367-369] Slowly varying Currents 323 



integral of the original equation for which V is a function of u only. This 

 integral is easily obtained, for from equation (304), 



d (. dV\ 



-?- 1 -j- ) = - 

 du\ & duj 



whence d ^ =C 



du 



in which is a constant of integration. 



Integrating this, we find that the solution for V is 



in which the lower limit to the integral is a second constant of integration. 

 Introducing a new variable y such that y 2 = ^KRu z , and changing the con- 

 stants of integration, we may write the solution in the form 



r 



* 



~y dy .................. (305). 



369. We must remember that this is not the general solution of equa- 

 tion (303), but is simply one particular solution. Thus the solution cannot 

 be adjusted to satisfy any initial and boundary conditions we please, but will 

 represent only the solution corresponding to one definite set of initial and 

 boundary conditions. We now proceed to examine what these conditions are. 



At time t = 0, the value of xj\/t is infinite except at the point x 0. 

 Thus except at this point, we have V=V when t = 0. At this point the 

 value of xj*Jt is indeterminate at the actual instant t = 0, but immediately 

 after this instant assumes the value zero, which it retains through all time. 

 Thus at x = 0, the potential has the constant value 



or, say, F= V lt where C' = 



At x = oo , the value of V is V= V through all time. 



Thus equation (305) expresses the solution for a line of infinite length 

 which is initially at potential V=V , and of which the end x= oo remains at 

 this potential all the time, while the end x = is raised to potential K by 

 being suddenly connected to a battery-terminal at the instant t = 0. 



The current at any instant is given by 



i = -^ ~ , from equation (301), 



c' i /IKR 



~~ ~R 2 V ~T~ 



4T ................................. (306). 



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