. In the previous section we showed that if the primary magnetic field is a step function, the current passing through any vertical strip (Fig. 1.9b) is independent of time. Now, we derive a similar result for the complex amplitude of the current. In fact, making use of Eq. (1.333), we find that the integral of the complex amplitude of the current density along the semi-infinite strip is

1o * dz = -2ikB 1 00

1* =

00.

j

o

11-

e

,k z

2Bo dz = - - f.L

0

(1'.338)

Thus, the function 1* is defined by only the primary magnetic field B o, and, in particular, it is independent of the conductivity and frequency. Since j*

= je-i'Pe = a - ib

(1.339)

we can rewrite Eq, (1.338) as 00

1o

2Bo adz= - - 11-

1o bdz 00

and

=

0

(1.340)

where lal and Ibl are amplitudes of the in-phase and quadrature components, respectively. Therefore, at any frequency, the distribution of the quadrature component of the current density along the z-axis is such that the sum of these currents with the positive and negative directions is equal to zero. It is almost obvious that Eqs. (1.338) and (1.340) remain valid in the more general case when the medium is a horizontally layered one and the field changes along the z-axis only. Making use of Eqs. (1.317), (1.331), and (1.339), we have j y = a sin co t

+ b cos w t

(1.341)

where a = In j *

and

b = Qj *

Thus, we can imagine that at each point of the conducting medium there

I.lO

Diffusion and Periodic Quasistationary Fields

137

are simultaneously two currents. One of them, a sin tot

changes synchronously with the current of the primary field, while the other, b cos wt

is shifted in phase by 'TT/2 with respect to i oy • In accordance with the Biot-Savart law, these currents generate the in-phase and quadrature components of the quasistationary magnetic field, respectively. The behavior of both components of the current density as functions of parameter p is shown in Figure I.11b. Next we study the magnetic field B x ' First of all, from Eq. (1.328) it follows that the field on the external side of the conductor surface is

B; = 2Box As was pointed out in the previous section, this peculiar feature of the field behavior is a consequence of the fact that we are dealing with a one-dimensional model of the medium and the field depends on the z-coordinate only. Also, taking into account the fact that the total magnetic field B; vanishes when z tends to infinity, we have to conclude that Eq. (1.328) remains valid even in a horizontally layered medium. Representing the exponent in Eq. 0.327) as a series, we obtain for the complex amplitude of the magnetic field 00

B x=2B o I:

n=O

(ikz) n

-n!

For instance, in the range of small parameter we have In B;» 2Bo - 2Bo(p -

tp3) Q B;» 2Bo( p - p2 + tp3),

if p < 1

(1.342)

Thus, in the low-frequency spectrum (p « 1), the quadrature component as well as the difference In B* - ZB o are almost directly proportional to the parameter p. This indicates that the field is mainly defined by induced currents beneath the conducting surface where distances satisfy the condition z < 8. And this phenomenon occurs in spite of the fact that every elementary current layer with thickness Llz creates above and beneath it a uniform field LlB x ' Correspondingly, the small influence of currents located beyond the range of small parameters (z > 8) can only be explained

138

I

Electromagnetic Fields

by oscillations of the amplitudes a(z) and bt;z), This conclusion also follows from Eqs. (1.340). Let us make two more comments concerning the low-frequency spectrum. 1. The quadrature component of the field and the difference In B x 2Box are very small compared to the field on the conductor surface. 2. The leading term of both these functions is the same, and it is directly proportional to the parameter p. This behavior of the lowfrequency spectrum is the exception and, in general, when the primary field is caused by real generators, these terms do not coincide with each other. Now, we discuss the high-frequency part of the spectrum. In accordance with Eq. (1.331), the field B; oscillates rapidly and decays exponentially with distance. In particular, at relatively high frequencies the field is exponentially small, even in the vicinity of the conductor surface. This means that induced currents are mainly located near the surface, and, in the limit when the frequency tends to infinity, the field B; becomes a discontinuous function. In fact, from the external side of the surface, the field is always equal to 2B ox regardless of the frequency, while on its internal side the field approaches zero. As follows from Eq. (1.331), for the in-phase and quadrature components we have InB* =2Boxe-Pcosp

Q B*

=

2B oxe- P sin p

(1.343)

It is obvious that both components are related to each other in a very simple manner, and we have

In B* Q B*

= =

COt(2W7')

1/2

tan(2£1)7)

1/2

Q B*

(1.344)

In B*

Thus, knowing one component of the field at some frequency, the other component is easily calculated at the same point of the conducting medium. Of course, this result is not surprising, and it follows from more general relations described later.

1.11 Distribution of the Electromagnetic Energy; Poynting Vector In previous sections we investigated the propagation of electromagnetic fields in an insulator and their diffusion in a conducting medium.

1.11 Distribution of the Electromagnetic Energy; Poynting Vector

139

Now we begin to study both phenomena together and, with this purpose in mind, to consider factors which determine a change of the electromagnetic energy. Suppose that in some volume V this energy is distributed with density w. Then the total amount of this energy in V is equal to

jWdV v and correspondingly its change is characterized by the quantity dw

j v -dV=jwdV dt v

(1.345)

Also, we assume that in general the medium in this volume is conductive and that both electric and extraneous forces can be present. As is well known (Part A), the two functions Q = E .j

and

P

=

E ext • j

(1.346)

describe the work performed by the electric and extraneous forces in a unit volume during 1 sec, respectively. In other words, Q is the amount of electromagnetic energy which is transformed into heat every second, while P is the amount of this energy arising due to the extraneous forces during the same time. In accordance with Eqs. (1.346), the amounts of these energies are

jvE· j dV

and

(1.347)

It is essential that j is the density of conduction currents only; that is, displacement currents are not directly involved in transformation of the energy of extraneous forces into electromagnetic energy and the latter into heat. At the same time, it is proper to point out that displacement currents have an influence on the electric field E. Next we consider one more factor which also produces a change of the energy; this factor is caused by the propagation of the electromagnetic field. It is natural to expect that this phenomenon can be described in terms of the movement of electromagnetic energy. For this reason we introduce the concept of the flux of this energy through the surface S, surrounding the volume V. By definition, the flux can be written as

~Y'dS s

(1.348)

where Y is a vector that points in the direction of the movement of the energy and dS = n dS, where n is directed outward, as shown in Figure L11~

140

I

Electromagnetic Fields

The magnitude of Y is equal to the amount of energy which passes through a unit surface area during 1 sec, where this elementary surface is perpendicular to the vector Y. This means that Y is the velocity of the flux of the electromagnetic energy. In those parts of the closed surface S where the vector Y is directed inward, the flux is negative, and this leads to an increase of the energy in the volume V. In contrast, a positive flux of the vector Y results in a decrease of this energy inside V. Thus, there are three phenomena which can cause a change of the amount of electromagnetic energy in the volume V. 1. The work of extraneous sources with power P. 2. Transformation of electromagnetic energy into heat. 3. Movement of energy through the medium.

Now we can formulate the principle of conservation of energy in the following way: (1.349) where w= dw/dt. This equation represents the principle of energy conservation in integral form. In order to derive its differential form, we make use of Gauss's theorem:

~Y . dS = s

fv divY dV

Then, instead of Eq. (1.349), we obtain

fvwdV fv{j . E =

ext

j . E - divY} dV

-

Taking into account the fact that the volume V is arbitrary, we finally have

w=

dw -

dt

= j .

E ext

-

j . E - divY

(1.350)

Of course, the third term on the right-hand side of Eq. 1.350 describes the flux of the electromagnetic energy through the surface surrounding the elementary volume. In this light, it is appropriate to emphasize that the principle of conservation of this energy implies existence of its flux. In other words, a decrease of the electromagnetic energy in one part of the medium and an increase in others is always accompanied by the movement of energy between these parts.

1.11 Distribution of the Electromagnetic Energy; Poynting Vector

141

Inasmuch as both the density of energy wand the flux of the energy are related to the field, it is natural to express then in terms of E and B. With this purpose in mind, let us assume that extraneous forces E ext are absent in the volume. Then, Eq. (1.350) can be written as

w- div Y

j .E= -

(1.351)

and we will attempt to represent the product j . E as the sum of two terms in the same way as those on the right-hand side of Eq. (1.351). From the second of Maxwell's equations we have curlB . j=---EE J.L

Therefore, E,VXB

J: E =

. - EEE

(1.352)

J.L

Taking into account the equality div(E X B) = B . V X E - E . V X B or E .VXB

B . V XE

div(E X B)

J.L

f.L

u.

we have j'E=

B .V XE

-

div(E X B)

J.L

. -EE'E

(1.353)

J.L

As follows from the first of Maxwell's equations, VxE=

-13

and therefore Eq. (1.353) can be written as B'B

1

J.L

J.L

J: E = - - - - EE· E- - div(E X B) or j . E = - -1a(B'B - - - + EE . E ) - -1 div(E X B) 2

at

J.L

J.L

(1.354 )

It is easy to see that our problem is almost solved. In fact, comparison of

142

I

Electromagnetic Fields

Eqs. 0.351) and (1.354) allows us to assume that expressions for the energy density and the density of its flux are

w=2"1

(B'B ~+EE'E

)

(I.355)

and 1 Y=-(EXB)

(1.356)

J.L

Let us make several comments: 1. We have assumed that the medium in the volume V is piecewise uniform and that surface currents are absent. Otherwise, we have to take into account the change of energy due to these currents. 2. Even though we have studied a nonmagnetic medium, Eqs. 0.355) and (1.356) are also applied when remanent magnetization is absent and the relationship between the magnetic field B and the induced magnetization Mind is

where X is the magnetic susceptibility. 3. The approach which was used for deriving Eqs, (1.355) and 0.356), as well as other approaches, does not allow us to uniquely express the density wand vector Y in terms of the electric and magnetic fields. However, numerous experimental studies of the conservation of energy in the case of a closed surface and calculations based on the use of Eqs, 0.355) and (1.356) give the same result. It is proper to note that we met a similar ambiguity earlier. For instance, this happened when the Biot-Savart law was derived from experimental studies of interaction of closed currents. 4. The vector Y is called the Poynting vector and its dimension is watt

W

[y]=-=2 2 m

m

while the density of the electromagnetic energy w has dimension joule

J

m

m

[w]=-=3 3

143

1.11 Distribution of the Electromagnetic Energy; Poynting Vector

Now, substituting Eq, (1.356) into Eqs, (1.349) and (1.350), We obtain

1vwdV 1vPdV - 1vQdV - ~ -EXB . =

8J.L

dS

and

(1.357)

EXB

w=P-Q-div-J.L

where w is given by Eq. (1.355). In essence, the study performed in this section can be treated as a generalization of results derived earlier, since it indicates that in a conducting medium we observe simultaneously two phenomena-namely, propagation and diffusion. In the next chapter we consider a field behavior in which both of these phenomena vividly manifest themselves, but now let us return to Eqs. (1.357) and discuss some cases which illustrate the distribution of the energy.

Case 1 Suppose that the field is time-invariant. Then, in accordance with Eq, (1.357), we have

1v(P - Q) dV = ~

Y • dS 8

or

1P dV + f. Y . dS 1Q dV + f. Y . dS =

V

where 5 = 51

81

V

(1.358)

82

+ 52 and the integrals

characterize the amount of electromagnetic energy which arrives and leaves the volume V, respectively. Thus, in a time-invariant field the increase of energy in V due to its arrival from outside and due to extraneous forces is compensated by the amount of energy which leaves the volume and is transformed into heat. This study suggests to us again that the constant field can be interpreted as a system of impulses with the same magnitude and sign, continuously following each other.

144

I

Electromagnetic Fields

In accordance with Eq, (1.357), in the absence of extraneous forces we have

~Y'dS=

(1.359)

-f.QdV

v

s

This means that for the time-invariant field the difference between the electromagnetic energy which arrives and leaves the volume transforms into the heat if E ext = O.

Case 2

In the simplest model, when the medium is nonconducting and extraneous forces are absent we have dw

1v dt -

dV = - ~ Y . dS = S

-

J. Y' dS - J. Y . dS Sl

(1.360)

S2

In particular, if both fluxes on the right-hand side of Eq. (1.360) are equal to each other, then the amount of energy does not change even if the field varies with time.

Case 3

Let us assume that the influence of displacement currents is negligible, that is, that the field is quasistationary. Then, as follows from Eq, (1.355), we have dw

1

,

1v -dV=1vB'BdV dt J.L

(1.361)

Thus, the entire energy of a quasistationary field is stored in the form of magnetic energy. At the same time, as in the general case, the change of energy is defined by all three terms which are present on the right-hand side of Eq. (1.349).

Now we describe some features of propagation of the electromagnetic energy in two relatively simple models when conduction current is surrounded by an insulator.

1.11 Distribution of the Electromagnetic Energy; Poynting Vector

145

Example 1 Consider the current circuit shown in Figure Ll ld. Within its internal part, the current density j and the electric field E have opposite directions. Therefore, the Poynting vector Y is directed outside the internal part of the circuit. On the other hand, in the external part both vectors E and j have the same direction, and correspondingly the Poynting vector is directed inside the circuit. This means that the flux of the electromagnetic energy is negative in this part of the circuit, and, in accordance with Eq. (1.349), the arriving energy results in an increase of density os, This considerations shows that the electromagnetic energy leaves the internal part of the circuit and travels through a surrounding medium. Then it returns into the external part of the circuit and transforms partly into heat. In the case of a time-invariant field (w = 0), the differences P - Q represents the amount of energy which leaves the internal part. This energy moves in the surrounding medium with a density of flux equal to Y and then completely transforms into heat in the external part. It is essential that in this case the electromagnetic energy does not travel inside the conducting circuit.

Example 2' Suppose we have the current circuit shown in Figure 1.12a, which consists of three parts. 1. The internal part where extraneous forces produce the work which

results in an increase of the electromagnetic energy. 2. The long and conductive transmission line. 3. The relatively resistive load. As was demonstrated earlier, the electromagnetic energy leaves the internal part of the circuit and travels through the surrounding medium. Now we consider the behavior of the field and Poynting vector in the vicinity of the transmission line and the load. Inasmuch as the line has very low resistivity, the tangential component of the electric field E is very small inside the line. In fact, from Ohm's law we have Et=pj

Due to continuity of the tangential component, it is also small on the external side of the conductor. At the same time, surface charges create

a

b

E Sp S12

y

n

2 -'

1

v

So

c d

Im 00

In/oo+oool 00 - 00 0

."cP,

R/Y 1/

_

...!Q....

roo

Fig. 1.12 (a) Propagation energy between generator and loading; (b) illustration for deriving Eq. (1.374); (c) path of integration of Eq, 0.398); (d) behavior of weighting factor.

1.12 Determination of Electromagnetic Fields

147

outside this line a nonnal component of the field En which is much greater than the tangential component:

Then, as is seen from Figure I. 12a, the Poynting vector is practically tangential to the transmission line. This means that the electromagnetic energy travels along this line; that is, the line plays the role of a guide, defining the direction of the movement of the energy into the load. Of course, due to the presence of the tangential component of the electric field, a small amount of the electromagnetic energy moves in the transmission line and transforms into heat. This is a pure loss which reduces the amount of energy arriving to the load. Unlike the transmission line, the load is relatively resistive, and correspondingly the tangential component of the electric field prevails over the normal component:

Et>E n Therefore, the Poynting vector is mainly directed inward, and the electromagnetic energy transforms there into heat or other types of energy.

1.12 Determination of Electromagnetic Fields

To develop the theory of electromagnetic methods applied in geophysics, we have to determine the field in different media. In other words, it is necessary to establish the relationship between the field and the parameters of the medium. With this purpose in mind we must formulate a boundary value problem, since a part of the field generators remains unknown until the electromagnetic field is calculated. It is proper to emphasize that only these unknown charges and currents contain information about the distribution of the physical parameters of the medium. As in the case, for instance, of a time-invariant electric field, the solution of a boundary value problem is the only way to calculate the electromagnetic field. In principle, it is possible to formulate this problem for an arbitrary electromagnetic field regardless of how the generators of the primary field change with time. However, we use a special approach based on the assumption that the generators of the primary field vary with time as sinusoidal functions. As was pointed out, this type of excitation has one

148

I

Electromagnetic Fields

remarkable feature, namely, that the secondary field caused by the generators arising in the medium is also sinusoidal and has the same frequency as the primary field. Moreover, the use of sinusoidal oscillations implies that we deal with an established field. From the physical point of view this means that the behavior of the field at the moment of its appearance does not have any influence on the field when it is measured. Taking into account this consideration, we formulate the boundary value problem for sinusoidal oscillations. Then, applying Fourier's integral, we can determine the field for an arbitrary excitation of the primary field. Bearing in mind that in geophysical applications different electromagnetic fields are used, we study both the frequency and transient responses. It is proper to note that in considering the transient field we will pay special attention to the case when the primary magnetic or electric field changes as a step function. It will also be assumed that the medium is piecewise uniform and the magnetic permeability is constant and equal to /-La' In accordance with Eqs. 0.114), the complex vectors describing the electromagnetic field

at regular points satisfy the system curlE = icuB,

curlB = /-L(j - icu€E)

(1.362)

and interfaces (1.363)

where E and B are still complex field vectors, but for simplicity the index "*,, is omitted. In most cases we assume that the surface density of currents i is equal to zero, and then in place of Eqs. (I.362) and (I.363) we have curIE=icuB,

E lt =EZI

curl B = /-L( 'Y - iCUE)E

B lt=B 2 ! ,

ifi=O

(1.364)

Thus, the behavior of all possible electromagnetic fields, which vary with time as sinusoidal functions, should be described by Eqs. (I.364), provided that /-L = constant and i = O. In other words, this system has an infinite number of solutions, and correspondingly it constitutes only part of the boundary value problem. Of course, this fact is not surprising, and it is inherent to the system of equations of any field.

1.12 Determination of Electromagnetic Fields

149

Now it is appropriate to remember the following. From the integral form of Maxwell's equations,

~Bt

dt= Jl.,

f ('Y - iw€)E dS S

L

n

and continuity of tangential components of fields E and B, we have to conclude that the normal components of the magnetic field and the total current density are also continuous functions: (1.365)

Our next step in formulating the boundary value problem is almost obvious. In fact, changing the position and parameters of generators of the primary field, we also change the distribution of charges and currents arising in a medium. In other words, the total electromagnetic field also changes. Therefore, we have to know the type, intensity, and position of generators of the primary field. Taking into account the fact that in geophysical applications these generators can usually be treated as either electric or magnetic dipoles or current lines with an infinitely small cross section, we restrict ourselves to only such types of generators of the primary field. Certainly, these are an approximation to real generators and because of this the primary field can have some peculiar features which as a rule manifest tlrernselves near its generators. Considering the time-invariant fields of electric and magnetic dipoles, as well as current lines, we established that in approaching these generators their fields increase without limit (Part A). Therefore, the total electromagnetic field tends to the primary one when the observation point approaches its generators, that is Eo + E,

---7

Eo

B = B o + B,

---7

Bo

E

=

(1.366)

where E, and B, are the electric and magnetic fields caused by secondary generators which appear in the medium. We can say that Eq, (1.366) defines the field behavior on a surface surrounding the primary generators and located in their vicinity. Let us mentally imagine an arbitrary generator of the field with given parameters and known location. Then, from the physical point of view, it is clear that this generator of the primary field creates only one electromagnetic field in the medium with a given distribution of parameters.

150

I

Electromagnetic Fields

As follows from Eqs. 0.364), this system contains information about the medium and the relationship between the electric and magnetic fields. At the same time, it does not describe the field behavior at infinity, while in most cases models of media considered in geophysics have at least one unlimited dimension. Again, from the physical point of view it is natural to expect that at infinity both the electric and magnetic field tend to zero: E~O,

if L

B~O,

~

(1.367)

00

where L is the distance between generators of the primary field and the observation point. Of course, Eqs. (1.367) do not tell us how the electromagnetic field decreases at infinity, and this question will be considered in each specific problem. Thus, Eqs. (1.364), (1.366), and 0.367) describe the field everywhere and correspondingly the boundary value problem is formulated as follows. 1. At regular points of the medium: curlE

=

iwB,

curl B = J.L( y - iWE)E

2. At interfaces:

if i

=

0

and

J.L = const.

(1.368)

3. Near generators of the primary field:

4. At infinity: E~O,

B~O,

if L

~

00

Often it is more useful to replace the system (1.364) by equations which describe either the electric or magnetic fields. In Section 1.5 we demonstrated that vectors E and B satisfy Helmholtz's equation at regular points of the piecewise uniform medium:

where k 2 = iyJ.Lw

+ W2EJ.L

Thus, the other form of the boundary value problem is as follows. 1. At regular points of the medium:

1.12 Determination of Electromagnetic Fields

151

2. At interfaces, where the surface current is absent, tangential components of the field are continuous functions: if J.L = const.

(1.369)

3. In the vicinity of generators of the primary field: 4. At infinity: E~O,

B~O

Certainly both sets of equations, (1.368) and (1.369), are equivalent to each other. Earlier we pointed out that each set of these equations uniquely defines the electromagnetic field. In fact, they contain complete information about the medium and generators of the primary field, and it is impossible to imagine that the same generators create different fields in the same medium. Although the uniqueness of the solution of the boundary value problem given by either set (1.368) or (1.369) is obvious, we will still prove it. The main purpose of this step is to introduce the complex Poynting vector, which allows us to describe the distribution of energy in terms of complex amplitudes of the field. Now we show that conditions which constitute the boundary value problem given by either Eqs. (1.368) or (1.369) uniquely define the field. For simplicity, we assume that the medium inside the volume V has only one interface (Fig. I.12b). In order to prove the theorem of uniqueness we derive an equation which to some extent describes the principle of conservation of energy. With this purpose in mind, let us write down Maxwell's equations for both time dependence e- iwt and e'?" and perform some algebraic transformations. Then we have curlE = iwB curl B = /L( y - iWE)E curlE* = -iwB* curlB* =,u(y+iWE)E*

(1.370)

where E*e iwt and B*e iwt are conjugate to the functions Ee- iw t and Be- iwt , respectively. Next, multiplying the first equation of the set (1.370) by B * and the fourth by E and forming their difference, we obtain B* . curl E - E· curlB*

= iwB' B* -

/LyE'

- iWEJ.LE . E *

E*

152

I

Electromagnetic Fields

or

Inasmuch as

E = (Re Ej + i ImEo)e- iwt E* = (ReEo-ilmEo)eiwt

B = (Re Bj + i ImBo)e- iwt

(1.372)

B* = (ReB o - iImBo)e iwt where Re Eo, 1m Eo, Re B o , and 1mBo are real vectors. Then we can conclude, after performing cross and dot product operations, that (1.373) Let us note that Eq. (1.371) reminds us of the equation describing the principle of conservation of the energy, but, as will be shown later, it does not have the same meaning. Now applying Gauss's theorem and taking into account the presence of an interface 5 12 in place of Eq. (1.373), we have

(1.374)

= - J.L

J.vy E . E * dV

where E 1 ,B 1 * and E 2 , B2 * are functions at the back and front sides of the surface 5 12 , respectively. The scalar (E X B *)n is the component of this cross product which is perpendicular to the surfaces 50 and 5 p • The latter surrounds the generators of the primary field and is located in their vicinity. At this point we are prepared to prove that the conditions of the boundary value problem uniquely define the field. First of all, Eq. (1.374) can be greatly simplified. In fact, the integral over the surface 50 vanishes, since the field tends to zero at infinity. Due to the continuity of tangential components of the electric and magnetic fields, the third integral on the left-hand side of Eq. 0.374) also disappears. Then, taking into account the fact that in ap-

1.12 Determination of Electromagnetic Fields

153

proaching generators of the primary field the total field tends to the primary one, Eq, (1.374) can be rewritten as (1.375)

It is important to emphasize that since the volume V does not contain

extraneous forces the integrand

is a positive number. In fact, we have yE' E* = y(ReE =

+ i ImE) . (ReE - i ImE)

y{(ReE)2 + (ImE)2) > 0

Suppose that there are two different electromagnetic fields caused by the same generators of the primary field; that is, at points of the surface Sp

and

Then it is obvious that Eq. (1.375) for the difference of these electric fields E(3) = E(2) - E(1) becomes

fvyE(3). E(;)dV= 0

(1.376)

Inasmuch as the integrand is positive, we have to conclude that the field E(3) is equal to zero at every point of the volume V. In other words, the boundary value problem [Eqs, (1.368) or (1.369)] uniquely defines the field. Making use of the first of Maxwell's equations curlE = iwB we see that the magnetic field is also uniquely determined. Let us make several comments: (a) The medium can have any number of interfaces. (b) We assume that surface currents are absent. Otherwise the third integral on the left-hand side of Eq, (1.374) is not equal to zero, but it has

154

I

Electromagnetic Fields

the form RefJ.-

f

i· E* dS

SJ2

which is also a positive number. (c) The integral over the surface So must be equal to zero, and in order to guarantee this we suppose that the medium has a nonzero conductivity, everywhere, even if it is infinitesimal. (d) The theorem of uniqueness was proved, assuming that the magnetic permeability is constant. The same approach can be used in a medium with interfaces where magnetic permeability is a discontinuous function. (e) In those cases when the primary field does not have singularities in the vicinity of their generators, we do not need to introduce the surface s.. Next, let us discuss the physical meaning of every team in Eq. (1.371). Taking into account the fact that the functions E . E * and B . B * are real, this equation can be represented as Rediv (

EX B* ) fJ.-

=

-yE· E* (1.377)

Imdiv (E XfJ.-B* )

=

2w(_B_~fJ.-_B_*

__ EE_~ E_*)

It seems that the meaning of Eqs, (1.377) is obvious. However, it is

necessary to be careful since the functions under consideration are complex. In fact, we have E

(Re Ej + i 1mEo)(cos tot - isin wt)

=

Eoe- i w t

=

(Re Eo cos cot + 1mEo sin wt) + i(Im Eo cos tot - Re Eo sin w)

=

aCt) +ib(t)

=

(1.378)

E * = Eo * e iw t = (Re Eo - i 1mE o)( cos wt + i sin w t ) =

(ReE o cos wt + Im E, sin w) - i(ImE o cos tot - ReE o sin wt)

=

aCt) - ib(t)

Thus,

E(t) = aCt) = ib(t) E*(t) = aCt) -ib(t)

(1.379)

1.12 Determination of Electromagnetic Fields

155

By analogy we have

B(t) =c(t) +id(t) B * (t) = c( t) - ide t)

(1.380)

Each vector function a(t}, b(t}, c(t}, and d(t) is a solution of Maxwell's equations, and in order to describe the real electric and magnetic field we have to choose either the real or imaginary part of the complex functions E and B. Correspondingly, the Poynting vector is 1

yet) = -(ReE x ReB)

(1.381)

JL

while the densities of the magnetic and electric energies are ReB-ReB

v=----2p,

and

(1.382) eReE'ReE

u=----2

where ReE

=

aCt) = Re Eg cos tat + Irn Ej sin cot

Re B = c( t)

=

Re B o cos tot + ImB o sin tot

Therefore, in general the electric and magnetic fields change both their magnitude and direction during the period of their oscillations. It is clear that the Poynting vector, as the cross product of these vectors, displays the same feature. Of course, the density of the electromagnetic energy also varies with time. In other words, the instantaneous values of these quantities are hardly useful in describing the density of the electromagnetic energy and its flux. For this reason we introduce the mean values of these functions from the relationship

Mm

liTM(t)dt

= -

T

(1.383)

0

where T is the period of oscillations. In particular, if M is a sinusoidal function, its mean value is equal to zero. Now we make an attempt to express the mean values of Y, v, and u in terms of complex vectors of the field.

156

I

Electromagnetic Fields

In accordance with Eqs. (1.378)-0.380), we have E( t)

+ E * (t)

ReE(t)

=

2

ReB(t)

=

B(t) +B*(t) 2

and

Therefore, we have

The sum of two terms

can be represented as N 1 = (a + ib) X (c + id) + (a - ib) X (c - id) =2(aXc-bXd)

Making use of Eqs. 0.378)-0.380) we find that N 1 = esin2wt

where e is some vector. As follows from Eq. (I.383), the mean value of N1 is equal to zero. Consider the second pair on the right-hand side of Eq. 0.384); N 2=E* XB+EXB*

It is clear that N2 = (a - ib) X (c + id) + (a + ib) X (c - id) = 2a Xc + 2b X d = 2{Re Eo X Re B o + Im Eo X 1m Bol

that is, this vector is independent of time and correspondingly (1.385)

By definition, the vector N2 can be written as (1.386)

1.12 Deterrnination of Electromagnetic Fields

157

Thus, the mean value of the Poynting vector is

1 1 1 -(ReE X Re Bj , = - N Z m = - N z ~

4~

4~

Then, taking into account Eq. 0.386), we have (1.387)

or

where (1.388)

is called the complex Poynting vector. Thus, we have expressed the mean value of the Poynting vector in terms of complex vector amplitudes. Now let us perform similar procedures with the dot products ReE· ReE

and

ReB' ReB

By analogy with Eq. (1.384) we have

and

Therefore, Eq (1.377) can be written in the form (1.389)

This means that the real part of the divergence of the vector Yc defines the average amount of heat which appears during 1 sec. At the same time, the imaginary part of the divergence by a factor 2 w exceeds the difference of mean values of the magnetic and electric energies.

158

I

Electromagnetic Fields

Applying Gauss's theorem we have

(1.390)

Certainly, there is a fundamental difference between the Poynting vector and the complex Poynting vector. The real Poynting vector Y characterizes the density and direction of the flux of the energy at any instant, while the other is a complex vector and the real and imaginary parts of its flux through a closed surface have clear but different physical meanings. Now we return to the main subject of this section, namely, the determination of the field. In Section 1.5 we introduced the vector potentials of the electromagnetic field A* and A and pointed out that often they allow us to simplify the field calculation. Taking this fact into account, we formulate the boundary problem in terms of these potentials. As follows from Eqs. (1.101) and 0.102), we have the following equations for the vector potential of the magnetic type:

E = curl A",

iwB = k 2A* + graddivA*

(1.391)

where A* is a complex vector. Therefore, the boundary value problem in terms of the. vector potential A* consists of the following elements. 1. At regular points:

2. In the vicinity of generators of the primary field: A*~A~

where A~ is the vector potential of the primary field. 3. At interface, the behavior of the function A* must provide the continuity of the tangential components of the electric and magnetic fields. 4. At infinity: A*~O

In general, the electromagnetic field is described by six components. At the same time, in many problems which are of great practical importance

1.13 Relationships between Different Responses of the Electromagnetic Field

159

in geophysics, it is possible to define the field using only one or two components of the vector potential. In accordance with Eqs. 0.104) and (1.106)-0.109) we have for the vector potential of the electric type B = curIA

E=iwA+

1

.

grad divA

'YJ-L - IWEJ-L

(1.392) and correspondingly the boundary value problem can be formulated for the vector potential A in the same manner as that for the function A*. Also, it is appropriate to note that in some cases it is necessary to use both vector potentials in order to determine the field. As is well known, in solving boundary value problems different methods are used, such as the separation of variables, integral equation, and finite differences. All of them have been used for developing the theory of electromagnetic methods in geophysics, and some results of their application are discussed in Part C.

1.13 Relationships between Different Responses of the Electromagnetic Field In this section we explore some general relationships between various responses of the electromagnetic field. First of all we start from the relationship between the quadrature and in-phase components of the field. Suppose M represents the complex vector of either the electric or magnetic fields and M = InM

+ i(QM)

(1.393)

where In M and Q M are the in-phase and quadrature components, respectively. Substituting Eq. 0.393) into Helmholtz's equation,

we have

160

I

Electromagnetic Fields

or

and

(1.394)

Thus, there is a linkage between the in-phase and quadrature components of the spectrum. Now we examine this subject in more detail. Let us assume that a solution of Maxwell's equation has the form

where M is the complex amplitude of the field in an arbitrary direction. Of course, in order to obtain an actual sinusoidal field, it is necessary to take the imaginary part of the product

As we know, if the solution contains a complex amplitude term, from the physical point of view this means that there is a phase shift between the field and, for instance, the current which generates the primary field. Correspondingly, the field can be represented as being the sum of the quadrature (Q) and the in-phase (In) components, and we have

M = In M + i(Q M) = Mo sin cp + iMo cos cp

(1.395)

where M o and cp are the amplitude and phase, respectively. Making use of the conventional symbols for representing a complex function, we can write M as

M(z)

=

U(z) + iV(z)

(1.396)

where Ui z ) and V(z) are the real and imaginary parts of the function

Mt z), respectively, and z is an argument defined as z =x + iy where x and yare coordinates on the complex plane z. In our case, the complex variable z is the frequency w

=

Re w

+ i 1m w

and the complex amplitude M of the electromagnetic field is usually an analytic function of frequency w in the upper part of the w-plane (Im w > 0).

1.13

Relationships between Different Responses of the Electromagnetic Field

161

As is well known, the Cauchy- Riemann conditions

au av

au

av

ax

ay

ax

ay'

(1.397)

are necessary and sufficient for analyticity of a function. These conditions express the relationship that exists between the real and imaginary parts of an analytic function in the vicinity of a point z; that is, they represent analyticity in differential form. Our purpose is to describe the relationship between the quadrature and in-phase components of the field for real values of lV, because the electromagnetic field is observed only at real frequencies. Moreover, unlike the Cauchy-Riemann conditions, we will establish such relationships that both components will be represented in an explicit form. With this purpose in mind, let us make use of the Cauchy formula. It shows that if the function M(z) is analytic within a contour C, as well as along this contour, and a is any point in the z-plane, then if a

M(z) tf.. - - dz = 21TiM( a) ~ z-a

E

C

if a on C if a

(1.398)

~C

It is clear that the Cauchy formula allows us to evaluate M(a) at any point within the contour C when the values of M(z) are known along this contour. This relationship is a consequence of the close connection which exists between values of an analytic function on the complex z-plane. We will consider a path, consisting of the x-axis and a semicircle with an infinitely large radius. Its center is located at the coordinate origin (Figure I.l2c). The internal area of this contour includes the upper half plane. We will attempt to find the quadrature component of the function M = U + iV by assuming that the in-phase component U is known along the x-axis or vice versa. Applying the Cauchy formula we have

M(n

1 =

-;-ptf..

M(z) --dz

11T ~ z - (

(1.399)

The point (= E + iYJ is located on the path of integration, and the symbol p indicates the principle value of the integral. Inasmuch as the path of

162

I

Electromagnetic Fields

integration coincides with the x-axis, (TJ = 0), we obtain 1

00

M(E,O)=-;-pf 1'Tr

M(x,O) X -

-00

(1.400)

dx

E

In developing Eq. (1.400) it has been assumed that the value for the integral along the semicircular part vanishes when the radius of the circle increases with out limit. This conclusion follows from the fact that the function M(z) representing the field is analytic at infinity and with an increase of z the ratio M(z)/z tends to zero. Because

M(E,O) = U(E,O) +iV(E,O) and

M(x,O) = U(x,O) +iV(x,O) from Eq. 0.400) it follows that 1

U(E,O)=-P

foo V(x,O)

'Tr

X -

-00

1

00

V(E,O)=--pf 'Tr

-00

(1.401)

dx

E

U(x,O) dx X-E

(1.402)

The integrands in these expressions are characterized by a singularity at E, which can be readily removed by making use of the identity

x =

00

dx

pf - = 0 -ooX-E Now we can rewrite Eqs. 0.401) and 0.402) in the form 1 foo V(x,O) - V(E,O)

U(E,O)=-

'Tr

-00

X-E

dx

(1.403)

and

V(E,O)

1 foo U(x,O) - U(E,O) =

-'Tr

-00

X -

E

dx

(1.404)

1.13 Relationships between Different Responses of the Electromagnetic Field

163

since

f -dx 00

V(E,O)

dx f -=0 00

= U(E,O)

-ooX-E

-ooX-E

Equations (J.403) and (1.404) establish the relationship between the real and imaginary parts of an analytic function, and the integrands on the right-hand side of these expressions do not have singularities. Let us return to consideration of the complex amplitude of the field M(w) =lnM(w) +iQM(w)

In accordance with Eqs. (J.403) and (J.404), the relationships between the quadrature and in-phase components are: 1 InM(wo)=7T

foo

QM(w) - QM(wo) w-w o

-00

1

QM(w o)=-7T

foo -00

dto

In M( w) - In M( wo)

w-w o

(1.405)

dto

(1.406)

Thus, when the spectrum of one of the components is known, the other component' of the field can be calculated by making use of either Eq. (J.405) or Eq. 0.406). It is now a simple matter to find the relationship between the amplitude and phase responses of the field component. Taking the logarithm of the complex amplitude M = Moe-i'l' we have

In M

=

In M o - itp

From this equation we see that the relationship between the amplitude and phase responses is the same as that for the quadrature and in-phase components. For instance, for the phase we have

(1.407)

Often, it is preferable to express the right-hand side of Eq. (J.407) in

164

I

Electromagnetic Fields

another form. After some algebraic operations we obtain

f-ooudL I

III2

'P( Wo) = - -1°o -d lncoth - du 7T

(1.408)

where L = In M o ,

W

u=lnWo

It can be seen from Eq. (1.408) that the phase response depends on the slope of the amplitude response curve plotted on a logarithmic scale. Inasmuch as the integration is carried out over the entire frequency range, the phase at any particular frequency W o depends on the slope of the amplitude response curve over the entire frequency spectrum. However, the relative importance of the slope over various portions of the spectrum is controlled by a weighting factor [Incoth u /21, which can also be written as:

w +w o In--w -w o

The behavior of this factor is shown in Figure I.l2d. It increases as the frequency approaches W o and is logarithmically infinite at that point. Therefore, the slope of the amplitude response near the frequency for which the phase is to be calculated is much more important than the slope of the amplitude response at more distant frequencies. From a geophysical point of view, Eqs, (1.405)-(1.408) lead us to the following conclusions. First of all, measurements of the phase response do not provide additional information about parameters of the medium when the amplitude response is already known. However, it may well be that the shape of the phase response curve more clearly reflects some diagnostic features of the geoelectric section of a medium than the amplitude response curve. It is also important to stress that while there is in essence a unique relationship between the quadrature and in-phase responses, as well as between the phase and amplitude responses, this does not mean that there is a point-by-point relationship between them. In fact, measurements of both amplitude and phase at one or a few frequencies provide two types of information characterizing the geoelectric section in a different manner. The same conclusion can be reached for measurements of the quadrature and in-phase components.

1.13

165

Relationships between Different Responses of the Electromagnetic Field

Next, we investigate the relationship between the frequency and transient responses. The information we need is obtained through the use of the Fourier transform, which has the well known form

M(t)

=

1 -f 27T

00

M(w)e-iwtdw

-00

M(w) =

(1.409)

M(t)e iwt dt

foo -00

where M(t) and M(w) are the transient response and its spectrum, respectively. Assuming that the transient response appears at some instant t = 0, we can rewrite the last equation of set (1.409) as Re M( w) + i 1m M( w) = l°OM(t)eiwt dt o =

100M( t ) cos wt dt + i 100M( t) sin wt dt o 0

or Re M ( w) 1m M( w)

=

=

100M( t) cos wt dt o

(1.410)

l°OM(t) sin wtdt o

Therefore, the real and imaginary parts of the complex spectrum are even and odd functions, of w, respectively:

ReM(w) = ReM( -w) 1m M(w)

=

(1.411)

-1m M( -w)

Let us note that Eqs. (l.41O) allow us to calculate the in-phase and quadrature components of the spectrum as well as their derivatives with respect to w when the transient response is known. This procedure may be useful for reduction of different types of noise. In most cases considered in this monograph, it is assumed that a transient electromagnetic field is excited by a step function current.

166

I

Electromagnetic Fields

Correspondingly, the primary magnetic field accompanying this current does likewise:

t 0

(1.412)

Now we demonstrate that the transient response can be expressed either in terms of the quadrature or in-phase component of the spectrum. Similar results can be derived for the electric field, too. As follows from Eq. (1.409) and Eq. (1.412) the spectrum for the primary magnetic field is

Bo(w)

Bo

=-.

(1.413)

IW

The amplitude of this spectrum decreases at a rate inversely proportional to frequency, while the phase remains constant. Inasmuch as the energy of the primary field is mainly concentrated at the low-frequency part of the spectrum, when a step function excitation is applied, its use is often preferable in the practice of geophysical methods. This is related to the fact that with a decrease of frequency the field penetrates to greater depth. In accordance with the Fourier transform formula, the primary field can be written as

Bo(t) =

Bo

-f 27T

00

-00

e- iw t

-.-dw

(1.414)

IW

where the path of integration is not permitted to pass through the point w = O. Let us write the right-hand integral of Eq. (1.414) as a sum:

1 00 e- iw t 1 E e- iw t - f -dw=-f -dw 27T -00 iw 27T -00 iw iw t 1 +E e1 00 e- i w t +-f -dw+-f -dw 27T -E ito 27T +E ico

We select a semicircular path of integration around the origin w whose radius tends to zero. In calculating the middle integral we introduce a new variable tp: Then we have

dw

=

ire'" de:

=

0,

1.13 Relationships between Different Responses of the Electromagnetic Field

167

and 1 +€ e -iWI 1 271" ire'" 1 - f - d w = - j - . dcp=21T - € ita 21Ti 71" re'" 2

Respectively, the second expression for the primary field when the variable of integration w takes only real values is Bo

Bo(t)

= -

2

iWI

Bo e-f -. -dw 21T lW 00

+

(1.415)

-00

Next, making use of the principle of superposition, we obtain the following expressions for a nonstationary field: 1 B(t)=-.f 21Tl

B( w) . --e-,w1dw

00

(1.416)

w

-00

and B(t)

s; + -1. foo __ B ( w) IW . e- I dw

(1.417)

= -

2

21Tl

W

-00

where B ( w)

=

In B ( w)

+ i Q B ( w)

is the complex amplitude of the spectrum. In other words, B(w) characterizes the field in the medium when the primary field varies as the sinusoidal function B o sin wt. As is well known, the derivative of a step function with respect to time is a delta function, and, in accordance with Eq, (1.415), we then have B(t)

1

=

00

- - f B(w)e-iW'dw 21T

(1.418)

-00

where B(t) is the transient response caused by a delta function excitation. Therefore, we can treat B(w) as the spectrum of the field, provided the primary field is a delta function. Let us represent Eq. (1.417) in the form

e; 1 foo Q B( w) cos wt -In B( w) sin wt B(t)=-+dw 2 21T W -00

i

-21T

foo Q B( w) sin wt + In B( w) cos wt -00

W

dw

(1.419)

168

I

Electromagnetic Fields

From the physical point of view, as well as from Eqs. (1.411), it follows that the second integral in Eq. (1.419) is equal to zero and we obtain

s;

B(t) = -

2

1

1'"

Q B( w) cos tot - In B( w) sin cot

rr

0

w

+-

dco (1.420)

If the time t is negative, then

B(t) =Bo

and therefore Bo =

s; -

2

'" Q B( w) cos 1

1

+-

tot

7T 0

+ In B( w) sin tat

dw

W

or

0=

e;

1

--+2

00

1

Q B( w) cos cat + In B( w) sin cot

7T 0

dto

(1.421)

W

It is proper to note that in these last expressions the time t is taken as

positive. Combining Eqs. (1.420) and (1.421), we obtain B(t)

2

ooQB(w)

= ( 7T

io

w

cos cot dca

ana

(1.422) 2 B(t)=B o-- ( tr

00InB(w)

io

w

sinwtdw

Equations (1.422) permit us to calculate the transient response when either the quadrature or in-phase components of the spectrum are known. Of course, making use of Eq. (1.422) it is a simple matter to express derivatives of the field with respect to time in terms of the spectrum. In particular, we have aB

-

at

2

00

= - - ( Q B ( w ) sin w t d co tr i o (1.423)

or aB

2

00

-=--(

at

tr

io

InB(w)wtdw

It is obvious that similar equations for the magnetic and electric fields can

be derived

for other types of excitation of the

primary field.

References

169

References Bursian, V. R. (1972). "Theory of Electromagnetic Fields Applied in Electroprospecting." Nedra, Leningrad. Frederiks, V. K. (1933). "Electrodynamics and Theory of Light." Kubuch, Leningrad. Kaufman, A. A. (1992). "Geophysical Field Theory and Method, Part A." Academic Press, San Diego. Stretton, J. A. (1941). "Theory of Electromagnetism." McGraw-Hill, New York. Smythe, W. R. (1968). "Static and Dynamic Electricity, 3d ed.," McGraw-Hill, New York. Tamm, I. E. (1946). "Foundation of Theory of Electricity." GITIL, Moscow.

Chapter II

The Magnetic Dipole in a Uniform Medium

11.1 Frequency Responses of the Field Caused by the Magnetic Dipole 11.2 The Transient Responses of the Field Caused by a Magnetic Dipole References

In developing the theory of electromagnetic methods, we are mainly concerned with the behavior of fields observed either on the earth's surface or in the borehole. However, in order to understand their behavior better it is useful at the beginning to investigate in detail the field and currents in a uniform medium. This approach will permit us to obtain some insight into the physical principles on which electromagnetic methods are based, even though the effect of boundaries between various media cannot be discussed. In this chapter we assume that the field is caused by a magnetic dipole; and this choice is related to the fact that an inductive excitation of the primary field is used in most electromagnetic methods.

11.1 Frequency Responses of the Field Caused by the Magnetic Dipole

Let us define a magnetic dipole with the moment (11.1)

where M o = IoSn is the magnitude of the moment and 1= Ioe- i w t is the complex function describing the current in the dipole, w is the frequency in radians per second, n is number of turns in the loop, S is the area enclosed by one turn of the loop, and k is the unit vector along the z-axis. Now we formulate the boundary value problem in terms of the vector potential A*. As follows from Section 1.12, this function must satisfy three 170

II.1

Frequency Responses of the Field Caused by the Magnetic Dipole

171

conditions: 1. At regular points of the medium A* is a solution of Helmholtz's equation:

(11.2) where k 2 is the square of the wave number, and A* is the complex vector potential. 2. In the vicinity of the dipole the vector potential A* approaches A~ , which describes the primary magnetic field (11.3)

A*~A~

3. At infinity, the vector potential tends to zero:

(11.4)

A*~O

In accordance with Eqs. (1.391), the electromagnetic field is expressed in terms of the potential A* as

+ graddivA*

iwB =k 2A*

E = curlA*,

(II .5)

We take a spherical system of coordinates, R, e, and 'P, and a cylindrical coordinate system, r, 'P, and z, having a common origin. The dipole is located at the origin and its moment is directed along the z-axis (Fig. 1I.1a). As was shown in Section 1.7, the primary electric field has only the component, Eo",. Therefore, it is natural to assume that the induced conduction and displacement currents in the medium are described by only this azimuthal component. Correspondingly, the total electric field has the single component E",. Inasmuch as E = V X A*, we suppose that the components A~ and are equal to zero so that the field is described by a single component of the vector potential A~ , which is a function of the distance R. In solving the boundary value problem we will see that both assumptions are correct. Equation (II.2) takes the following form in a spherical system of coordinates:

A:

1 _ d _ R 2 dR

( R2 _ dA* ) +k 2A*=O _ 2 dR 2

(II .6)

The solution for this equation is well known:

»»

e- i k R

R

R

A*=C--+D-2

where C and D are constants.

(11.7)

172

The Magnetic Dipole in a Uniform Medium

II

a

b 100

Z

--1 --2 10

R

:;><.:::::-b

t>;" ,// M

»::

--- <, ....!L <,be

./

.1

10

.1

.01

100

c 1
bR X=2

1()2 6

10

14

10-2

d 18

QbR

1()'4

0.6

22 0.4

10"6 0.2

10"6 1Q"'4

~

10"2

1()2

1()4

Fig. II.l (a) Magnetic dipole in uniform medium; (b) behavior of wave number as function of a; (c) dependence of field b R on parameter f3; (d) behavior of quadrature components QbRo

P

II.1

Frequency Responses of the Field Caused by the Magnetic Dipole

173

It is obvious that only the first term on the right-hand side of Eq (II.7) satisfies the condition at infinity, and consequently we represent the vector potential as

eik R A*=C-

(II .8)

R

Z

As is seen from Figure 1I.1a, we have the following expressions for components of the vector potential in a spherical system of coordinates: A~ =A~

cosO,

A~

= -A~ sin 0,

A*

Substituting these expressions into the first equation of the set (11.5), we have 1 ( E

a(

R aR

RA~)

aA~) ee

- --

(11.9)

Carrying out some relatively simple algebraic operations we obtain e ik R E

(11.10)

At the same time, the primary electric field as was demonstrated in Section 1.7, ~

(II.ll) As follows from the boundary value problem, the electromagnetic field in the vicinity of the dipole coincides with the primary one. Therefore, comparing Eqs. 01.10) and rn.in we have to conclude that

iwp.M C=-47T

(11.12)

where M is the magnitude of the moment. Thus, the components of the electric field are E

iwp.M .


= _ _ e' k R (

47TR 2

E R =EIJ

=

0

1 - ikR) sin 0

(11.13)

174

II

The Magnetic Dipole in a Uniform Medium

From the second equation of set (II.5) it follows that iwBR = k 2A*R

a

+ -aR divA*

and iwB =k 2A* (J

(J

a

+- divA* Rae

Since divA*

aA*

aA* aR

= __ Z = __ Z

az

_

aR

aA*

= __ Z

az

cos e

aR

or iWJLM .

divA* = --e'kR(ikR -1) cos () 2

47TR

we have for the magnetic field BR = B

= (J

~

2JLM .

- - 3 e 'kR(1

47TR

- ikR) cos() (II.14)

JLM e ikR(1 _ ikR _ k 2R 2 ) sin e

47TR 3

we know (Section 1.7), the primary electromagnetic field is E~ =

iWJLM --2

47TR

2JLM

sin e,

B~ =

--3

47TR

cos e, (11.15)

In other words, Eq. (I1.15) describes the field when the influence of the conduction and displacement currents in the medium can be ignored. Usually such a field is called quasistatic. Thus, the equations of the electromagnetic field, when normalized to the primary field, take the form

E EO

.

e = ---.:£.. = e 'kR(1 - ikR)

'"

(II.16)

'"

(11.17)

(11.18)

Il.I

175

Frequency Responses of the Field Caused by the Magnetic Dipole

The amplitude coefficients in these equations, ecp' bR , and bo' are complex and can be rewritten in terms of amplitude and phase or the quadrature and in-phase components. For example, in the case of the electric field ecp as well as bR , the amplitude and phase are given by

Acp

=

e- b R [ (1

1/ 2

+ bR)2 + a 2R2]

(11.19)

and

aR cP = aR - tan " ! - - cp 1 +bR

(11.20)

The in-phase and quadrature components are In ecp = In bR = Acpcos cPcp

(11.21)

Q ecp = Q bR =Acpsin cPcp

(11.22)

Similarly, for the component of the magnetic field b o ' we have

2

A o= [(1+bR+b 2R 2 - a 2R 2 ) +(aR+2abR 2 )

2] 1/2 e- b R

(11.23)

and cp = aR - tan -

o

aR + 2abR 2 1

(11.24)

----~~---::----=-

1 + bR

+ b 2R 2 - a 2R 2

Correspondingly, the in-phase and quadrature components of bo are In bo =Aocos cPo,

Q bo =Aosin cPo

(11.25)

In Eqs. (11.19)-(11.24) the parameters a and b are the real and imaginary parts of the wave number, respectively. Expressions for a and b can be obtained from the equality

(11.26) Squaring both sides of the equation and separating the real and imaginary parts, we have

yp.,w

=

2ab

(11.27)

Solving this pair of equations for a and b we have /2[

a=w (

2iu: ) 1

b=w (

li E)1/2[ (1+a T

(l+( 2 )

1/ 2 ] 1/2

+1

2 ) 1/ 2 _

]112

1

( 11.28) (II .29)

176

II

The Magnetic Dipole in a Unifonn Medium

where y a=-

(II.30)

we

The parameter a represents the ratio of the conduction currents to displacement currents in the medium. It is useful to examine the behavior of the real and imaginary parts of the wave number a and b as functions of a. First we assume that the displacement currents are significantly larger than the conduction currents, that is, a is much less than unity. Expanding the expressions on the right-hand side of Eqs, 01.28) and 01.29) in a power series and neglecting higher order terms, we obtain

(11.31 ) ifa«:1

where (II.32) and

1 ( JL ) 1/2 b = -y e 2 e

(11.33)

Thus, in the limiting case when the parameter a tends to zero, the real part of the wave number a 6 is independent of the conductivity and it is directly proportional to the frequency. It is clear that w

ae = V-

(II.34)

where v is the velocity of the propagation of the wave. As follows from Eqs. (11.19) and (11.23), the imaginary part be characterizes the rate of decrease of the field. It is directly proportional to the conductivity, inversely proportional to the square root of the dielectric permittivity, and is independent of the frequency. The ratio of a, and b; is

be

1 2

-=-a

a;

(11.35)

As may be seen from Eq. 01.31), even when the value of a becomes nearly as large as unity, the real and imaginary parts of the wave number k differ

11.1

Frequency Responses of the Field Caused by the Magnetic Dipole

177

only slightly from the limiting values a E and bE' We also see from Eq. (1.33) that conductivity has an influence at all frequencies. It should be emphasized that in these expressions the conductivity, as well as dielectric permittivity, is independent of the frequency, while in some rocks this assumption may not always be warranted. Next, consider the case in which conduction currents dominate over displacement currents. Expanding the right-hand side of Eqs. (II.28) and (11.29) in power series in terms of 11a, we have a

=::

b

v

(1 + b (1 __ 1) a

y

_1 ) 2a

y

2a

(II.36) (11.37)

where a » 1 and 'Y/-LW ) 1/2

a=b= ( Y Y 2

1 8

=-

and 8 is the skin depth. From Eqs. (11.36) and (II.37) it is apparent that displacement currents have a relatively small effect. In the limiting case when a approaches infinity, the wave number k does not depend on the dielectric constant and then, as we know, 1+i k=8

(11.38)

Figure II.lb shows a set of curves illustrating the behavior of the real and imaginary parts of the wave number k as a function of a. Let us now introduce two new parameters, namely, a characteristic length R o and characteristic frequency W o ' The parameter R o is defined from the relationship

or Ro=

~ ( ~. I

r-

) 1/2

pF;

= 188.5'

(II.39)

The characteristic frequency is defined as being the frequency at which the

178

II

The Magnetic Dipole in

a Uniform Medium

displacement and conduction currents are equal, that is,

(11.40) We then treat components of the electromagnetic field bR , be, and ee as functions of two dimensionless parameters, namely, W

{3=-

and

R X=-

Wo

Ro

(11.41)

where R is the distance from the dipole to the location where the field is observed. A set of typical curves which illustrate the dependence of the function bR on the frequency parameter {3 is given in Figure 1I.1c. The index of each curve is the parameter X. These curves show that with an increase in frequency the electromagnetic field first decreases in a conducting medium. Then, near a characteristic frequency the amplitude of the field passes through a minimum value and with further increase of the frequency the field grows. When the characteristic length R o become smaller, the minimum value of bR decreases and shifts toward higher frequencies. If frequencies are lower than the characteristic frequency w o,

, the field is practically independent of the dielectric permittivity. Correspondingly, in this part of the spectrum it behaves as a quasistationary field. At frequencies which are higher than the characteristic frequency, w > wo, the field magnitude becomes greater and can be many times larger than the primary field, caused by the current of the dipole. Over this part of the spectrum the field essentially depends on both the conductivity of dielectric constant of the medium. Next, we consider in detail the quasistationary field of the magnetic dipole in a uniform medium (w «w o)' Assuming that displacement currents are small compared to the conduction currents, we can neglect by the propagation effect and deal only with the diffusion. Correspondingly, from Eqs. (II. 16)-0I.18) we have erp = e ik R ( l _ ikR)

bR = e ik R ( l - ikR) be = e i k R(l - ikR - k 2R 2 )

(II,42)

11.1

Frequency Responses of the Field Caused by the Magnetic Dipole

179

In these expressions 1+i k=-

o '

2 )1 /2 lO3 0= - = _(lOpT)1/2 ( YJLW 27T

(11.43)

and A 0=27T

where T is the period of oscillations and A is the wavelength. As was pointed out earlier, even though the term wavelength is used for the quasistationary field, it does not imply that the propagation effect is considered. Making use of Eq, (11.43), the magnetic field bR can be represented as the sum of two components. One of them is the quadrature component, which is shifted in phase by 90° with respect to the current in the dipole. The second one is the in-phase component, which is shifted in phase by 0° or 180° with respect to the source current. Then we have

= e- p [( l + p) sin p - p cos p] In bR = e-p[(l + p) cos p + p sin p] Q bR

(11.44) (11.45)

where

(11.46) is the parameter characterizing the distance between the dipole and the observation point, expressed in units of the skin depth. In Chapter I we noted that this parameter is sometimes called the induction number. According to the Biot-Savart law, the quadrature component of the magnetic field arises due to currents which are shifted in the phase by 90° with respect to the current in the dipole, while the in-phase component is the algebraic sum of the primary and secondary fields. The in-phase component of the secondary field is contributed by induction currents in the medium shifted by either 180° or 0° with respect to the dipole current. Thus, we assume that at each point of the medium there are quadrature and in-phase components of the current, and this approach is very useful for understanding electromagnetic methods when such components of the magnetic field are measured. Let us discuss the asymptotic behavior of the quasistationary field. First, consider the low-frequency spectrum. Expanding the exponential e i k R in

180

II

The Magnetic Dipole in a Uniform Medium

the series in the form of a power series eik R =

00

(ikR(

n=O

n!

E ---

and substituting this into Eq. (11.42), after some simple algebra we have

bR = 1 +

00

E

I-n

00

1-n

--,-(ikR( = 1 + E __,_2nI2pnei(3rrnl/4 n=2 n. n=2 n.

(II.47)

Taking into account the fact that 'YJ1- ) 1/2

p=R ( T

w

l/ 2

(II.4S)

we see that the series describing the low-frequency spectrum contains whole and fractional powers of w. Later we will discover that a similar representation of the spectrum holds even in a nonuniform medium when p ~ O. As follows from Eq. (11.47), Qb

2

R

::::p2 _ _ p3

3 2 In bR :::: 1 - 3P3

(11.49)

As is seen from Eqs. (11.49), at low frequencies the quadrature and in-phase components are related to the frequency, conductivity, and the distance from the dipole in a completely different manner. In fact, from Eqs, (11.48) and (II.49) we obtain for the field B R

(II .50)

It is clear that the in-phase component of the secondary field is more sensitive to a change of the conductivity, and in this approximation it is independent of the separation. This interesting fact indicates that the in-phase component has a much greater depth of investigation than the quadrature component within this part of the spectrum.

n.1

Frequency Responses of the Field Caused by the Magnetic Dipole

181

Our analysis was based on the study of the component bR , but it is obvious that the main results remain valid for the component be' too. As follows from Eqs. (11.44) and (II AS), in the opposite case, that is, at high frequencies, we have (II.51)

or

In other words, in this part of the spectrum we observe the internal skin effect, when induced currents concentrate in the vicinity of the dipole. Correspondingly, the secondary in-phase component In b'k tends to its limit -b~. Now we consider curves, illustrating the dependence of the quadrature and in-phase components of the field on the parameter p (Figs. I1.1d and II.2a). The quadrature component Q b R passes through a maximum value with an increase in the parameter p and then decreases to zero. The absolute value of the in-phase component of the secondary field also increases as the parameter p increases and then approaches b~. At large values of the parameter p, both components oscillate around their asymptotic values. Let us examine the low-frequency part of the spectrum in more detail. According to Eq, (IIA9), the quadrature component prevails over the in-phase component In bk, and from Eq, (11.50) we have 2p,M yp,wR 2 p,M QBR = 47TR3 2 cos e >- 47TRyp,wcos(J,

if p « 1

(11.52)

Hence, in the range of small parameter values, the quadrature component is directly proportional to the conductivity and the frequency and inversely proportional to the distance from the magnetic dipole. As will be shown later, these features of the field also remain valid in a nonuniform conducting medium. From Eq. (II.50) we see that over this part of the spectrum the secondary field is much smaller than the primary field: (II .53)

At this point it is appropriate to explain the behavior of the field in terms of distribution of induced currents. Making use of Eq, (11.42) and Ohm's law in differential form, j =yE

182

II

The Magnetic Dipole in a Uniform Medinrn

b

z

d

0.8 '0

lcp

c 0.6

0.4

0.2

0 -.01

-0.2 Fig. 11.2 (a) Behavior of in-phase component In bR ; (b) current field as system of current toroids; (c) current density j~ as function of distance; (d) behavior of quadrature component of current density.

P

ILl

Frequency Responses of the Field Caused by the Magnetic Dipole

183

we have the following expression for the current density at every point in the uniform medium: j",=

iYfLwM . ' k R(I-ikR)sin(J 4 2 e

rrR

(11.54)

As in the case of the magnetic field, we can represent the current density as the sum of the quadrature and in-phase components, and, in accordance with Eq. 11.54, we obtain Q j'l'

YfLW rM

= - - --3 e -p

4rr R

[(1 + P ) cos P + P sin p]

(11.55)

and YfLW rM

In j'l' = - - - --3 e -p [( 1 + p) sin p - p cos p] 4rr R

(11.56)

where r

- = sin (J R It is clear that the current field can be imagined as a system of current

toroids or rings which have a common axis with that of the dipole, and they are located in places perpendicular to this axis, as shown in Figure 11.2b. First of all, we determine the induced currents which arise due to the primary electric field only. As follows from Eq, (I1.15), the density of these currents is

(11.57) and it is shifted in phase by 90° with respect to the dipole current. If we could neglect the influence of the magnetic field accompanying the induced currents in the medium, the character of the current distribution would be defined by Eq, (11.57). In such a case the current density at any point in the medium is a function which is described by the product of two terms. One of them depends on the dipole moment, frequency, and conductivity, while the other is the function of the geometric parameters only. The behavior of j~ in planes perpendicular to the dipole axis is shown in Figure II.2c. It can be seen that with increasing z the distance from the z-axis to the ring with the maximal current also increases.

184

II

The Magnetic Dipole in a Uniform Medium

Let us introduce the notation yfLwM r

jo =

47T

R3

and rewrite Eqs. 01.55) and (II.56) as Q jcp = joe - P [ (1

+ p) cos p + P sin p]

In jcp = -joe- p [( 1 + p) sin p - p cos p]

(II .58)

An analysis of these functions permits us to explore how the actual current

density i, differs from jo for various values of the parameter p, especially for different distances from the dipole. Curves for the quadrature and in-phase components of the current density, normalized by i«. are shown in Figures II.2d and Il.3a. For small values of the parameter p, the quadrature component of the current density is essentially the same as the current density jo , that is, the interaction between currents is negligible in this case. With an increase in the parameter p, the ratio Q jcp/jo decreases, passes through zero, and, for larger values of the parameter p, approaches zero in an oscillating manner. The curve for the ratio of the in-phase component of the current density to jo has a completely different character. At small values of p the ratio In jcp/jo approaches zero, then increases to a maximum when p is about 1.5; and for larger values of p, it tends to zero again in an oscillatory manner. Therefore, the actual distribution of currents, in contrast to the behavior of t«. is determined by both geometric factors and interaction of currents. This last factor is taken into account in the case of a uniform medium by the parameter p. Comparing the curves in Figures II.2d and II.3a, we can see that for small values of p the quadrature component of the current density dominates. However, there is a range of values of p over which the in-phase component is significantly larger. The curve in Figure II.2d can be analyzed from two points of view. If the conductivity and frequency are held constant, the curve shows a change in the quadrature component of the current density related to j() when the distance from the dipole to an observation point increases. On the other hand, the position of the observation point can be fixed. Then this curve illustrates the frequency responses of the ratio Q jcp/jo or its behavior when the conductivity changes. This approach allows us to explain the main features of the quadrature component of the magnetic field proceeding from the distribution of the quadrature component of the current density. As can be seen from Figure Il.2d, for relatively small

11.1 Frequency Responses of the Field Caused by the Magnetic Dipole

185

values of p the current density Q i