A spherical conducting shell of inner radius r₁ and outer radius r₂ has a charge Q. 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.

The charge of magnitude +q present inside the spherical shell induces a charge of magnitude −q.On the inner part of the shell. Therefore, for the charge on the inner part, the surface charge density will be due to −q.

It is given by:

σ₁​=Total charge​/surface area of inner part

⇒$\frac{-q}{4\pi r_1^2}$

The charge on the outer shell will be +q and the surface charge density at the outer part will be due to sum of q and Q charges present on the outer shell.

σ₂​=Total charge/Outer surface area​=$\frac{\mathrm{Q}+\mathrm{q}}{4{\mathrm{\pi r}}_2^2}$