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Q.

The capacity of an isolated copper sphere increases ‘ $n$ ’ times, when it is enclosed by an earthed concentric thin copper shell. The ratio of the radius of the outer shell to that of the inner sphere is

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a

$n^{2} / (n - 1)$

b

$n / (n - 1)$

c

$(2 n + 1) / (n + 1)$

d

$(2 n) / (n + 1)$

answer is B.

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$\begin{array}{l} C_{1} = 4 {πε}_{0} R_{1} \\ \begin{aligned} C_{2} & = 4 {πε}_{0} \frac{R_{1} R_{2}}{R_{2} - R_{1}} \\ = {nC}_{1} \end{aligned} \\ ∴ \frac{R_{1} R_{2}}{R_{2} - R_{1}} = {nR}_{1} \\ or R_{2} = {nR}_{2} - {nR}_{1} \\ or {nR}_{1} = (n - 1) R_{2} \\ ∴ \frac{R_{2}}{R_{1}} = \frac{n}{n - 1} \end{array}$

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