A charge `q` is uniformly distributed over the volume of a shpere of radius `R`. Assuming the permittively to be equal to unity throughout, find the potential
(a) at the centre of the sphere,
(b) inside the sphere as a function of the distance `r` from its centre.

Explanation

Inside `(r lt R)`
`E = (1)/(4 pi in_(0)) (Qr)/(R^(3))`
Outside sphere `(r gt R)`
`E = (1)/(4 pi in_(0)) (Q)/(r^(2))`
Electric potential `(r gt R)`
`V_(B) - V_(00) = - int_(oo)^(r ) vec(E ) . d vec( r) = - int_(oo)^(r ) E dr`
`= -(Q)/(4 pi in_(0)) int_(oo)^(r ) r^(-2) dr = - (Q)/(4 pi in_(0)) |(r^(-1))/(-1)|_(oo)^(r )`
`= -(Q)/(4 pi in_(0)) |(1)/(r )|_(oo)^(r ) = (Q)/( 4 pi in_(0)) |(1)/(r ) - (1)/(oo)|`
`V_(B) = (1)/(4 pi in_(0)) (Q)/( r )`
At surface `(r = R)`
`V_(P) = (1)/(4 pi in_(0)) (Q)/(R)`
Electric potential `(r lt R)`
`V_(A) - V_(oo) = - int_(oo)^(r ) vec(E ) . d vec( r) = - int_(oo)^(r ) E dr`
`= - [int_(oo)^(R) Edr + int_(R )^(r ) Edr]`
`= -[int_(oo)^(R) (1)/(4 pi in_(0)) (Q)/(r^(2)) dr + int _(R )^(r ) (1)/(4pi in_(0)) (Qr)/(R^(3)) dr]`
`= - [ (1)/(4 pi in_(0)) (Q)/(r ) + (1)/(4 pi in_(0)) (Q)/(R^(3)) |(r^(2))/(2)|_(R )^(r )]`
`V_(A) = (1)/(4 pi in_(0)) (Q(3 R^(2) - r^(2)))/(2R^(3))`
At center , `r = 0`
`V_(0) = (1)/(4 pi in_(0)) (3Q)/(2R)`