: : HJCCTRICAL CAPAOITT OF A LONO NARROW CYLINDER, 



Lei Q be the potential enetfy of any arbitrary distribution of electricity 



on the cylinder. 



The charge m &' 



Let us now suppose this charge to distribute itself so as to pass into 

 the state of equilibrium ; then the potential will become uniform, and equal, 

 ay, to *. and <?. = W.# 



If AT is the capacity of the conductor, 



= and A'=' 



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

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

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



2Q 



r 



j -i 



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

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

 approximation to the truth. Thus, if we suppose, in the case of the cylinder, 

 A to b uniform, 



where < /" f 4/' + o > . For a long narrow cylinder, 



I 



To obtain a closer approximation, let us suppose the distribution to be of 

 any form, and to be expressed in the form of a series of harmonics. 



The potential due to any such distribution at a given point may be 

 expressed in terms of spherical harmonics of the second kind. See 

 'ical Harmonics, Chap. v. 



