ELECTROMAGNETIC INDUCTION. 499 



If both sides of this equation are multiplied by I, and we 

 integrate from /=0 to the time t when the shell takes up its final 

 position, we get 



ft ft / , \ t 



(13) 



The first member of this equation represents the excess of the 

 chemical energy, furnished by the battery during the time /, over the 

 energy which appears as heat in the circuit during the same time. 



The first term of the second member is the total work of the 

 electromagnetic actions ; this work depends on the law of the 

 motion. We may imagine, for instance, that the shell may have 

 been approached very slowly, so that the induction is very feeble, 

 and the principal current differs very little from its initial value I ; 

 in this case, if M is the flow of force corresponding to the final 

 position of the shell, the electromagnetic work will be equal 

 to 3>MI . 



This latter term represents the change in the potential energy of 

 the current ; it is zero if the two limiting values of the current are 

 the same that is, if the magnetic shell is in a state of rest in its 

 final position. 



521. If the power of the magnetic shell, while still at rest, 

 was variable, the equation would be quite analogous : 



(El - RI 2 X/= M I - dt+ ( L 



and would lead to the same conclusions. 



522. The most general case is that in which the three functions 

 M, < and L vary simultaneously that is to say, when the magnetic 

 shell changes its strength, its form, and relative position, and that 

 the circuit itself is deformed. Equation (12) gives therefore the 

 electromotive force of induction 



LI) d& ,dU dl d"L 



(14) e = --^ = M + $--- + L-- + I . 



at at at dt dt 



If the induced circuit contains no electromotive force inde- 

 pendent of induction, we need only make E = in the preceding 

 equations. 



K K 2 



