514-516] Alternating Current 447 



The electromotive force produced by the change in the number of tubes 

 of the external field is 



dN d . ~ n . 



-j- = -T.(Gcos2)t) = pCsinpt. 

 ctt ctt 



Thus, if self-induction were neglected, the current, as given by Ohm's 



Law, would be 



pC . 

 *jl sin pt, 



and this of course would agree with that which would be given by equation 

 (436) if L were zero. 



The modifications produced by the existence of self-induction are repre- 

 sented by the presence of L in expression (436), and are two in number. In 

 the first place the phase of the current lags behind that of the impressed 



electromotive force by tan" 1 - , and in the second place the apparent resist- 

 ance is increased from R to V R* -f- L*p* . 



515. The conditions assumed in this problem are sufficiently close to 

 those which occur in the working of a dynamo to illustrate this working. A 

 coil which forms part of a complete circuit is caused to rotate rapidly in a 

 magnetic field in such a way as to cut a varying number of lines of induction. 



*D 



The quantity ^ may be supposed to represent the number of alterna- 

 tions per second i.e. the number of revolutions of the engine by which the 

 dynamo is driven. We see that the current sent through the circuit will be 

 an " alternating " current of frequency equal to that of the engine. In the 

 example given, the rate at which heat is generated is (p cos ^) 2 R, and the 

 average rate, averaged over a large number of alternations, is ^p*R or 



This, then, would be the rate at which the engine driving the dynamo 

 would have to perform work. 



Discharge of a Condenser. 



516. A further example of the effect of induction in a single circuit which 

 is of extreme interest is supplied by the phenomenon of the discharge of a 

 condenser. 



Let us suppose that the charges on the two plates at any instant are Q 

 and Q, the plates being connected by a wire of resistance R and of self- 

 induction L. If G is the capacity of the condenser, the difference of potential 



