Mb maxwell, ON FARADAY'S LINES OF FORCE. 77 



VIII. Electro-magnetic spherical shell. 



Let us take as an example of the magnetic effects of electric currents, an electro-magnet 

 in the form of a thin spherical shell. Let its radius be a, and its thickness t, and let its 

 external effect be that of a magnet whose moment is la^. Both within and without the shell 

 the magnetic effect may be represented by a potential, but within the substance of the shell, 

 where there are electric currents, the magnetic effects cannot be represented by a potential. 

 Let p, p, be the external and internal potentials, 



3 



p' = /— ■ cos 9, p, = Ar cos 9, 



r 



dp' dpi 

 and since there is no permanent magnetism, — = —r- , when r = a, 



A = -2l. 

 If we draw any closed curve cutting the shell at the equator, and at some other 

 point for which 9 is known, then the total magnetic intensity round this curve will be 

 Sla cos 9, and as this is a measure of the total electric current which flows through it, 

 the quantity of the current at any point may be found by differentiation. The quantity 

 which flows through the element td9 is - 37a sin 9d9, so that the quantity of the current 

 referred to unit of area of section is 



-31- sin 9. 

 t 



If the shell be composed of a wire coiled round the sphere so that the number of coils 



to the inch varies as the sine of 9, then the external effect will be nearly the same as if 



the shell had been made of a uniform conducting substance, and the currents had been 



distributed according to the law we have just given. 



If a wire conducting a current of strength I^ be wound round a sphere of radius a 



2a 

 so that the distance between successive coils measured along the axis of a? is — , then 



n 



there will be n coils altogether, and the value of /j for the resulting electro-magnet will be 



/i = 

 The potentials, external and internal, will be 



ba 



, ^ n a' ^ ^ n r ^ 



^^ = -'2 S -J cos 9, Pi = - S-^a 7. - cos 9. 



or 6 a 



The interior of the shell is therefore a uniform magnetic field. . 



IX. Effect of the core of the electro-magnet. 



Now let us suppose a sphere of diamagnetic or paramagnetic matter introduced into the 



electro-magnetic coil. The result may be obtained as in the last case, and the potentials become 



n Sk' a^ ^ ^ n 3k r ^ 



p = I„- — J — cos 9, Pi=- 2/, - -, 77 - cos 9. 



'^ ^ 6 2k + k r" ^' ^6 2k+k a 



The external effect is greater or less than before, according as k' is greater or less 



thai) k, that is, according as the interior of the sphere is magnetic or diamagnetic with 



