Question

A point charge q is placed at a distance r from the centre O of an uncharged spherical shell of inner radius R and outer radius 2R. The distance is r < R. What will be the electric potential at the centre of shell.

Answer 1

It is given in the comment section that $R>r$

The point charge $q$ is inside the sphere.

In static condition, there is no electric field and no net charge at any point inside a conductor. The charge much appear at the surface of the conductor, potential surface of the conductor, potential $V$ at the centre or any point inside it is same and its value is that at its surface (a conductor is an equipotential volume). Hence

$V=\frac{1}{4\pi E_0}\frac{q}{R}$

If it is a thin shell:-

The charge $q$ placed inside the shell causes an induced charge $-q$ on the inner surface of the shell and $+q$ charge on the outer surface. The potential at the centre is now due to $\left(i\right)$ charge $q$ placed inside the shell $\left(ii\right)$ charge $-q$ on the inner surface of the shell and $\left(iii\right)$ charge $+q$ on the outer surface of the shell. Since the shell is thin its inner and outer radii can be considered equal each to $R$

$V=\frac{1}{4\pi E_0}\frac{q}{r}-\frac{1}{4\pi E_0}\frac{q}{R}+\frac{1}{4\pi E_0}\frac{q}{R}=\frac{1}{4\pi E_0}\frac{q}{r}$