INDUCTIVE CIRCUITS. 53 



the current. Also, the E.M.F. of self-induction lags a right 

 angle behind the current. For the proof of this, see 

 Chap. III., 21. 



Proposition 2. If a condenser of capacity C is placed in 

 an alternating-current circuit, an E.M.F. whose maximum value is 



-~> where /is the maximum value of the current flowing through 



the circuit, is generated in consequence, and this capacity E.M.F. 

 leads before the current by a right angle. The 

 proof of this is given in Chap. IV., 22. 



Proposition 3. If two circuits, A and B, have a mutual 

 induction, M, the maxima values of the consequent E.M.F.s in 

 the circuits A and B are respectively pMI% and pMI\, where I\ is 

 the maximum value of the current in the circuit A, and 7 2 that in 

 the circuit B\ and these induced E.M.F.s lag a right 

 angle behind the currents I 2 and Ij respectively. 



Proof. Let the current in the circuit B be given by 



i z = 7 2 sin pt 



then the E.M.F. ^ induced in the circuit A due to mutual induc- 

 tion, is given by 



ei = M-AIt sin pt) 



= pMI% cos pt 



sin (pt 5 



The maximum value of this is pMI^, and it lags by an angle -. 

 behind the current i%\ similarly the E.M.F. induced by mutual 

 induction in circuit B lags by an angle behind the current i\. 



a 



Thus the proposition is established. 



Having established these propositions, we are in a position to 

 proceed to the solution of some typical problems by the aid of the 

 vector calculus. 



CIRCUITS CONTAINING KESISTANCE AND SELF-INDUCTION ONLY. 



42. To determine the Current flowing in a 

 Circuit of Resistance r and Self-induction L 9 



