414-418] Magnetic Field of Force 365 



Thus Q,Q**jjfda;dydz+ ff-dS (348), 



where p, cr are given by 



dB dC 



0- = lA+mB + nC (350). 



Thus the potential of the magnet at any external point Q is the same as 

 if there were a distribution of magnetic charges throughout the interior, of 

 volume density p given by equation (349), together with a distribution over 

 the surface, of surface-density cr given by equation (350). 



Potential of a Magnetic Shell. 



417. A magnetised body which is so thin that its thickness at every point 

 may be treated as infinitesimal, is called a " magnetic shell." Throughout 

 the small thickness of a shell we shall suppose the magnetisation to remain 

 constant in magnitude and direction, so that to specify the magnetisation of 

 a shell we require to know the thickness of the shell and the intensity and 

 direction of the magnetisation at every point. 



Shells in which the magnetisation is in the direction of the normal to the 

 surface of the shell are spoken of as " normally-magnetised shells." These 

 form the only class of magnetic shells of any importance, so that we shall deal 

 only with normally-magnetised shells, and it will be unnecessary to repeat in 

 every case the statement that normal magnetisation is intended. 



If / is the intensity of magnetisation at any point inside a shell of this 

 kind, and if r is its thickness at this point, the product IT is spoken of as the 

 "strength" of the shell at this point. Any element dS of the shell will 

 behave as a magnetic particle of moment IrdS, so that the strength of a 

 shell is the magnetic moment per unit area, just as the intensity of magneti- 

 sation of a body is the magnetic moment per unit volume. 



Any element dS of a shell of strength <f> behaves like a magnetic particle of 

 strength < dS of which the axis is normal to dS. 



The magnetisation of a magnetic shell may often be conveniently pictured 

 as being due to the presence of layers of positive and negative poles on its 

 two faces. Clearly if <f> is the strength and r the thickness of a shell at 



any point, the surface density of these poles must be taken to be - . 



418. To obtain the potential of a shell at an external point, we regard 

 any element dS of the shell as a magnetic particle of moment $dS and axis 

 in the direction of the normal to the shell at this point, it being agreed that 

 this normal must be drawn in the direction of magnetisation of the shell. 



