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 $(\frac{R_{2}}{R_{1}})$ is:

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$\frac{n}{n - 1}$

$\frac{2 n}{2 n + 1}$

$\frac{n + 1}{n}$

$\frac{2 n + 1}{n}$

answer is A.

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Detailed Solution

Capacitance of isolated Conducting sphere $= 4 {πε}_{0} R_{1}$

By enclosing inside another sphere of radius

$R_{2}, new capacitance = \frac{4 {πε}_{0} R_{1} R_{2}}{(R_{2} - R_{1})}$

$Given : \frac{4 {πε}_{0} R_{1} R_{2}}{(R_{2} - R_{1})} = n \times 4 {πε}_{0} R_{1}$

$\begin{array}{l} \Rightarrow \frac{R_{2}}{(R_{2} - R_{1})} = n \Rightarrow \frac{\frac{R_{2}}{R_{1}}}{(\frac{R_{2}}{R_{1}} - 1)} = n \\ \Rightarrow \frac{R_{2}}{R_{1}} = n \frac{R_{2}}{R_{1}} - n \Rightarrow \frac{R_{2}}{R_{1}} = \frac{n}{(n - 1)} \end{array}$