Capacitance of an isolated conducting sphere of radius $R_{1}$ becomes $n$ times when it is enclosed by a concentric conducting sphere of radius R₂ connected to earth. The ratio of their radii is $(\frac{R_{2}}{R_{1}}) :$

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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 is given as $C_{1} = 4 π ε_{0} R_{1}$
By enclosing inside another sphere of radius R₂ ,

New capacitance, $C_{2} = \frac{4 π ε_{0} R_{1} R_{2}}{(R_{2} - R_{1})}$
Given, $C_{2} = n C_{1} = \frac{4 π ε_{0} R_{1} R_{2}}{(R_{2} - R_{1})} = n \times 4 π ε_{0} R_{1}$
$\begin{array}{l} \Rightarrow \frac{R_{2}}{(R_{2} - R_{1})} = n \Rightarrow \frac{R_{2} / R_{1}}{(\frac{R_{2}}{R_{1}} - 1)} = n \Rightarrow \frac{R_{2}}{R_{1}} = n \frac{R_{2}}{R_{1}} - n \\ \Rightarrow \frac{R_{2}}{R_{1}} = \frac{n}{n - 1} \end{array}$