Principles of Electromagnetic Methods in Surface Geophysics

Quasi-Stationary Field of the Electric Dipole in a Uniform Medium

Derivation of Equations for the Field

Next we consider the general case of a quasi-stationary field proceeding from Eq. (13.3). As in the case of the magnetic dipole, in order to simplify the solution of the boundary value problem, we introduce a vector potential of the electric field, based on the fact that div B = 0, in the following way:

=curlA

(13.14)

Of course, the function A is not uniquely defined by Eq. (13.14). Substituting the last expression in the first Maxwell's equation,

E=-?B?t

we have

E=-curl?A?torcurl(E+?A?t)=0

Whence,

=-?A?t-gradU

(13.15)

Here, U is the scalar potential. Replacement of E and B in the second Maxwell's equation by the functions A and U gives

curlA=-µ0??A?t-?µ0gradU

(13.16)

Assuming that the electromagnetic field is sinusoidal,

=Re[E*exp(-i?t)],B=Re[B*exp(-i?t)]

and

=Re[A*exp(-i?t)],U=Re[U*exp(-i?t)]

and making use of the vector identity

curlA=graddivA-?A

we obtain from Eq. (13.16) the following equation for the complex amplitudes:

divA*-?A*=i?µ0?A*-?µ0gradU*

(13.17)

As in the case of potentials describing the field of the magnetic dipole, the functions A* and U* are not uniquely defined from Eqs (13.14) and (13.15), and therefore, there is freedom to choose a pair of functions in such a way as to simplify Eq. (13.17) and eliminate from further consideration the complex amplitude of the scalar potential. Choosing a pair of functions U* and A* that satisfy the gauge condition,

A*=-?µ0U*,

(13.18)

we obtain the Helmholtz equation for the complex amplitude of the vector potential A*

2A*+k2A*=0,

(13.19)

where k2 = i?µ0? is the square of the wave number. In accord with Eqs (13.14), (13.15), and (13.18), the electromagnetic field has been expressed solely in terms of the vector potential:

*=curlA*E*=i?A*+1?µ0graddivA*

(13.20)

The physical intuition suggests that the quasi-stationary field has as the constant field the f-component of the magnetic field only. Then bearing in mind Eq. (13.14), it is reasonable to find an expression for all components of electromagnetic field with help of only one component of the vector potential Az. Besides we suppose that this component is a function of the coordinate R; that is, z*(k,R). Then in the spherical system of coordinates, Eq. (13.19) has the form

R2ddRR2dAz*dR+k2Az*=0,

(13.21)

since it is assumed that

Az??=?Az?f=0.

As was shown earlier, the solution of Eq. (13.21) which decreases as a function of R is

z*=Cexp(ikR)/R.

(13.22)

It is clear that this expression is the same as that for the complex amplitude of the vector potential for the magnetic dipole but the constants may be different. From Eq. (13.22), we have

A*=?Az*?z=Cexp(ikR)R2(ikR-1)cos?.

(13.23)

In the spherical system of coordinates, the vector potential is characterized by two components AR and A?:

R=Azcos?,A?=-Azsin?,

and in accordance with Eq. (13.14), the magnetic field can be written as

*=1R2sin?|1RR1?Rsin?1f??R?????fAz*cos?-Az*Rsin?0|.

Whence,

R*=B?*=0

and

f*=CR2(1-ikR)exp(ikR)sin?.

(13.24)

As the frequency goes to zero, Eq. (13.24) becomes equivalent to Eq. (13.12), so that we are able to find value for the constant C:

=µ0Idl4p.

(13.25)

Thus, we have the following expression for the magnetic field:

f*=µ0Idl4pR2(1-ikR)exp(ikR)sin?

(13.26)

Making use of Eq. (13.20), expressions for the complex amplitudes of the electric field are

R*=2p04pe0R3exp(ikR)(1-ikR)cos?

and

?*=p04pe0R3exp(ikR)(1-ikR-k2R2)sin?

(13.27)

Inasmuch as the electromagnetic field, described by Eqs (13.26) and (13.27), satisfies the Helmholtz equation for the complex amplitude as well as boundary conditions near the source and at great distances, one can say that a unique solution has been found. In other words, our assumptions about the behavior of the vector potential were correct. In accord with Eq. (13.26), the geometry of the magnetic field is remarkably simple: its vector lines form circles situated in horizontal planes centered on the z-axis. As in the case of constant field currents, which are the sources of the magnetic field, are located in the meridian planes and in accord with Eq. (13.27),

R*=Idl2pR3exp(ikR)(1-ikR)sin?,j?*=Idl4pR3exp(ikR)(1-ikR-k2R2)sin?.

(13.28)

In contrast to the case of the magnetic dipole, there are two sources for the electric field from the electric dipole, namely, the electric charges on the surface of the dipole electrodes and time variations of the magnetic field Bf. In order to study the field of the magnetic dipole, it is convenient to normalize it by the primary field caused by the current in a loop. Here it is natural to consider the ratio between the total field and the stationary (direct current) field; that is, let us represent Eqs (13.26) and (13.27) as

f*=µ0Idl4pR2bf*sin?ER*=2p04pe0R3eR*cos?E?*=p04pe0R3e?*sin?,

(13.29)

where

f*=eR*=exp(ikR)(1-ikR)e?*=exp(ikR)(1-ikR-k2R2)

(13.30)

and

R=Rd(1+i)=p(1+i)

It is proper to notice that expressions for the electric and magnetic fields caused by the electric dipole with the accuracy of a constant are the same as those for the magnetic and electric fields caused by the magnetic dipole. As usual, we distinguish three zones where the behavior of the sinusoidal waves in a conducting medium is different: the near, intermediate, and wave zones. Let us first consider in detail the range of small values for induction number p.

The Near Zone p < 1

Representing exp (ikR) as a series and substituting this into the first equation of the set (Eq. (13.30)), we obtain

R*=bf*=1+Sn=281-nn!(ikR)

(13.31)

Thus for the quadrature and in-phase components R*, we have

eR*~p2-23p3,Inef*~1-23p3,

(13.32)

and correspondingly,

ER*~2p04pe0R3cos?(p2-23p3)

or

ER*~Idl2pcos?[µ0?2R-13·21/2(µ0?)3/2?1/2]

(13.33)

and

nER*~2p04pe0R3cos?(1-23p3)

or

nER*~Idl2pcos?[?R3-13·21/2(µ0?)3/2?1/2]

(13.34)

Applying the same approach to the expression for the component ?*, we have

?*=1+Sn=28(n-1)2n!(ikR)n

(13.35)

Hence,

e?*~-p2+43p3,Ine?*~1+43p3

(13.36)

and therefore,

E?*~Idl4psin?[-µ0?2R+21/23(µ0?)3/2?1/2]

and

nE?*~Idl4psin?[?R3+21/23(µ0?)3/2?1/2]

(13.37)

By analogy,

Bf*~µ0Idl4psin?[?µ0?2-13·21/2(?µ0?)3/2R]

and

nBf*µ0Idl4psin?[1R2-13·21/2(?µ0?)3/2R]

(13.38)

Equations (13.33)-(13.38) allow one to understand better the range of small parameters (near zone) or the low-frequency part of the spectrum. The in-phase component of the electric field can be thought of as a sum of the galvanic and vortex ones. The first one is directly proportional to the resistivity of the medium and coincides with the stationary field...