Two concentric, conducting, thin spherical shells A and B have radii r and R as shown in Fig. Shell A carries a charge q and B carries a charge Q the potential difference $\left(V_A-V_B\right)$ between the inner shell and the outer shell is

 

For points outside the outer shell, the two shells behave as if all their charge were concentrated at their common centre O. At a point x > R,
$V=\frac{q+Q}{4\pi {\in }_{0}x}$

If we put x = R, we get the potential on the surface of B. Thus 
$$\begin{gathered} V_B=\frac{q+Q}{4\pi {\in }_{0}R} \\ \because V_A=\frac{{\displaystyle 1}}{{\displaystyle 4\pi {\in }_0}}\left(\frac{{\displaystyle q}}{{\displaystyle r}}+\frac{{\displaystyle Q}}{{\displaystyle R}}\right) \\ \therefore V_A-V_B=\frac{1}{4\pi {ϵ}_0}\left(\frac{q}{r}+\frac{Q}{R}\right)-\frac{1}{4\pi {\in }_0}\left(\frac{q+Q}{R}\right) \\ =\frac{q}{4\pi {ϵ}_0}\left(\frac{1}{r}-\frac{1}{R}\right) \end{gathered}$$