Capacitance of an isolated conducting sphere of radius R₁ becomes n times when it is enclosed by a concentric conducting sphere of radius R₂ connected to earth. The ratio of their radii $\left(\frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}\right)$ is:

Capacitance of isolated Conducting sphere$=4{\mathrm{\pi \epsilon }}_0{\mathrm{R}}_1$

By enclosing inside another sphere of radius

${\mathrm{R}}_2,\text{ new capacitance }=\frac{4{\mathrm{\pi \epsilon }}_0{\mathrm{R}}_1{\mathrm{R}}_2}{\left({\mathrm{R}}_2-{\mathrm{R}}_1\right)}$

$\mathrm{Given} : \frac{4{\mathrm{\pi \epsilon }}_0{\mathrm{R}}_1{\mathrm{R}}_2}{\left({\mathrm{R}}_2-{\mathrm{R}}_1\right)}=\mathrm{n}\times 4{\mathrm{\pi \epsilon }}_0{\mathrm{R}}_1$

$\begin{array}{l}\Rightarrow \frac{{\mathrm{R}}_2}{\left({\mathrm{R}}_2-{\mathrm{R}}_1\right)}=\mathrm{n}\Rightarrow \frac{\frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}}{\left(\frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}-1\right)}=\mathrm{n}\\ \Rightarrow \frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}=\mathrm{n}\frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}-\mathrm{n}\Rightarrow \frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}=\frac{\mathrm{n}}{(\mathrm{n}-1)}\end{array}$