511-513] Single Circuit 445 



Let us use this equation first to find the effect of closing a circuit pre- 

 viously broken. Suppose that before the time = the circuit has been 

 open, but that at this instant it is suddenly closed with a key, so that the 

 current is free to flow under the action of the electromotive force E. 



The first step will be to determine the conditions immediately after the 

 circuit is closed. Since -^-(Li } ) is, by equation (433), a finite quantity, it 



follows that Li^ must increase or decrease continuously, so that immediately 

 after closing the circuit the value of Li^ must be zero. 



To find the way in which ^ increases, we have now to solve equation (433), 

 in which E, L arid R are all constants, subject to the initial condition that 

 ij = when t = 0. Writing the equation .in the form 



we see that the general solution is 



where C is a constant, and in order that i\ may vanish when = 0, we must 

 have C = E, so that the solution is 



.(434). 



The graph of ^ as a function of t is shewn in fig. 131. It will be seen 

 that the current rises gradually to its final 

 value E/R given by Ohm's Law, this rise 

 being rapid if L is small, but slow if L is 

 great. Thus we may say that the increase in 

 the current is retarded by its self-induction. 

 We can see why this should be. The energy 

 of the current ^ is ^Lif, and this is large when 



L is large. This energy represents work per- FlG 131 



formed by the electric forces: when the current 



is ij, the rate at which these forces perform work is Ei lt a quantity which 



does not depend on L. Thus when L is large, a great time is required for 



the electric forces to establish the great amount of energy Lif. 



A simple analogy may make the effect of this self-induction clearer. Let the flow of 

 the current be represented by the turning of a mill-wheel, the action of the electric forces 

 being represented by the falling of the water by which the mill-wheel is turned. A large 

 value of L means large energy for a finite current, and must therefore be represented by 

 supposing the mill-wheel to have a large moment of inertia. Clearly a wheel with a small 

 moment of inertia will increase its speed up to its maximum speed with great rapidity, 

 while for a wheel with a large moment of inertia the speed will only increase slowly. 



