Find the electric potential energy of a uniformly charged sphere.

Explanation

Consider a uniformly charged sphere of radius `R` having a total charge `q_0`. The volume charge density is :
`rho=q_0/(4/3piR^3)=(3q_0)/(4piR^3)`
When the radius of the sphere is `r`, the charge contained in it is
`q=(4/3pir^3)rho=(q_0/R^3)r^3`
The potential at the surface is
`V=q/(4piepsilon_0r)=q_0/(4piepsilon_0R^3)r^2`
The charge needed to increase the radius from `r` to `r+dr` is
`dq=(4pir^2)drrho`
`=(3q_0)/R^3.r^2 dr`
`:.` The self energy of the sphere is
`U_s=int_0^RVdq`
`=int_0^R(q_0/(4piepsilon_0R^3)r^2)(3q_0)/R^3.r^2dr`
`=(3q_0^2)/(20piepsilon_0R)`