ELECTRICAL METHODS 



481 



( 1 ) A source of current 6" is placed at a distance A below a plane xy 

 which delimits a semi-infinite medium of resistivity p. (Figure 290.) To 

 obtain the solution of La- 

 place's equation, it will be 

 convenient to introduce an 

 image source S' (equal to S) 

 at a distance A on the other 

 side of the xy plane. It will 

 also be convenient to assume, 

 for the time being, that the 

 medium of resistivity p ex- 

 tends beyond the xy plane. If 

 S' is set equal to S, the value 

 of the potential due to the two 

 sources may be written 



1 



Fig. 290. — Current source 5 located in a semi- 

 infinite medium of resistivity p at a distance A 

 below the xy plane. 



v = s 



+ 



1 



2]%f 



(9) 



[r2+ (.s-^)2]V. ' [^2+ {z-\- Ay] 



where r~ = x^ + y^. 



This value of the potential satisfies the boundary condition that the cur- 

 rent crossing the xy plane is zero, for 



(2 + A} 



( 1 '^\ = ^ r (^-A) 



+ 



[r^- + (s + A)''} 



Lt' 



Also, Laplace's equation V"^' = is satisfied throughout the region below 



the plane xy except at 6*. The value of V given by the right hand side of 



Equation 9 does not hold for the 

 region above the plane xy, because 

 it makes V infinite at S\ There- 

 fore, for the region investigated, 

 i.e., for the region below the plane 

 xy, Equation 9 is the required so- 

 lution. 



It is of interest to point out 

 that this same procedure may be 

 used to deduce the solution of the 

 problem of two electrodes, a source 

 and a sink, both buried at a dis- 

 tance A below a horizontal plane 

 and thus arrive at a generalized 



solution for the problem treated under Case III. 



(2) A plane xx separates two media having resistivities pi and po. 



(Figure 291.) 



(p = pi for £r < 0, p = p2 for 2 > 0) 



A small source of current i^" is placed at a distance A from the xy plane 



Fig. 291. — Small source of current \S placed at 

 a distance A from the xy plane. 



