A charge Q is distributed over two concentric hollow spheres of radii r and R (R > r) such that the surface densities are equal. Find the potential at the common centre.

$\frac{1}{4 \pi \varepsilon_0} \frac{Q(R+r)}{R^2+r^2}$

$q_1+q_2=Q$                . . . (i)

$\sigma=\frac{q_1}{4 \pi r^2}=\frac{q_2}{4 \pi R^2}$        . . . (ii)

from (i) and (ii)

$q_1=\frac{Q r^2}{\left(r^2+R^2\right)} \quad q_2=\frac{Q R^2}{\left(r^2+R^2\right)}$

$V_{\text {centre }}=V_1+V_2=\frac{1}{4 \pi \varepsilon_0}\left(\frac{q_1}{r}+\frac{q_2}{R}\right)$

$=\frac{1}{4 \pi \varepsilon_0} \frac{Q(R+r)}{R^2+r^2}$