IILRCTRIC CIRCLir TJILURV 353 



Equatinp,- the two expressions and integrating we get 



CoV(t)= rs,,(T)dT- l'v(T)S,,{l-T)dT (276) 



whicli is the integral ecjuation of the problem. In order to get an 

 approxiniale solution without detailed coni])ulation we assume that 

 the cable is long. In this case the leading terms of (274) and (275) 

 are large compared with the terms following: Furthermore Si-^it) 

 builds up very slowly while 522(0 is a rapidly varying function. A 

 good approximation therefore results if we take V{t) outside the 

 integral sign in (270) and write 



CoVO)^ f's,,{T)dr-v{t) l's,,{T)dT 



whence 



, rSn{t)dt 

 n/) = 7t-^y-^ • (277) 



^1+^ / S22mt 



This approximation is quite good for long cables and shows the way 

 V(t) builds up quite truthfully. We see that V is initially zero, and 

 builds up ultimately to unity. For large values of /, it becomes 



/ Sn{t)dt 

 no =4^ . (278) 



/ S22{t)dt 



Jo 



This is the approximate formula also for the open circuit voltage, as 

 may be seen by setting Co = in (277). 



In electric circuit problems, it is often sufficient, as implied above, 

 to know qualitatively the behavior of an electric system without going 

 through the labor of detailed computation. For this purpose the 

 formulation of the problem as a Poisson Integral Equation is par- 

 ticularly well adapted. A simple example will be given, which can 

 be checked from the known solution. Suppose that we require the 

 voltage V at point x of an infinitely long transmission line {L,R,C) 

 in response to a unit e.m.f. impressed at .v = 0. This is, of course, 

 known from formula (211-a): we shall here be concerned, however, 

 with approximate solutions from the Poisson integral equation of 

 the problem. 



