384 Note 12: electric capacity 



The potential at the curved edge is approximately 



(8) 



This is the smallest value of the potential for any point of the cylinder. 

 The capacity of the cylinder cannot therefore be less than 



E 2.1 + b 



= 2 l gg-V' 



E 2l + b 

 nor greater than -j = . (10) 



Cavendish does not take into account the flat ends of the cylinder, but in 

 other respects these limits are the same as those between which he shows that 

 the capacity must lie. The approximation, however, is by no means a close 

 one, for when the cylinder is very narrow the upper limit is nearly double the 

 lower. Indeed Cavendish, in Arts. 479, 682, has recourse to experiment to 

 determine the best form of the logarithmic expression. 



We may obtain a much closer approximation by the following method, 

 which is applicable to many cases in which we cannot obtain a complete solution. 



Let W be the potential energy of any arbitrary distribution of electricity 

 on a body of any form W= &(*{,), (11) 



where e is the charge of any element of the body, and ifj the potential at that 

 element. 



The charge is E = S (e). (12) 



Let us now suppose the electricity to become moveable and to distribute 

 itself so as to be in equilibrium. The potential will then be uniform. Let its 

 value be ^ , and since the charge remains the same the potential energy of the 

 electrification in the state of equilibrium will be 



pp^ IjA . (13) 



If K is the capacity of the conductor, 



and K ^ 



Since W, the potential energy due to any arbitrary distribution of the 

 charge, may be greater, but cannot be less than W , the energy of the same 

 charge when in equilibrium, the capacity may be greater, but cannot be less, than 



1 2 (I, (e)] 2 



2 W r I. ' 



This inferior limit of the capacity is greater than that derived from the 

 maximum value of the potential, and, as we shall see, sometimes gives a very 

 close approximation to the true capacity. 



