54 TREATISE ON ALTERNATING CURRENTS. 



'when an Alternating 1 P.D. 'whose Maximum 

 Value is e and Frequency is n, is applied be- 

 tween its Terminals. 



and / be a vector representing the maximum value of the current 

 flowing through the circuit. 



Then by proposition 1 and 38 the vector representing the 

 maximum value of the E.M.F. of self-induction in the circuit in 

 which the current / is flowing is 



The applied potential difference has to drive the current against 

 the resistance of the circuit and also to balance the E.M.F. of self- 

 induction. The vector e has therefore to be capable of resolution 

 into two components one equal to rl, and the other equal to 

 -f IcpLI. We thus arrive at the vector equation 



rr+kpLF=c ...... (1) 



which is essentially a vector equation of E.M.F.s. 



If we reduce the lengths of each of these vectors in the ratio 

 \/2 : 1, we may regard the vectors as representing root mean 

 square values ; but this would not produce any change in equa- 

 tion (1), which only depends upon ratios of magnitudes of vectors, 

 and therefore holds good if each vector in the equation is multi- 

 plied by the same constant. 



Let OP (Fig. 17) represent the vector rl. 



Then the vector OQ' = - kpLI represents the E.M.F. of self- 

 induction ; and the vector PQ = 4- kpLI, which is equal in length 

 to Off, parallel to it, but of opposite sense, is the vector repre- 

 senting the E.M.F. necessary to overcome self-induction ; therefore 

 the vector 



OQ = or + PQ 



is the vector representing the applied potential difference e. 



Further, the vector OP, which is in phase with the current, 

 lags behind the vector representing e by an angle, 0, where 



tan0=-^ 



T 



Also (see 35) the magnitude of e is given by 

 e = 



