ELEMENTARY ELECTRICITY AND MAGNETISM. 33 



increases, and the increase of this magnetic flux during a given 

 time is equal to the number of lines of flux swept over by AB 

 during that time ; or in other words, the rate of increase of the 

 magnetic flux through ABCD is equal to the number of lines 

 of force cut by AB per second. Therefore the induced electro- 

 motive force in a circuit is equal to the rate at which the magnetic 

 flux through the circuit is changing, electromotive force being ex- 

 pressed in abvolts. This result is here derived for the very spe- 

 cial case represented in Fig. 15, but it is true for any shape of 

 circuit in any magnetic field uniform or non-uniform, and it is true 

 whether the change of flux through the circuit be due to motion 

 or to changing magnetism or to both. 



Let the magnetic flux through a circuit be represented by <l>, 

 then its rate of change is represented by d3> jdt, and we have 



d$> 



E=^ (20*) 



in which E is the induced electromotive force in a circuit expressed 

 in abvolts. 



If the circuit has Z turns of wire then the electromotive force 

 induced in each turn is d^jdt, and the total induced electromo- 

 tive force expressed in abvolts is : 



Equations (2Oa) and (2o) are more properly written with the negative sign, 

 thus: 



/r d * A *> 7 <H> 



= -=- and = Z -- 



dt dt 



inasmuch as an increasing flux produces an induced electromotive force which is in 

 the direction in which a left handed screw would have to be turned to cause the screw 

 to travel in the direction of the magnetic field upon which the flux depends. 



In any given case an induced electromotive force may be 



thought of as being produced by either (a) the cutting of lines of 



force at a definite rate by a wire, or (ff) the change at a definite rate 



of the magnetic flux enclosed by a loop or coil of wire. These 



3 



