78 Mr maxwell, ON FARADAY'S LINES OF FORCE. 



respect to the external medium, and the internal effect is altered in the opposite direction, 

 being greatest for a diamagnetic medium. 



This investigation explains the effect of introducing an iron core into an electro-magnet. 

 If the value of k for the core were to vanish altogether, the effect of the electro-magnet 

 would be three times that which it has without the core. As k has always a finite value, 

 the effect of the core is less than this. 



In the interior of the electro-magnet we have a uniform field of magnetic force, the 

 intensity of which may be increased by surrounding the coil with a shell of iron. If 

 k' = 0, and the shell infinitely thick, the effect on internal points would be tripled. 



The effect of the core is greater in the case of a cylindric magnet, and greatest of all when 

 the core is a ring of soft iron. 



X. Electro-tonic functions in spherical electro-magnet. 



Let us now find the electro-tonic functions due to this electro-magnet. 



They will be of the form 



Co = 0> ^0 = t>)X, 7o = - <^yi 



where to is some function of r. Where there are no electric currents, we must have Ojj ^a' <^3 

 each = 0, and this implies 



the solution of which is 



/ d(i)\ 



(3. + r.-j=0. 



d f dc 



dr 



c. + ^ 



Within the shell w cannot become infinite ; therefore w = C, is the solution, and outside 

 a must vanish at an infinite distance, so that 



0} = 



r. 



3 



is the solution outside. The magnetic quantity within the shell is found by last article to be 



-2 

 therefore within the sphere 



^6a 2k + k ' dr dy 



Wq= - 



I^n 1 



2a 3k + k' ■ 



Outside the sphere we must determine w so as to coincide at the surface with the internal 

 value. The external value is therefore 



•■2" 

 W = 



2a Sh + A' r^ ' 



where the shell containing the currents is made up of n coils of wire, conducting a current of 

 total quantity I^. 



Let another wire be coiled round the shell according to the same law, and let the total 

 number of coils be n ; then the total electro-tonic intensity EI^ round the second coil is 

 found by integrating 



EI^ = / a)ffl sin Qds, 



''a 



