Law of Electromagnetic Induction, 507 



does not vanish we Lave, dividing by 0, 



R 



showing that the current is increased by ^— R, or that 



dt I rfN 

 there is an additional E.M.F. in the circuit equal to -,— 



± clt ' 



If E = 0, we can only say that if a current is produced 

 when the circuit is moved (and experiment can be appealed 

 to in support of this), its value, in accordance with the con- 

 servation of energy, must be such that 



d'N 

 = C 2 R-r-0^7, 

 at 



» G =-f/ R - 



The following elementary method of arriving at the law 

 is based on the existence of electrons in matter. It is so 

 simple that one can hardly believe that it has not been 

 already noticed, though I have failed to find any specific 

 mention of it. 



If £ is a charge of electricity moving with velocity v in a 

 magnetic field, the force on it is Bev sin perpendicular to 

 the plane By in the direction of translation of a right- 

 handed screw turning from v to B the magnetic induction, 

 being the angle between B and v. Suppose now, instead 

 of a charge e, we have an element PQ of a linear conductor 

 moving in the same way. There will be forces on its posi- 

 tive atoms and free electrons, displacements will be produced, 

 and a current will flow round a closed circuit of which PQ 

 is an element. The force per unit charge will be Bv sin 

 and this resolved along PQ is Bv sin 6 cos c/>, <*> being the 

 angle between PQ and the normal to the plane Br. The 

 E.M.F. round a circuit of which PQ is an element ds is, 

 consequently, J Bv sin 6 cos <f>ds, taken round the circuit. 



Let PQ move into the position P ; Q' in the time St. Then 

 the number of lines of induction cut by PQ in this time is 

 Bv cos ^ sin -^ ds 67, where ^ is the angle made by B with the 

 normal to the plane PQt 1 , and ty the angle that PQ makes 

 with v. But it is an easy problem in spherical trigonometry 

 to show that sin -v/rcos%=sin 6 cos <f>. 



Hence f Bv sin cos</> ds is the number of lines of induction 



2 M 2 



