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

If $q_1$  and $q_2$  are the charge on spheres of radius r and R, respectively, in accordance with conservation of charge,

$Q=q_1+q_2$                         ….. (i)

and according to the given problem, ${\sigma }_1={\sigma }_2$

i.e, $\frac{q_1}{4\pi r^2}=\frac{q_2}{4\pi R^2}\Rightarrow \frac{q_1}{q_2}=\frac{r^2}{R^2}$                                 …… (ii)

So, (i) and (ii) give

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

Potential at common center

$V=\frac{1}{4\pi {\epsilon }_0}\left[\frac{q_1}{r}+\frac{q_2}{R}\right]$

$=\frac{1}{4\pi {\epsilon }_0}\left[\frac{Qr}{\left(R^2+r^2\right)}+\frac{QR}{\left(R^2+r^2\right)}\right]=\frac{1}{4\pi {\epsilon }_0}.\frac{Q\left(R+r\right)}{\left(R^2+r^2\right)}$