76 



Mr maxwell, ON FARADAY'S LINES OF FORCE. 



the coefficient of this quantity can be but little altered by changing the value of k' to k, the 

 value in space. The expression then becomes 



1 Kj" K, 



k 



Pa? sin 9.9, 



independent of the external medium *. 



VII. Permanent magnetism in a spherical shell. 



The case of a homogeneous shell of a diamagnetic or paramagnetic substance presents no 

 difficulty. The intensity within the shell is less than what it would have been if the shell 

 were away, whether the substance of the shell be diamagnetic or paramagnetic. When the 

 resistance of the shell is infinite, and when it vanishes, the intensity within the shell is zero. 



In the case of no resistance the entire effect of the shell on any point, internal or external, 

 may be represented by supposing a superficial stratum of magnetic matter spread over the 

 outer surface, the density being given by the equation 



p = 3l cos 0. 

 Suppose the shell now to be converted into a permanent magnet, so that the distribution of 

 imaginary magnetic matter is invariable, then the external potential due to the shell will be 



p = - / - cos 0, 



and the internal potential pi= — Ir cos 0. 



Now let us investigate the effect of filling up the shell with some substance of which 



the resistance is k, the resistance in the external medium being k'. The thickness of the 



magnetized shell may be neglected. Let the magnetic moment of the permanent magnetism 



be /a^ and that of the imaginary superficial distribution due to the medium k = Aa'. Then 



the potentials are 



a' 

 external p'= {I + A) -- cos 6, internal pi= (I + A) r cos 9. 



T 



The distribution of real magnetism is the same before and after the introduction of the 

 medium Ic, so that 



i,/4/ = l(/+^)+|(/H-^). 



or A^ — -, /. 



9.k + k 



The external effect of the magnetized shell is increased or diminished according as X: is 

 greater or less than k'. It is therefore increased by filling up the shell with diamagnetic 

 matter, and diminished by filling it with paramagnetic matter, such as iron. 



* Taking the more general case of magnetic induction re- 

 ferred to in Art. (28), we find, in the expression for the moment 

 of the magnetic forces, a constant term depending on T, besides 

 those terms which depend on sines and cosines of 8. The result 

 is, that in every complete revolution in the negative direction 

 round the axis of T, a certain positive amount of work is 

 gained ; but, since no inexhaustible source of work can exist 



in nature, we must admit that ^=0 in all substances, with 

 respect to magnetic induction. This argument does not hold 

 in the case of electric conduction, or in the case of a body 

 through which heat or electricity is passing, for such states are 

 maintained by the continual expenditure of work. See Prof. 

 Thomson, Phil. Mag. March, 1851, p. 186. 



