237] INDUCTION OF CURRENTS. 473 



quantity due to the induction current is 



If the current be passed through an electrodynamometer, that is 

 an instrument containing a fixed and a movable coil, the mechani- 

 cal action between them is proportional to the square of the 

 current and the momentum imparted to the movable coil is pro- 

 portional to the time integral of the square of the current. The 

 effect due to the whole induced current is 



= </, - /o 2 f 



Jo 



These two integrals have the same values that would be obtained 

 from a steady current of strength ( J /i)/2 passing for a time 2r. 



(2) Two CIRCUITS. In the case of two circuits which are 

 closed at the same instant, or which have their electromotive 

 forces or resistances suddenly changed simultaneously, we have 

 during the subsequent period the differential equations 



(4) 



Again calling the final steady currents 1^ =E l /R 1 , / 2 (1) 



we have for the induced currents 1-^ = I 1^, I 2 (i) = / a / 2 (1) , 





These equations are typical of all those in this chapter, and are 

 readily integrated by the assumption 



/!<*> = Ae xt , IJb = BeV, 



where A t B and X are constants to be determined. Inserting 

 these values in the differential equations (5), the factor eP appear- 

 ing in every term may be omitted, giving us the simultaneous 

 equations 



(L l \ + R l )A+M\B = 0, 



M\A + L\ + RB = 0. 



