ELECTRICAL METHODS MZ 



and 



Hence, 



dV_^S_ 

 dr r" 

 and 



V = - - +C 

 r 



The values of 5" and C must be determined from boundary conditions of the 

 problem. If it is assumed that the value of V at infinity is zero, C vanishes. 

 Also, the constant 6^ may be expressed in terms of the total current / which 

 flows out of the source. Consider a small sphere surrounding the source. 

 The current which flows through one sq. cm. of the surface of the sphere 

 in an outivard direction is 



4^ ^'^ 



Equation 5 expresses the value of the potential at any point in an infinite 

 isotropic homogeneous medium due to a small source of current. 



It will be noticed that the solution is of the form V = S/r. (The par- 

 ticular value given for the constant : viz., 5" == Ip/^v, is due to the geometric 

 form of the electrode.) The general conclusion which can be drawn from 

 the above analysis, therefore, is that the potential due to a small current 

 source or sink is S/r, where vS is chosen to satisfy the boundary conditions. 



Practically, this condition is approximately realized in that type of 

 electrical bore-hole exploration in which one current electrode is lowered 

 a considerable distance into the earth while the other current electrode is 

 fixed at a great distance from the first. 



