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)

${\text{C}}_1=4\pi {ϵ}_0{\text{R}}_1$
and ${\mathrm{C}}_2=4\pi {\epsilon }_0\left(\frac{{\mathrm{R}}_1{\mathrm{R}}_2}{{\mathrm{R}}_2-{\mathrm{R}}_1}\right)$
Given that ${\text{C}}_{\text{2}}={\text{nC}}_1$
$\therefore \frac{{\text{R}}_1{\text{R}}_2}{{\text{R}}_2-{\text{R}}_1}={\text{nR}}_1$
or $\frac{{\text{R}}_2/{\text{R}}_1}{{\text{R}}_2/{\text{R}}_1-1}=\text{n}$
or $\frac{{\text{R}}_2}{{\text{R}}_1}=\frac{\text{n}}{\text{n}-1}$