T , , , / 7 an an am 



I dxdydz (I ^ + m -=- + n -=- , 

 V dx dy dzj 



370 Permanent Magnetism [CH. xi 



Potential Energy of any Magnetised Body in a Magnetic Field of Force. 



425. Let / be the intensity of magnetisation and I, m, n the direction- 

 cosines of the direction of magnetisation at any point #, y, z of a magnetised 

 body, and let n be the potential, at this point, of an external field of magnetic 

 force. The element dxdydz of the magnetised body is a magnetic particle 

 of strength Idxdydz, of which the axis is in the direction I, m, n. Thus its 

 potential energy in the field of force is, by formula (353), 



an an a 



-=- 

 dy 



and by integration the potential of the whole magnet is 



an an an 





FORCE INSIDE A MAGNETISED BODY. 



426. So far the magnetic force has been defined and discussed only in 

 regions not occupied by magnetised matter: it is now necessary to consider 

 the more difficult question of the measurement of force at points inside a 

 magnetised body. 



At the outset we are confronted with a difficulty of the same kind as that 

 encountered in discussing the measurement of electric force inside a dielec- 

 tric, on the molecular hypothesis explained in 143. We found that the 

 molecules of a dielectric could be regarded as each possessing two equal 

 and opposite charges of electricity on two opposite faces. If we replace 

 " electricity " by " magnetism " the state is very similar to what we believe 

 to be the state of the ultimate magnetic particles. In the electric problem 

 a difficulty arose from the fact that the electric force inside matter varied 

 rapidly as we passed from one molecule to another, because the intensity of 

 the field set up by the charges on the molecules nearest to any point was 

 comparable with the whole field. A similar difficulty arises in the magnetic 

 problem, but will be handled in a way slightly different from that previously 

 adopted. There are two reasons for this difference of treatment in the first 

 place, we are not willing to identify the ultimate magnetic particles with 

 the molecules of the matter, and in the second place, we are not willing to 

 assume that the magnetism of an ultimate particle may be localised in the 

 form of charges on the two opposite faces. We shall follow a method which 

 rests on no assumptions as to the connection between molecular structure 

 and magnetic properties, beyond the well-established fact that on cutting 

 a magnet new magnetic poles appear on the surfaces created by cutting. 



