476 



EXPLORATION GEOPHYSICS 



and a sink of current are assumed to exist at a finite distance from each 

 other. Consider the following problem. A source ^i and a sink ^2 are 

 located in a plane P delimiting a semi-finite, isotropic, homogeneous 

 conductor of resistivity p. ( See Figure 282. ) 



HORIZONTAL DISTANCE 



Fig. 283. — Variation of potential along line through current 

 electrodes S\ and ^o. (It should be noted that the potential curve 

 is not correctly described by Equation 8 in the immediate vicinity 

 of S\ and ^2. It is evident from physical considerations that the 

 potential must be finite at these points.) 



The potential at any point p located at distances ri and r2 from 6"! and 

 5*2 is obtained by adding the potentials due to Si and S2. The potential 



c ^ 



due to source ^i is V\ = — , and the potential due to sink 6^2 is F2 = — 



ri 



^2 



Hence, the total potential V at p is 



F=Fi+F2=— + — 

 ri ^2 



Since both Vi and V2 satisfy Equation 4, their sum satisfies Equation 4 

 at all points except the points where the source 5^1 and the sink ^'2 are 

 located. 



Also, 



i^.Oat.=0 

 P d^ 



If it is assumed that the total current / leaving from the source St is 

 equal to that entering the sink S2, evaluation of the constants yields 



Si = —S2 and 5"i = ■:^ as in case II. Hence, the potential may be ex- 

 pressed by the equation 



TT _ :^ r j^— ^P ^P 



ri ^2 2irri 2-Kr2 



2ir \ri To / 



(8) 



