POTENTIAL COEFFICIENTS FOR GROUND RETURN CIRCUITS 367 



It is assumed that n is everywhere unity in electromagnetic units. 



7 = a: + 1/3 is the desired propagation constant 

 The electric field parallel to the wire is 



— — iO^ ^12 — — — 



(2) 

 dx 



It has previously been sho\vn that^ 



Zi2 = M2 log p'7p' + MQ - iP)] (3) 



where 



Q - iP = (^^ _{_ ^^2 _ j;)^-^''' cos y'v dv, 



w' = IV \/q: and y' = y \/q;. 



To get the potential coefficient for a ground return circuit it is necessary- 

 to compute the scalar potential. 



V = io:k-'(^U.+ ^Uy + ^u) 



\dx dy dz I ^^^ 



As in previous work the propagation constant y is assigned the value ik 

 as a first approximation. This is an ideal value for y but the following 

 considerations make it an imperative choice: (l) to assume that the current 

 is propagated down the line with a velocity less than that of light makes the 

 integrals very hard to evaluate, (2) to assume that the attenuation is not 

 zero on an infinite line amounts to assuming an infinite source of energy and 

 makes the integrals diverge. 



It should not be inferred that the resulting formulas are necessarily poor 

 if the physical system does not closely approximate the ideal one in which 

 7 is ik. ik is employed as a convenient first approximation in evaluating the 

 correction terms in Zn and pn. Eventually, if there were but one wire, 

 one would compute y = \/(2 + Zii)(G + io/pn), wherein Zn and pn 

 have been evaluated with ik for y, and this would be a second approxima- 

 tion to y. Past experience with the second approximation so obtained has 

 justified the expectation that it would be a satisfactory final result. Since 



