12 TREATISE ON ALTERNATING CURRENTS. 



E.M.F. of Mutual Induction. If two closed circuits 

 A and B, carry respectively currents ii and H the number of lines 

 of force linking A due to mutual induction is Mi% (see 7, Chap. 

 I.), and the number linking B due to mutual induction is Mi L . If 

 the two currents vary there will be E.M.F.s due to mutual 

 induction in A and B respectively equal to 



,, 



rf(Jft) 



d 



or if the coefficient of mutual induction, M, is taken to be constant, 

 these become respectively 



-"**-** ..... (4) 



which mean that the induced E.M.F. of mutual induction in either 

 coil is numerically equal to the rate at which that coil cuts the 

 lines of magnetic force due to the current flowing in the other. 



ENERGY OF A MAGNETIC FIELD DUE TO ELECTRIC CURRENTS. 



14. CASE I. Field due to a single electric circuit, 



If a conducting wire carries an electric current, a magnetic 

 field is produced. Suppose that at that time t after the circuit is 

 made the value of the electric current is i and that the coefficient 

 of self-induction of the circuit is L. 



The magnitude of the E.M.F. which opposes the growth of the 

 current is 



tin by 13, equation (3) 



and the rate at which work is being done which is the product of the 

 corresponding instantaneous values of the current and E.M.F. is 



. T di 

 i.L 



dt 



If / is the maximum value of the current (when the steady 

 state is attained) the total work done against the counter E.M.F. 

 is the sum of the products 



..jdi*. 



iL-j-bt 



at 



where $t is a small element of time corresponding to the value i 



