Chap. 10] ELECTRICAL METHODS 683 



determination of the potential distribution for the horizontal or vertical 

 plane is possible by calculating the potentials of each electrode separately 

 from Ohm's law for the semi-infinite space and by combining them for any 

 given point. The resistance of a hemispherical shell with the radius r, the 

 thickness dr, and the resistivity p \s dR = p dr/2irr , and the potential 

 drops from the inside of the shell to the outside by the amount —dV = IdR. 

 Therefore, from an integration of this expression and a similar one for the 

 second electrode, the potential at any point is 



The "equipotential surfaces" are defined by the expression 1/r — 1/r' = 

 constant and are surfaces of revolution of the fourth order about the 

 base AB. In the vicinity of either electrode 1/r' is negligible compared 

 with 1/r and the equipotential surfaces are nearly spherical. The poten- 

 tial gradient or electrical field strength is proportional to the inverse square 

 of the distance, or 



E = :^ = ^(|l| + |l|). (10-216) 



dr 27r \\r^ \ | rf 1/ 



The addition is vectorial. As applied to the surface, this becomes 



E = el(i + ^-^,) (10-210) 



(algebraic addition). In the center between two points of the distance 

 (base length) b, 



Ec = fdo = p^^, (10-21d) 



arid the current density i at any point below the center in the vertical 

 plane is 



id = to -cos <p, (10-21e) 



where <p is the angle subtended by a ray from the electrode to this point 

 with the horizontal plane. 



For given fractions of the current density at depth d in terms of the 

 density at the surface center, the corresponding depths can be determined. 

 C. H. Knaebel^® and W. Weaver" have calculated the portion of the 

 current penetrating below a depth d through a section at right angles to 

 the electrode basis (see Fig. 10-32). 



2«Mich. CoU. Min. Bull., NS, 5(2) (Jan., 1932). 

 " W. Weaver, A.I.M.E. Geophys. Pros., 70 (1929). 



