A charge `Q` is uniformly distributed inside a non- conducting sphere of radius `R`. Find the electric potential energy stored in the system.

Explanation

The electric potential energy is equal to work done in bringing charges from infinity and place them on the shell .
Let at some instant , a charge `q` be placed on the shell. The potential at the surface is
`V = (1)/(4 pi in_(0)) .(q)/(R )`
The work done in bringing a charge `dq` from infinity to this shell.
`dW = (dq)V = (qdq)/(4 pi in_(0) R)`
Total work done
`W = int_(0)^(Q) (qdq)/(4pi in_(0) R) = (1)/(4 pi in_(0)R) |(q^(2))/(2)|_(0)^(Q)`
`= U = (Q^(2))/(8 pi in_(0) R)`
Alternative method : The energy stored per unit in electric field i.e. energy density
`u = (1)/(2) in_(0) E_(2)`
Inside shell `(r lt R) , E = 0`
Outside shell `(r gt R) , E = (1)/(4 pi in_(0)) .(Q)/(r^(2))`
The energy density outside shell : `u = (1)/(2) in_(0) E^(2)`
Take a small volume (a spherical shell of radius `r` and thickness `dr`)
`dV = 4 pi r^(2) dr`
Small energy stored in this volume
`dU = udV = (1)/(2) in_(0) E^(2) . 4 pi r^(2) dr`
`= (1)/(2) in_(0)((1)/(4 pi in_(0)) (Q)/(r^(2)))^(2) . 4 pi r^(2) dr`
`U = (Q^(2))/(8 pi in_(0)) int_(R )^(oo) (dr)/(r^(2)) = (Q^(2))/(8 pi in_(0) R)`