364-367] Slowly varying Currents 321 



I. Discharge of a Condenser through a high Resistance. 



366. Let the two plates A, B of a condenser of capacity C be connected 

 by a conductor of high resistance R, and let the condenser be discharged by 

 leakage through this conductor. At any instant let the potentials of the two 

 plates be V A , V B , so that the charges on these plates will be C(V A V B ). 

 Let i be the current in the wire, measured in the direction from A to B. 



Then, by Ohm's Law, 



V A -V B = Ri, 



whence we find that the charges on plates A and B are respectively 4- CRi 

 and CRi. Since i units leave plate A per unit time, we must have 



a differential equation of which the solution is 





where i is the current at time t = 0. The condition that the strength of the 

 current shall only vary slowly is now seen a posteriori to be that OR shall be 

 large. 



At time t the charge on the plate A is CRi or 



t 

 CRi e~CR. 



This may be written as 



where Q Q is the charge at time t = 0. Thus both the charge and the current 

 are seen to fall off exponentially with the time, both having the same 

 modulus of decay CR. 



Later (516) we shall examine the same problem but without the limita- 

 tion that the current only varies slowly. 



II. Transmission of Signals along a Cable. 



367. It has already been mentioned that a cable acts as an electrostatic 

 condenser of considerable capacity. This fact retards the transmission of 

 signals, and in a cable of high-capacity, the rate of transmission may be so 

 slow that the analysis of the present chapter can be used without serious 

 error. 



Let x be a coordinate which measures distances along the cable, let V, i 

 be the potential at x and the current in the direction of a?- increasing, and let 

 j. 21 



