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. $\frac{2n}{n+1}$
• D. $\frac{2n+1}{n+1}$

Answer 1

Answer: (B)

Capacity of an isolated sphere $C_1=4\pi {ϵ}_0{R}_1$
and Capacity of a sphere when enclosed by an earthed concentric sphere
$C_2=4\pi {\epsilon }_0\left(\frac{R_1{R}_2}{R_1+R_2}\right)$
Given that $C_2=n{C}_1$
or $\frac{R_2{R}_1}{R_2-R_1}=n{R}_1$
or $\frac{\frac{R_2}{R_1}}{\frac{R_2}{R_1}-1}=n$
or $\frac{R_2}{R_1}=\frac{n}{n-1}$