448 Induction of Currents in Linear Circuits [CH. xiv 



of the two plates will be ~, and this will now play the same part as the 

 electromotive force of a battery. The equation is accordingly 



g- -r,(Li) = Ri (437). 



The quantities Q and i are not independent, for i measures the rate of 

 flow of electricity to or from either plate, and therefore the rate of diminution 



of Q. We accordingly have i = -~^ , and on substituting this expression for 

 i, equation (437) becomes 



The solution is known to be 



Q = Ae-^ + Be-*** ........................ (438), 



where A, B are arbitrary constants, and X 1? X 2 are the roots of 



La; 2 - Rx + ^ = ........................... (439). 



o 



If the circuit is completed at time t = 0, the charge on each plate being 

 initially Q , we must have, at time 2 = 0, 



and these conditions determine the constants A and B. The equations 

 giving these quantities are 



If the roots of equation (439) are real, it is clear, since both their sum 

 and their product are positive, that they must themselves be positive quanti- 

 ties. Thus the value of Q given by equation (438) will gradually sink from 

 Qo to zero. The current at any instant is 



- d - = A^e-> 

 dt 



and this starts by being zero, rises to a maximum and then falls again to 

 zero. The current is always in the same direction, so that Q is always of the 

 same sign. 



It is, however, possible for equation (439) to have imaginary roots. This 

 will be the case if 



7W 4 



H ^y- 



is negative. Denoting R* ~^- , when negative, by 2 , the roots will be 



7? _i_ ' 



Jt "T IfC 



Al ' A2 = ~2T~' 



