792 



ELECTRICAL METHODS 



[Chap. 10 



ponent is likewise readily obtainable; and a phase shift beyond 180° does 

 not occur (disregarding absorption through the overburden) As a rule, 

 however, conditions are complicated by the finite conductivity of the 

 sheet. The fundamental equations for the potentials of the electrical and 

 electromagnetic fields of a cable in the presence of a conductive sheet have 

 been derived by Levi-Civita. The electrical forces are virtually inter- 

 cepted by the sheet and need not be considered. The potential U of the 

 electromagnetic field is given by 



U = -2/oe 



i"'--2) 



Yl0gA-l0gl)+^-^(l0ge^) 



(10-56) 



in which /o sin u)t is the current in the primary cable, ri and ri are distances 

 (as in Fig. 10-111), and q is an induction factor. The first part of this 

 expression indicates the direct effect of the cable at the point F if the sheet 

 is not present; the second is the effect of a perfectly conductive sheet, 

 equivalent to a field 180° out of phase caused by the cable image; the third 

 is a quadrature term indicating the phase shift resulting from finite 

 conductivity. 



The induction factor q is defined by 



q = 



4^ 



4tVs 



Pohm— cm 



10" 



(10-57) 



where / is the frequency in cycles per second and R is the resistance of 

 1 cm^ of the sheet. With s as the thickness of the sheet and p as the 

 resistivity, R — p/8. The horizontal and vertical field components are 

 then obtained by differentiation of the potential given in eq. (10-56). 

 There is no strike component; X = 0, Y = - dU/dz, and Z = dU/dy. 

 Hence, 



Y = 2/oe 



i(ut — 



Since rl ^ y' -^ (z - d)' and rl = y- -\- (z -\- df for z > 0, 



{z + d z-d\ ^^ ./ {z-Vdy-y' \ ] 



Y = 27o sm (at ( — 2 V~ J — 2/o cos at I —4 J 



/y y\ /2y{z-\-d)\ 



Z = 2/0 sin con -2 - ^2 ) + 27o cos oit ( —1 1 . 



\r2 ri/ \ qra / 



> (10-58a) 



' (10-586) 



