428 On the Dimensions of a Magnetic Pole. 



it is to deduce it from the expression given by Maxwell for 

 the magnetic force produced by an electric current. Maxwell. 

 in his 'Electricity and Magnetism,' 2nd edit. §§ 498,499, 

 makes the following statements about the magnetic force due 

 to a current: — 



" If any closed curve be drawn and the line-integral of mag- 

 netic force taken completely round it, then if the closed curve 

 be not linked with the circuit the line-integral is zero ; but if 

 it is linked with the circuit so that the current i flows through 

 the closed curve, the line-integral is Airi. The line-integral 

 47rl depends solely on the quantity of current and not on any 

 other thing whatever. It does not depend on the conductor 

 through which the current is passing. Again, the line-integral 

 Air I does not depend on the medium in which the closed curve 

 is drawn. It is the same whether the closed curve is drawn 

 entirely through air, or passes through a magnet or soft iron, 

 or any other substance whether paramagnetic or diamagnetic." 



Hence, if F be the magnetic force due to an infinitely long 

 straight current of strength i at a distance r from the current, 

 it may without anv ambiguity be written 



F =r 



Xow the work done when a magnetic pole of strength m is 

 carried round the current so as to remain at a constant dis- 

 tance r from it, is niF'/.7rr, or, by the above equation, Airmi. 

 Hence the product mi is of the dimension of energy ; but on 

 the electrostatic system i is of the dimension M*L*T -2 : hence 

 m must be of dimension MJ L*, the value given by Maxwell. 



The dimensions of F the magnetic force are M*L*T~ 2 , 

 the magnetic induction m/r 2 is of dimension M*L~*; hence 

 the magnetic permeability /x, which is the ratio of magnetic 

 induction to magnetic force, is of dimensions L~ 2 T 2 . 



If II be the magnetic force due to a pole of strength m at a 

 distance r from it, the magnetic induction is /xR. Now the 

 magnetic induction integrated over any surface containing the 

 pole and no other magnetic matter =Airm. Hence, taking the 

 induction across the surface of a sphere whose centre is the 

 pole and radius r, we get 



\X . K . 47Tf 2 = ±TTlii ; 



.-.B=™ 



fit- 



Hence the force between two poles ///,//< is mm' fxr. Thus the 

 force between two magnetic poles depends on the medium in 



