Potential energy of a cylinder ***
A cylindrical volume of radius a is filled with charge of uniform density ρ. We want to know the potential energy per unit length of this cylinder of charge, that is, the work done per unit length in assembling it. Calculate this by building up the cylinder layer by layer, making use of the fact that the field outside a cylindrical distribution of charge is the same as if all the charge were located on the axis. You will find that the energy per unit length is infinite if the charges are brought in from infinity, so instead assume that they are initially distributed uniformly over a hollow cylinder with large radius R. Write your answer in terms of the charge per unit length of the cylinder, which is λ = ρπa2. (See Exercise 1.83 for a different method of solving this problem.)