Chap. 10] 



ELECTRICAL METHODS 



699 



proportion to its volume. The second derivative of the potential with 

 respect to x indicates that the current density is a minimum directly over 

 the sphere and that a maximum occurs at re = ±1.22^,, (where h. is the 

 depth to the center of the sphere). If the sphere were not present, the 

 equipotential lines would be parallel to one another (for line electrodes). 

 The potential of a line at a distance x' would be —Ex'. With the sphere, 

 the line is shifted to a position x with the potential given by eq. (10-28e), 

 so that 



a; = Aa; = c . 



R^ 



(10-29) 



Fig. 10-41. Current densities in conductors of various relative dimensions (e 

 axis ratio) as functions of resistivity ratio (adapted from Hummel). 



By differentiation of this expression with respect to x it can be shown that 

 the maximum displacement of the equipotential lines occurs at a dis- 

 tance X = 0.707/1. 



The effect of other bodies on the distribution of equipotential lines may 

 be calculated if they are of simple geometric shape.'" However, in most 

 cases, it is more convenient to determine the effects of such bodies by 

 model experiments as discussed below. 



2. Equipotential-line anomalies in stratified^ ground. Horizontally strati- 

 fied formations do not permit the application of equipotential-line methods. 

 Other potential methods must be used, the most important ones being the 

 resistivity and potential-drop-ratio methods discussed in sections v and vi. 



"See J. N. Hummel, Zeit. Geophys., 4(2) 67-75 (1928); Gerl. Beitr., 21(2/3), 

 204-214 (1929). 



