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

From Gauss' theorem intensity at a point, inside the sphere at a distance `r` from the centre is given by, `E_(r) = (rho r)/(3 epsilon_(0))` and outside it, given by `E_(r) = (1)/(4pi epsilon_(0)) (q)/(r^(2))`
(a) Potential at the centre of the sphere,
`varphi_(0) = int_(0)^(oo) vec(E). vec(dr) = int_(0)^(R ) (rho r)/(3epsilon_(0)) dr + int_(R )^(oo) (q)/(4pi epsilon_(0) r^(2)) dr = (rho)/(3 epsilon) (R^(2))/(2) + (q)/(4pi epsilon_(0) R)`
as `= (q)/(8pi epsilon_(0)R) + (q)/(4pi epsilon_(0) R) = (3q)/(8pi epsilon_(0) R)` `("as " rho = (3q)/(4pi R^(3)))`
(b) Now, potential at any point, inside the sphere, at a distance `r` from it `s` centre.
`varphi (r) = int_(r )^(R ) (rho)/(3epsilon_(0)) r dr int_(r)^(oo) (q)/(4pi epsilon_(0)) (dr)/(r^(2))`
On intergation `: varphi (r) = (3q)/(8pi epsilon_(0)R) [1 (r^(2))/(3 R^(2))] = varphi_(0) [1 - (r^(2))/(3R^(2))]`