50 ADVANCED ELECTRICITY AND MAGNETISM. 



is the number of turns of wire in a coil and <!>' is the average flux 

 per turn. Differentiating* this equation, we have: 



*-* 



Now $' has here exactly the same meaning that $ has in 



d& 

 equation (5) of Art. 29. Therefore -7- is the electromotive 



d& 



force which is induced in each turn of wire, and Z -7 is the 



at 



electromotive force e which is induced in the entire coil by the 

 changing current. Therefore from equation (i) we have: 



When the current in a circuit is increasing, then is positive 



dt 



and the negative sign indicates that e is opposed to the current. 

 That is, an increasing current induces in a circuit an opposing 

 electromotive force which is equal to L times the rate of increase 

 of the current. A decreasing current induces in a circuit a 

 helping electromotive force which is equal to L times the rate of 

 decrease of the current. 

 33. Electromotive force required to make a current increase. 



Imagine a boat to move 

 without frictional opposition; 

 then the propelling force would 

 be used wholly to cause the 

 velocity to increase (to produce 

 acceleration), and we would 



Imagine a circuit having no 

 resistance; then an electro- 

 motive force acting on the 

 circuit would be used wholly 

 to cause the current to increase, 

 and we would have : 



have: 



T a - dl 



E = L Tt (I) 



* This simple case of differentiation occurs so frequently in physical arguments 

 that it is very important that the student understand it as a proposition in arith- 

 metic. If ^ is always - times as large as /, then it must change ^ times as 



d& L dl d& dl 



fast as /. That is we must have = or Z = L . 



