Currents in Concentric Cables. 6< 



where 



c fi /i . s, .^/a/^cos oit -fa> sin &>f 



v .. s s ...W' 2 cos (ot + (o sin &>£ 

 s = b - ( 1 — mvr) 2, cos i 6 . 



v — fx y _«, a' + i'T 



§ 5. If an alternating current 27rS cos &>£ be introduced 

 into the concentric conductor at A, the charge s f at any 

 point P =Si+s/ } where 



, f$v(l — umr) * .nfii 2 cos cot 4 m sin cat 



S = _ -1 1 L % coSld™ 2 , g . 4 , 



CO V — fl _„ CO^ + f^t 4 



, Sitfl — vmr) s .,,. W 2 cos &>£ + ft> sin a>£ 



*2 = - — 2 cos «0 - . g . 



CO [JL — V _ 00 &r + ' / * 



§6. Consider the charge s' induced in the concentric 

 conductor by the charge *i + * 2 in the central conductor. 

 From §2, 



( d 2 d\, s , , d , _ 



_#_ _ Id rf2_ _l'd 



dd 2Sl ~~fidt Si > d6 2Sz ~vdt S2 ' J 



i l—viurdsi 1 — mvrdso ,ds' 



hence — *- -=-* h __f+mr--=0. 



yu, rfi v eft at 



Integrating, ?nr's'= (1— rnfir)si (l — mvr)s 2 + some 



r' " 



expression © independent of t. 



© is periodic in d, satisfies (^ - ^)(v^ 2 - ^)© = 0, 



and has a mean value zero. Hence ©=0, and 



mr's'= (l — mar)s l (1— mvr)s . 



In the same way the charge s induced in the central 

 conductor by the distribution s^ + s.J in the concentric con- 

 ductor is given by the equations 



{w 2 -^ m+n) 'iit} { - h ' +s2 ' )+mr it s=o ^ 



d 2 ,_ld , d 2 Id , 



d&* l -~jidt 8l > dd 2 ^-vdt S " 



and mrs = - ( 1 — miv)$i + - (1 — mrfi)s^. 



F2 



