AN ELECTRICAL EXAMPLE 871 



Let v = Ac"* be the solution of this equation. 



dv 

 --oA,- 



= 00} 



Hence a - - 7-77 



kR 



t 

 and v = A.e~%& 



But when t = 0, z> = V Q , and = A 



< 



Thus v = r e~ ** is the relation expressing i> the voltage in 

 terms of t the time. 



If the connecting wire is of resistance R and inductance L, 



Then Q = kv, C= -, and i; = RC + 



ill at 



Thus C=-^ 



at 



dC_ _ d*v 

 dt = "dt* 



, . dv ,- d^v 



Hence v = - RA; ^ kl^-r 



at at* 



d?v Rdv v 



and ^2- + Tj7 + rr = 



dt 2 L, dt kL 



Let v = Ae"' be the solution. 



Then -j- 



at 



R 1 



Hence a 2 + ^a + 77- = 



JL< /, 1. 



, R L R 2 1 , R 2 



ha + ~ ~kL 



Thus giving the required values of a. 



