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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
122
I
Electromagnetic Fields
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,) .. 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*
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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
:;>;" ,// M
»::
--- 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
O. I 1[q(t 2 - T5)1/2] is the modified Bessel function of the first order, 8Ct - TO) is the Dirac function, defined from the relationship
tf(x')8(n)(x -x') dx' a
= {( -l({n)(X)
a <x b
(11.68)
and t is time counted from the moment when the transmitter current is turned on.
II.2
The Transient Responses of the Field Caused by a Magnetic Dipole
193
In accordance with Eq. (11.66), the field arises at any point at the moment
and correspondingly with an increase of the distance from the dipole, the field, traveling with velocity
1
u= (
c
)1/2 = -(--)---;-1/-=2 EJ.L
ErJ.L r
appears at later times. Now we determine the components of the electromagnetic field. As follows from Eq. (11.5), the electric field E'P is related to the vector potential of the magnetic type A~ by
aA*
E = - __Z sin () 'I'
aR
Omitting intermediate transformations, we have if t < TO
E'P=O, and
E
(u) = ( 7T is the probability integral.
)1 /2 -Mp - u 5e-
f:
47T R 4
e-
x 2/ 2
dx
ll
2
12
sin e
202
II
The Magnetic Dipole in a Uniform Medium
a b,eq>
.1
.01
u
.2
.5
2
5
Fig. 11.4 (a) Transient responses of quasistationary field; (b) behavior of function N.
It is obvious that Eqs. (II,S7) are valid when displacement currents are negligible with respect to conduction currents and the field is measured at times significantly exceeding 70' which is the time required for the signal to arrive at the observation point. The behavior of the functions bR , be' and e