A spherical conducting shell of inner radius `r_1` and outer radius `r_2` has a charge Q.
(a) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell ?

Explanation

When charge Q is given to spherical conducting shell, whole of it spreads uniformly on the outer surface of shell. On placing a charge +q at the centre of the shell, a charge -q is induced on the inner surface of shell and in turn charge +q is induced on its outer surface.
`:.` Surface charge density of inner surface of the shell
`sigma_("inner") = (-q)/(4pi r_1^2)`
and surface charge density of outer surface of the shell
`sigma_("outer") = ((Q + q))/(4pi r_2^2)`