Two spherical conductors A₁ and A₂ of radii r₁ and r₂ (r₂> r₁) are placed concentrically in air. A₁ is given a charge +Q while A₂ is earthed. Then the equivalent capacitance of the system is -

By Gauss's law the electric field at a distance r in between the shells is

$\mathrm{E}.4\pi {\mathrm{r}}^2=\frac{\mathrm{Q}}{{ϵ}_0}\Rightarrow \mathrm{E}=\frac{\mathrm{Q}}{4\pi {ϵ}_0{\mathrm{r}}^2}$

The potential difference between shells is

 ${\int }_0^{\mathrm{V}} \mathrm{dV}=-{\int }_{{\mathrm{r}}_2}^{{\mathrm{r}}_1} \mathrm{Edr}$

$\begin{array}{r}\text{ or }\mathrm{V}=-\frac{\mathrm{Q}}{4\pi {ϵ}_0}{\int }_{{\mathrm{r}}_2}^{{\mathrm{r}}_1} \frac{\mathrm{dr}}{{\mathrm{r}}^2}=\frac{\mathrm{Q}}{4\pi {ϵ}_0}\left(\frac{1}{{\mathrm{r}}_1}-\frac{1}{{\mathrm{r}}_2}\right)\\ \text{ or }\mathrm{V}=\frac{\mathrm{Q}}{4\pi {ϵ}_0}\cdot \frac{{\mathrm{r}}_2-{\mathrm{r}}_1}{{\mathrm{r}}_1{\mathrm{r}}_2}\end{array}$

$\therefore \text{ Capacitance, }\mathrm{C}=\frac{\mathrm{Q}}{\mathrm{V}}=\frac{4\pi {ϵ}_0{\mathrm{r}}_1{\mathrm{r}}_2}{{\mathrm{r}}_2-{\mathrm{r}}_1}$