508 Neumann's Law of Electromagnetic Induction. 



cut per unit time by the linear conductor. It follows, there- 



fore, that the E.M.F. round the circuit is — -^- , and thus 



Neumann's law follows immediately from that of Ampere. 



The magnetic field has been supposed at rest, but the 



E.M.F. depends only on the relative motion of the circuit 



and the field. Hence it is given by \ Bv sin 6 cos <f> ds or 



\ Bv cos ^ sin y\r ds, where v is the velocity of the element 



ds of the circuit relative to the line of induction through it. 



^N 

 and this is equal to j- , for, the vector B being solenoidal, 



there is no way of increasing or diminishing N except by 

 lines of induction cutting the bounding circuit. 

 The above may be put shortly as follows : — 

 Let a surface bounded by the circuit move in a magnetic 

 field whose lines of induction are also in motion. The motion 

 of the circuit and the surface of which it is the boundary 

 is the resultant of two motions, the first such that the velocity 

 u of any point of the circuit or any point of the surface is 

 equal to that of the line of induction through it, and the 

 second such that the velocity v of any point of the circuit or 

 any point of the surface is equal to its velocity relative to 

 the line of induction passing through it. 



We may thus consider three near positions of the surface 

 Si, S 2 , S3. Applying Gauss's theorem to the space traced 

 out between S x and S 2 , we have 



JB^S - j B n dS + St j [>, B]ds = 8t \divBu n d$ + St P 



DT /S ' 



ctsb 



But, since B is solenoidal, l^-?dS = 0; also div B = 0, and 



I B n dS = I B^S. Hence J [u, B]ds — Q, or there is no 



s 2 J s L J 



E.M.F. round the circuit. Applying the same theorem to 

 the space traced out between S 2 and S 3 and remembering 

 that in the motion the lines of induction are at rest, we have 



f B a dS - fB n ^S + St f [r, B] ds=St f div B v n d8 + St fe rfS. 



« b, J s 2 J J J Ot 



But s— ^ =0, div B = 0, and, consequently, 



which is Neumann's law if §[v y B]ds fce identified with the 

 E M.F. round the circuit 



