482 EXPLORATION GEOPHYSICS 



in the negative direction. To arrive at the potential distribution, assume 

 an image source xM on the other side of the plane.* The potential Vx for 

 the region ^ < may be written 



' [r2+ (2 + ^)2] %^[r2+ (^-^)2]y. ^^^>> 



where r^ = x^ + y^, z is negative and A is positive. 



The potential V2 for the region ^r > is a function of the distance from 

 the source only ; that is, 



where 2-5" is a constant to be determined from the boundary conditions. 

 The boundary conditions are 



Fi = F2 1 



j, 9Fi^ l9Z2f for^r = 



pi ?)2 92 9-2 J 



If the boundary conditions are combined with Equations 10 and 10a, the 

 following relations are obtained : 



xS + xM = 2S (11) 



and 



1 / i^-C^^-^) xM{z-A) \ 



Pit [r'+{z + Ayy^ W+{B-Ayy'],=^ 



= 1 I 2S{z + A) \ 



P2I [r''+{z + Ay]^U=o 



or 



i^^ xMA 2SA xS xM 2S ,,^. 

 = or — — (Iz) 



pi pi P2 pi pi P2 



For Equations 11 and 12 to be true simultaneously, it is necessary that 



,5-= (-^) ^S and xM = f^^) i^" 



\P1T'P2/ \p2T"pl/ 



Hence, the potential may be expressed by the two equations 



l-S* / P2 ~ pi ^ xS_ 



[r2+(^ + ^)2]% + Vp2+pi/[^'+(<^-^)'] 



[r2+ (.2 + ^)2]% 



The vertical cross sections of the equipotential surfaces close to i^" are 

 plotted in Figure 291 and their behavior is typical of what happens at the 

 boundary of two conductors of different resistivities. In plotting these 

 curves, values of pi = 5, p2= I and i5" = 60 were assumed. 



* Note that in this case it is not assumed that iM and iS are equal in magnitude. 



"■ [r2+ (^ + ^)2]% + Vp2+pi/^2+ ri.->^^2^% ^^^ 



