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

Answer 1

Answer: (B)

$\text{C}_{1} = 4 \pi \epsilon _{0} \text{R}_{1}$
and $\mathrm{C}_2=4 \pi \varepsilon_0\left(\frac{\mathrm{R}_1 \mathrm{R}_2}{\mathrm{R}_2-\mathrm{R}_1}\right)$
Given that $\text{C}_{\text{2}} = \text{nC}_{1}$
$∴ \, \, \frac{\text{R}_{1} \text{R}_{2}}{\text{R}_{2} - \text{R}_{1}} = \text{nR}_{1}$
or $\frac{\text{R}_{2} / \text{R}_{1}}{\text{R}_{2} / \text{R}_{1} - 1} = \text{n}$
or $\frac{\text{R}_{2}}{\text{R}_{1}} = \frac{\text{n}}{\text{n} - 1}$