239, 240] INDUCTION OF CURRENTS. 489 



Proceeding as in 238, we write 

 (2) L + RI + 



and assume for the particular solution I Ae i(at , which inserted 

 in (2) gives 



(3) 



From this we get, by comparison with 238, for the impedance, 



and for the lag of the current behind the electromotive force, 



(5) 



so that the solution of (i) is 



j. E cos (a>t - a) 



In order to obtain the general solution we must add to this result 

 the solution of the equation with E = from the previous section. 

 An oscillation whose period is that of the force, as in our present 

 case, is called a forced oscillation or vibration, in contradistinction 

 to the case of the previous section, where, no force being applied, 

 the period is governed by the constants of the system, and the 

 oscillation is called a free oscillation. If there is damping, the 

 free oscillation soon dies away, leaving only the forced oscilla- 

 tion. We see by (6) that if there is no condenser, K oo , we 

 obtain the case of 238, and the current lags, while if on the 

 other hand L = 0, the lag is negative, or the current advances by 

 the phase-angle 



1 



a = tan" 1 



Ka>R' 



The reason of this is of course that in the differential equation 

 the inductance is multiplied by the derivative, and the capacity- 

 reciprocal by the integral of the current, which, when the 

 electromotive force is an exponential with imaginary exponent, 

 introduce the factor iw into the numerator or denominator 

 respectively, producing opposite effects on the argument of A. 



