Currents in Concentric Cables. 69 



Ac 

 [m {sin <*)tf s (/3) + cos (otf 4 (/3) } 



03 C 



Ac 

 + vr - — {sin atf z (y) + cos cotf^y) }, 



O) c 



+ ^?^{sin(^ + ^)/ 1 (/3)+cos(^ + 0)/ 2 ( / 8)} 



Bc 2 



+ vr- - {sin (at + i>)f x (y) + cos {at + 0)/ 2 (7) } . 



If a known current 2ttA cos <y£ be introduced tit A, and 

 the point B be kept at zero potential, the equation of the last 

 expression to zero for every value of t gives two equations to 

 determine B and 0, the amplitude and phase of the current 

 at the receiving end of the cable. These are 



A W 3 (/3) +vc/,( 7 )KBL^i{cos0/ 1 O9) ~ sin 0/*(£)} 

 + vc 2 {cos 0/1(7) -sin 0/3(7)}] =0; 



MfWifM +vc 2 / 4 ( 7 )} +B[>e 1 {sin <f>j\(/3) +cos 0/ 2 (/3)} 



+ vc 2 {sin 0/1(7) +cos 0/2(7)}] =0. 



Similarly, if the currents had been introduced into the 

 concentric conductor, the central one being insulated 

 throughout, we should have for the potential of the concentric 



conductor at B,- (5 + /), which is exactly the same expression 



c ' c 

 as for the central conductor substituting r' for r, — for — , and 



c c 



c l c °2 n n • n 1 c l c 2 i C 2 c/ •, • 



— tor-. Remembering that— = — rand— = —„ it is seen 

 c c ° c c' c. c 



that the equations for determining the phase and amplitude 

 of the current received at B when that end of the concentric 

 conductor is earthed are the same as in the case of the central 

 conductor, only interchanging Ci and c 2 . 



§8. So if an alternating current, 27rA cos (at + a), be 

 introduced into the central conductor at A, and a current 

 27rB cos (at + b) pass out of the concentric conductor at B, 

 the potentials of the conductors at A, B may be determined. 



The potential of the central conductor at A is 



— { (m + n) s + ms 1 } 

 mn 



where s s f are the charges on the two conductors. 



