ELECTRICAL PROSPECTING 11 
There are two simple primary fields for which the theoretical problems 
have been solved; the field of a circular coil, which may be considered to be an 
oscillating magnetic dipole with its axis perpendicular to the plane of the coil 
and the field of a long horizontal wire. Most electromagnetic methods have 
made use of one of these. 
The basic equations that govern alternating currents are Maxwell’s 
equations. The field components must satisfy the wave equation ? 
0A, 07A, 
Wy = pps =0. 15 
Ne up aE (15) 

If the impressed forces are simple harmonic functions of time, Ai can be 
represented as 
A, = real part of Ae“ 
where A is in general complex. Then Eq. (15) becomes 
V2A + iwpA(o — wp) = 0. (16) 
This equation shows the relative importance of the conductivity and the rel- 
ative permittivity or dielectric constant in determining the current flow in 
the earth. If c>wp the conductivity will be the determining factor. If wp>o 
the dielectric constant-will be the determining factor. If the conductivity is 
10-*ohms-cm and the dielectric constant is 10, then the two are of equal im- 
portance when the frequency is 
10-4 
f= ——_ = 11.8 X 10’. 
2x X 10 X 8.85 X 10-4 
It will be shown later, however, that the high frequency waves are very highly 
damped by the surface layers, and are thus not important in oil prospecting. 
At the lower frequencies, the conductivity governs the distribution ot current 
in the earth. Methods of locating oil directly by means of its dielectric con- 
stant are thus not very hopeful. 
In order to show the importance of using a low frequency, a particularly 
simple situation has been assumed. The primary field is due to a vertical oscil- 
lating electric dipole on the surface of the earth. The earth is homogeneous 
and of conductivity o to a depth h, below which there is a layer of very high 
conductivity which we shall assume to be infinite. The problem to be solved 
is—at what frequency will the highly conducting layer have the biggest in- 
fluence on the field at the surface of the earth? 
The field may be conveniently represented by the Hertzian function I, 
which satisfies the wave equation 
2 The rationalized practical system of units is used. Magnetic intensity H is expressed in 
ampere turns per cm. Magnetic flux density B is expressed in webers per cm?(=10* Maxwell’s 
per cm?). Permeability 1 =B/H=4z2 X10-® for free space. Electric intensity E is expressed in 
volts per cm. Displacement Dis expressed in coulombs per cm?. Permittivity p= D/E=8.85 - 10-“ 
for free space. Conductivity o is expressed in mhos per cm. 
8 Riemann-Webers, Diff. Gleich. der Phys. (Seventh ed. 1925) p. 542. 
