472 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. 



so that the induced current dies away in geometrical ratio as the 

 time increases in arithmetical progression. Since after an infinite 

 interval of time the total current has attained the steady value I lf 

 the value of the constant A is determined, and we have 



(3) 7 = /, + (7. - /O f * ' , / = (7. - /O -! ( . 



The induced current is always in such a direction as to oppose the 

 change in the total current. The effect of self-induction is accord- 

 ingly to make changes of strength less sudden. It is to be noticed 

 that the induced current varies in the same manner as the current 

 charging a condenser through a circuit without self-induction, as 

 treated in 207 (17). We shall here, as there, call the time in 

 which the current decreases in the ratio l/e the relaxation-time, 



r = L/R. 



Both in the case of the condenser and in the present case in- 

 creasing the capacity or the self-induction increases the relaxation- 

 time, but whereas in the former case increasing the resistance 

 increases the relaxation-time in the latter it produces the opposite 

 effect. 



In practical cases the relaxation-time is usually very short, so 

 that the induced current disappears almost entirely in a very 

 short time. Under these circumstances the total quantity of 

 electricity that has passed may be measured by a ballistic galvano- 

 meter. For as the force exerted by the current on a magnet is 



proportional to the strength of the current, the total quantity 



rt 

 passing, or the time integral I Idt, is proportional to the time 



Jo 



integral of the force on the magnet, or to the momentum imparted 

 to the magnet. If this momentum is all imparted before the 

 magnet has had time to move, it may be easily shown that it 

 may be measured by the first swing of the magnet. The quantity 

 passing in a time t is 



fl<B = I* {I, + (7 - /,) e -r] dt = IJ - T (7 - I,) (e J r - 1). 

 Jo Jo 



This formula was verified by Helmholtz* in 1851. The total 



* Helmholtz, " Ueber die Dauer und den Verlauf der durch Stromesschwan- 

 kungen inducirten elektrischen Strome," Pogg. Ann. Bd. 83, p. 505. Wiss. Abh. 

 Bd. 1, p. 429. 



