712 



ELECTRICAL METHODS 



[Chap. 10 



An observer facing the plate at B sees the source P at an intensity 

 reduced by transmission through the plate. The amount transmitted is 

 the original intensity minus the amount lost through reflection, which in 

 turn is proportional to k times the original intensity. Therefore, the light 

 and by analogy the potential at B is 



7, = -^ (1 - k). 



(10-326) 



Continuity of the potential requires that in the boundary plane where 



n = n = rz , Ya 

 Hence, 



and Fs be equal, so that ''— (1 



4Tr 



k = 



Pa ~ Pi 

 Pj + Pi' 



+ *) = ^(i-*). 



(10-32c) 



so that in eq. (10-326) 



1 - A; = 



2pi 



Pi + Pi 



and 



\-\-k = 



2p. 



PJ + Pi 



(10-32C?) 



k''1 



Fig. 10-50. Four points and images on 

 boundary. 



If the point A is considered 



The factor k expresses the 

 "electrification" of a plane be- 

 tween two media of different 

 resistivities due to a point 

 source and its image. The 

 application of this principle is 

 illustrated by a calculation of 

 the potential distribution re- 

 sulting from two current elec- 

 trodes near the ground surface, 

 corresponding to a Wenner- 

 Gish-Rooney arrangement 

 with buried electrodes (Fig. 

 as current source (/) and B 



10-50) 



as the corresponding "sink" ( — /), the potential distribution at the 

 surface is obtained by considering the effects of both sources, A and B, 

 and their images. A' and B' . The intensities of the images are k'l and 

 —k'l^ respectively. Applying eq. (10-32c), k' is seen to be equal to 1, 

 since pa = <» (air). Hence, the images have the strength I and — /, respec- 

 tively. The potential difference between P and Q due to a source / 



— — ^r ], and that due to an image / at A' is 



at A is AV 



Att \A1 



^ I -r-T^ — -rr?^ )• Similar relations apply to the potential difference at 

 47r \A P A QJ 



