Question

The capacity of an isolated sphere is increased $n$ times when it is enclosed by an earthed concentric sphere. The ratio of their radii is

• A. $\frac{n^{2}}{n - 1}$
• B. $\frac{n}{n - 1}$
• C. $\frac{2 n}{n + 1}$
• D. $\frac{2 n + 1}{n + 1}$

Answer 1

Answer: (B)

Capacity of an isolated sphere $C_{1}=4\pi \epsilon _{0}R_{1}$
and Capacity of a sphere when enclosed by an earthed concentric sphere
$C_2=4 \pi \varepsilon_0\left(\frac{R_1 R_2}{R_1+R_2}\right)$
Given that $C_{2}=nC_{1}$
or $\frac{R_{2} R_{1}}{R_{2} - R_{1}}=nR_{1}$
or $\frac{\frac{R_{2}}{R_{1}}}{\frac{R_{2}}{R_{1}} - 1}=n$
or $\frac{R_{2}}{R_{1}}=\frac{n}{n - 1}$