452 Induction of Currents in Linear Circuits [CH. xiv 



for all values of t, and for this to be satisfied the coefficients of e~ Kt and e~ k>t 

 must vanish separately. Thus we must have 



(8 - N\) B = MAX ........................ (447), 



(S-N\')B' = MA'\' ........................ (443), 



and if these relations are satisfied, and X, X' are the roots of equation (444), 

 then equation (442) will be satisfied identically. From equations (445), 

 (446), (447) and (448), we obtain 



B -5' A\ -A'\ r E, 



M " ~W ~ S - N\ ~ 8^N\' ~ MS (X- 1 - X'- ] ) ' 

 and the solution is found to be 



_ (S-N\')E t (S-N^E, 



* 1 ~ 1 -'- 1 + '->--> ( - 



ME, .,. ME, 



e~ xt -^ 



RS (X- 1 - X'- 1 ) R8 (X'- 1 - X- 1 ) 



rr 



We notice that the current in 1 rises to its steady value ^, the rise being 



similar in nature to that when only a single circuit is concerned ( 513). The 

 rise is quick if X and X 7 are large i.e. if the coefficients of induction are 

 small, and conversely. The current in 2 is initially zero, rises to a maximum 

 and then sinks again to zero. The changes in this current are quick or slow 

 according as those of current 1 are quick or slow. 



Sudden Breaking of Circuit. 



521. The breaking of a circuit may be represented mathematically by 

 supposing the resistance to become infinite. Thus if circuit 1 is broken, the 

 process occurring in the interval from t = to t = T, the value of R will 

 become infinite during this interval, while the value of ^ becomes zero. The 

 changes in ^ and i 2 are still determined by equations (440) arid (441), but we 

 can no longer treat .R as a constant, and we cannot assert that in the interval 

 from to T the value of Ri is always finite. 



It follows, however, from equation (441) that ^- (Mi^ Ni 2 ) remains finite 



throughout the short interval, so that we have, with the same notation as 

 before, 



Suppose for instance that before the circuit 1 was broken we had a steady 

 -p 



current ^ in circuit 1, and no current in circuit 2. We shall then have 



-that 



