Physics - Capacitor Question with Solution | TestHub

Capacitance of isolated conducting sphere $C=4\pi {\epsilon }_0{R}_1$

By enclosing inside another sphere of radius $R_2$, new capacitance $C^{\text{'}}=\frac{4\pi {\epsilon }_0{R}_1{R}_2}{\left(R_2-R_1\right)}$

Given: $\frac{4\pi {\epsilon }_0{R}_1{R}_2}{\left(R_2-R_1\right)}=n\times 4\pi {\epsilon }_0{R}_1$

$\Rightarrow \frac{R_2}{\left(R_2-R_1\right)}=n\Rightarrow \frac{\frac{R_2}{R_1}}{\left(\frac{R_2}{R_1}-1\right)}=n$

$$\begin{gathered} \Rightarrow \frac{R_2}{R_1}=n\frac{R_2}{R_1}-n \\ \Rightarrow \frac{R_2}{R_1}=\frac{n}{\left(n-1\right)} \end{gathered}$$