A small conducting sphere of radius is lying concentrically inside a bigger hollow conducting sphere of radius The bigger and smaller spheres are charged with and \((Q>q)\) and are insulated from each other. The potential difference between the spheres will be

The potential \(V_{1}\) of smaller sphere is given by

\(V_{1}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{r}+\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{R}\)
The potential \(\mathrm{V}_{2}\) of bigger sphere is given by
So, the potential difference between the plates \(\mathrm{V}=\mathrm{V}_{1}-\mathrm{V}_{2}\)
or \(\mathrm{V}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}}{\mathrm{r}}+\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{Q}}{\mathrm{R}}-\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{Q}}{\mathrm{R}}-\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}}{\mathrm{R}}\) \(=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{q}}{\mathrm{r}}-\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}}{\mathrm{R}}\)
\(=\frac{1}{4 \pi \varepsilon_{0}}\left(\frac{q}{r}-\frac{q}{R}\right)\)