474 



EXPLORATION GEOPHYSICS 



Case 11. — Consider next the case of a current source located in a plane 

 P which bounds a semi-infinite, isotropic, homogeneous conductor of resis- 

 tivity p. (Figure 281.) The boundary conditions in this case are V = 



1 r)V 



at r = CO and — - — — = at 2 = 0. The solution of this problem is 

 p d2 ^ 



v = ^ 



r 



Fig. 281. — Current source 5" in a plane xy 

 bounding a semi-infinite conductor. 



for this solution satisfies Laplace's equation. Also, V = when r = oo and 



p d^ 9 ?)s\r / p r^ 



The value of 6" in this case is not the same as that in Case I. Since the 

 total current / flows out of a hemisphere, 



/=-27rr2 



P dr 



ZttS 



p 



and 



Hence, 



^ 27r 



V = 



Jr. 



2irr 



(6) 



By using Equation 6, it is possible to calculate the resistivity of an 

 isotropic, homogeneous medium from the values of the potential observed 

 at the boundary surface and the total current / flowing from the source. 

 Suppose V is measured at a distance a and again at a distance 2a from 



