92 



BELL SYSTEM TECHNICAL JOURNAL 



The magnetic iield intensity can then be cbtained from the curl law, 



which gives 



dH 



dt 



dE 

 dr' 



Hj 



^^AjJo' (fj) + B,Ko' {pj)~j, 



(5) 



where the prime indicates differentiation with respect to pj. Taking 

 the line integral of both sides of (5) around circular paths in con- 

 ductor j lying close to the inner and outer surfaces of the cylinder we 

 obtain 



AjJo' (yj) + BjKo' (yj) = 



AjJo' (xj) + BjKo' (xj) = 



yj 



2fj.jip 



(/l + /2 + 

 (/l + /2 + 





(6) 



in which 



Ij = current in the jth conductor, 



Xj = ocjaj, 



yj = ocjbj, 



aj = external radius of jth conductor, 



bj = internal radius oi jth conductor. 



The values of the electric field intensity at the inner and outer sur- 

 faces of the/'' conductor can be written, from (4) 



E/ = AjJoiyj) ^BjKoiyj), 

 E'/ = AjJoixj) + BjKoixj). 



Combining, in turn, each of these equations with relations (6) to 



eliminate Aj and Bj, we obtain 



Ej' = Z;. /i + Z;2 /2 + . . . + Z^jTj (7) 



Ej' = Zj'i 1 1 -\- Z'ji /2 + • • • + Zjjij, 



in which 



7" _ 9 -v, n -/o (^i) Ko' (yj) - Jo' (yj) Ko jxj) 



Ajk - ^njip L^. j^, (^.^ ^^, (^.^ _ J, (^.^ ^^, (^.) 



1 Jo (Xj) Ko (Xj) - Jo' (Xj) Ko {Xj 



yj Jo' (xj) Ko' (yj) - Jo' (yj) K 

 _ 2fxjip r Jo (xj) Ko' (yj) - Jo jyj) Ko (x,) -| 

 ^''' Xj LJo' (Xj) Ko' (yj) - Jo' (yj) Ko' {xj)J ' 

 „, _ .^ r 1 -^o (>'.■) K^ {yi) - Jo {yi) Ko {yj) 



Zik= ^M. ■ 





\ ^j -J o \p^j 



Ko' {yj) - Jo' {yj) Ko' {xj) 

 1 Jo{yj)Ko'{xj) - Jo'{xj)Ko{yj)- 



(8) 



7- 



1 Jo {yj) Kg {Xj) - Jo {Xj) Ko (yjj -i , 

 y, Jo' {xj) Ko' {yj) - Jo' {yj) Ko' {xj) J • '^ ^ -^ 

 2pLjip f- Jo {yj) Ko' {yj) - Jo' {yj) Ko {yj) -] 

 Xj U'o {Xj) K'o {yj) - Jo' {yj) Ko' {xj)J 



