Calculate the self potential energy of charge q distributed over the surface of a hollow sphere of radius R. What happens to this self energy when R=0, or when the charge is an ideal point charge?

Explanation

Let `rho` is voluem charge density , such that `rho = ( 3Q)/( 4 pi R^(3))`
The procedure to find potential energy is same . We bring charges from inifinite distance to build up the sphere .
Let at any instant, the charge of sphere is q and its radius is r. The potential at its surface is , `V= ( q)/( 4pi epsilon_(0)r)`
When additional charge dq is given to the sphere, its radius increases by dr. Work done in the process is given by
`dW = Vdq `
Here ` dq= rho xx ( 4pi r^(2) dr)`
`implies dW = ( q)/( d pi epsilon_(0)r)xx rho xx 4pi r^(2) dr` ,brgt `implies dW = ((rho xx ( 4)/( 3) pi r^(3)) xx( rho xx 4pi r^(2) dr))/( 4pi epsilon_(0)r)[ :' q= rho xx ( 4)/(3) pi r^(3)]`
`implies U = (3)/(5) xx ( Q^(2))/( 4pi epsilon_(0)R) ` `[ :' rho = ( 3Q)/( 4pi R^(3)]`