390 Bedell and Crehore — Effects of Self-induction 



(•) °%- = - %■ 



v ' dt dx 



By Ohm's law the current in an element is equal to the 

 total B. M. E. (the sum of the impressed and that of self- 

 induction) divided by the resistance ; and, if R is the resis- 

 tance per unit length and we assume the back E. M. F. of self- 

 induction per unit length to be equal to the rate of change of 

 the current multiplied by a constant L, we may write 



fde T T di 7 \ 



\dx dx - L J t dx ) 



(2) i = - -^ 



Rdx 



In some cases this assumption may approximately represent 

 the true effect of self-induction, and the results obtained from 

 this particular assumption may show the nature of the effect 

 of self-induction even in cases where the assumption is not 

 justifiable. 



The differential equation for potential is obtained by elimi- 

 nating i from (1) and (2,) and is 



d*e T rJPe „ _ de 

 — 9 + LC-=^ - RC-- = 0. 

 dx~ dt dt 



The current equation, obtained in the same way, is similar 

 when i is written instead of e. 



The general solution of these equations is 



h,k 



hke. Gk +L 





A, c 



R 

 — (t-Gkx) 



CA 2 + L 



where e is the Naperian base, and A and k are constants to be 

 determined. 



If the impressed E. M. F. is harmonic, and at the origin 

 e=E sin cot, where to denotes angular velocity, the solution 

 for the potential at any point of the conductor at any time 

 becomes 



(3) e = 'Ee ±px sin (aot£z ax). 



The solution for the current at any time across any section of 

 the conductor is 



. -E^Cco ±px . / . p \ 



(4) ^ = == — £ sin ( cot ± ocx + tau — ). 



Vim- V a J 



