370 ELEMENTARY LESSONS ON [CHAP. x. 



4O5. Helmholtz's Equations. Helmholtz, who 

 investigated mathematically the effect of self-induction 

 upon the strength of a current, deduced the following 

 important equations to express the relation between the 

 self-induction of a circuit and the time required to 

 establish the current at full strength : 



The current of self-induction at any moment will be 

 proportional to the rate at which the current is increasing 

 in strength. Let r represent a very short interval of 

 time, and let the current increase during that short 

 interval from C to C -f c. The actual increase during 

 the interval is c, and the rate of increase in strength is 



Hence, if the coefficient of self-induction be L, the 



electromotive-force of self-induction will be - L-, and, if 



T' 



the whole resistance of the circuit be R, the strength of 

 the opposing extra-current will be - . - during the short 



interval r ; and hence the actual strength of current flow- 

 ing in the circuit during that short interval instead of 

 being (as by Ohm's Law it would be if the current were 

 steady) C = E -^ R, will be 



E _ L 

 " R R ' T ' 



To find out the strength at which the current will have 

 arrived after a time / made up of a number of such small 

 intervals added together requires an application of the 

 integral calculus, which at once gives the following 

 result : 



(where is the base of the natural logarithms). 



Put into words, this expression amounts to saying that 

 after a lapse of / seconds the self-induction in a circuit 

 on making contact has the effect of diminishing the 

 strength of the current by a qitantity, the logarithm of 

 whose reciprocal is inversely proportional to the coefficient 



