Capacitance of an isolated conducting sphere of radius $R_{1}$ becomes n times when it is enclosed by a concentric conducting sphere of radius $R_{2}$ 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 an isolated conducting

sphere of radius $R_{1}$ , $c_{1} = 4 {πε}_{o} R_{1}$
Capacitance of spherical shell of radius $R_{1} and R_{2}$

$c_{2} = 4 {πε}_{o} \frac{R_{2} R_{1}}{R_{2} - R_{1}}$
According to given situation,

$c_{2} = {nC}_{1}$

$\Rightarrow \frac{4 π ε_{o} R_{1} R_{2}}{R_{2} - R_{1}} = n .4 π ε_{o} R_{1}$

$\Rightarrow \frac{R_{2}}{R_{2} - R_{1}} = n$

$\Rightarrow \frac{R_{2} - R_{1}}{R_{2}} = \frac{1}{n} \Rightarrow 1 - \frac{R_{1}}{R_{2}} = \frac{1}{n}$

$\Rightarrow 1 - \frac{1}{n} = \frac{R_{1}}{R_{2}}$

$\Rightarrow \frac{n - 1}{n} = \frac{R_{1}}{R_{2}} \Rightarrow \frac{R_{2}}{R_{1}} = \frac{n}{n - 1} .$

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