ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS. 15 



This is au equation between the E.M.F.s acting in the circuit 

 at the instant at which the current is i, and if solved will give 

 the value of i in terms of c, t, and the constants L and r of the 

 circuit. Since c is usually capable of being expressed as a function 

 of t, the above equation gives the value of the current at every 

 instant of time. 



We dwell at length on this equation, because it is of funda- 

 mental importance in the theory of alternating currents, and 

 should be thoroughly understood. 



A solution is not possible until c is known in terms of t, and 

 this is left for consideration in a later chapter. As the equation 

 is of paramount importance, we proceed to consider it from a 

 different standpoint, and deduce it from the law of Conserva- 

 tion of Energy. 



The rate at which energy is being supplied to the circuit at the 

 instant at which the value of the current is i is the product 



ei, 



and this must equal the rate at which energy is being dissipated 

 in heating the circuit together with that 

 negative iield, that is by 14, must equal 



in heating the circuit together with that used in creating the 



Thus we arrive at the equation 



T*- tt/ V , , i 



iL ~ 7i 4- ri 2 = ei 

 at 



or, dividing by i ; 



It must be remembered that this equation is a relation between 

 the impressed potential difference, the E.M.F. of self-induction 

 and the E.M.F. necessary to drive the current against the resist- 

 ance of the circuit at that instant at ivhich the value of the 

 current is " i" 



CASE II. Electric Currents in Two Mutually In- 

 ductive Circuits. Let L lt L% be the coefficients of self- 

 induction of the two circuits, r\ t r% their resistances, and M their 

 coefficient of mutual induction, and let potential differences whose 



