POTENTIAL OF MAGNETIC SHELLS. 345 



The potential of the uniformly magnetised cylinder is then 



\ 



<)P ~br 



V = I = \2irr = \2Trx. 



ox ox 



Consequently the force in the interior is constant and equal to - 2?rl ; 

 its direction is opposite that of the magnetisation. 

 The induction is also constant and has the value 



F! = 477! - 2?rl = 27rl. 



360. POTENTIAL OF MAGNETIC SHELLS. We have seen that the 

 potential of a uniform magnetic shell is equal to the product of its 

 magnetic strength < by its apparent surface o> at the point in question. 

 If the shell is not uniform the expression for the potential is 



By a method like that of the preceding, the calculation of the 

 potential may be reduced to the potential of a magnetic layer, so as 

 to avoid the determination of solid angles. 



The potential at a point M, of an element of the shell d, is 

 equal to that of two magnetic layers o*/S, equal and of opposite 

 signs, the perpendicular distance of which satisfies the condition 

 vdn = <. 



Let us denote by n the distance of the element from a fixed 

 point on the perpendicular, on the same side as the negative face, 

 the potential of the layer <nS may be regarded as the product of 

 ds by a function of this distance n, so that the potential of the 

 element of the shell will be 



d$f(n) - d$f(n - dn) = d$dn . 



on 



As the distance dn of the two surfaces may be supposed to be 

 constant, the potential of the whole shell is 



^)n ^n 



J J 



