430 Discharge of Electricity in an Imperfect Insulator. 

 The instantaneous capacity 0, which is equal to ™, is given 



by 1 



C=r~^ r x ( 9 ) 



4:7r(/c 1 a l + k 2 a 2 -\-. . .) 



But dissipation at once sets in, and if the electromotive force 

 E be continued uniform, a steady state will ultimately be 

 reached in which the dissipation in each layer is equal to the 

 number of fresh tubes reaching that layer. The number of 

 tubes entering being the same throughout, the dissipation p 

 is also the same throughout. 

 We have then 



?^ = ^=&° (10) 



and substituting in (8) 



E = (r 1 a 1 + r 2 a 2 )p. 



Hence, if 11 = ^(2! + . . . r? 



P =l ; .. . ..... (ii) 



In this state we have the induction given by 



7l 4w£ 1 "~ 4,irk/~ 4tt^R ' ' ' A } 



If we now suddenly connect the extreme strata by means of 

 a conductor of small resistance, E will be suddenly changed 

 from the value E to zero, and Q / tubes will pass out from 

 each layer of the dielectric into the wire. If then X 1 be the 

 new value of the intensity, 



H 477-^ 4wkj, ' 

 whence 



X 1 1 = X 1 -4tt^ 1 Q / (13) 



Since, then, the difference of potential is zero, 

 ajXi 1 — a 2 X 2 1 + . . . =0 ; 

 substituting from (13) we get 



a 1 K 1 + a 2 ~K 2 + = 47r(a 1 & 1 + a 2 k 2 -f . . .)Q' ; 



or 



q -iisesfe)= cE =Q' fr ° m w and <»> • < u > 



Hence the instantaneous discharge is equal to the instantaneous 

 charge. 



By (10) and (11) we may put (13) in the form 



X 1 1 =pr 1 — 4:7rk 1 Q, 



= ^-47r/qQ (15) 



