and Resistance of Straight Conductors. 391 



We might commence with the investigation of steady cur- 

 rents ; but it will be sufficient to regard them as alternating 

 (e ipt ). The results applicable to steady currents can always 

 be deduced by simply putting p equal to zero. 



Assuming, then, that the component currents u, v vanish, 

 as well as the components of magnetic force y, «, we have, in 

 Maxwell's notation, the equations 



dV dR ^ <TI /OQN 



^ = -^-l^ = -^-^ H > ■ ' ' < 28 > 



-=4™, /«*— s-; • • ■ • • (29) 



so that 

 and 



w 



= 1 d/3__ 1 d 2 B. 

 4-7T dx 4:7TfjL da? ' 



d?H 47T/L6 d^ . . 4:7rup TT 

 dx* p dz p 



(30) 



(31) 



We will now apply (31) to that conducting strip which 

 lies on the positive side of the origin. Since, by hypothesis, 



d*P 



—j—, the rate at which the potential varies, is independent of 



x, the solution for regular periodic motion may be written 



H= - ^ + Ae^-^ + Be-^-^, . . . (32) 

 p dz 



fi^=z-mAe m(x ~ a ^mBe- m ^- a \ . . . (33) 

 in which A and B are constants, so far arbitrary, and 



m 2 =^^ (34) 



P 



One relation between A and B is supplied by the condition 

 that the magnetic force j3 must vanish at the external surface, 

 where x=a + b. Hence 



Ae mb = Be~ mb (35) 



If C be the total current corresponding to width y, we have 



C f« 



+6 u,rf,= i-(^ +s -^)=-i- A =^-(A-B), 



y J a 4>ir^ a+b ^ a) 4tt^ 4tt//, 



by (33) ; so that 



A _ B= ^c 



my ' 



These equations determine A and B, 



