A charge `q` is distributed uniformly over the volume of a ball of radius `R`. Assuming the permittivity to be equal to unity, find :
(a) the electrostatic self-energy of the ball,
(b) the ratio of the energy `W_(1)` stored in the ball to the energy `W_(s)` pervadinting the surrounding space.

Explanation

Electric fields inside and outside the sphere with the help of Gauss theorem :
`E_(1) = (varphi r)/(4pi epsilon R^(2)) (r le R), E_(2) = (q)/(4pi epsilon_(0)) (1)/(r^(2)) (r gt R)`
Sought self energy of the ball
`U = W_(1) + W_(2)`
`= int_(0)^(R) (epsilon_(0) E_(1)^(2))/(2) 4pi r^(2) dr + int_(R)^(oo) (epsilon_(0) E_(2)^(2))/(2) 4pi r^(2)n dr = (q^(2))/(8pi epsilon_(0) R) ((1)/(5) + 1)`
Hence, `U = (3q^(2))/(4pi epsilon_(0) 5R)` and `(W_(1))/(W_(2)) = (1)/(5)`