Question

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

The correct option is B $\frac{n}{n-1}$



Let $C_1$ be the capacitance of the isolated sphere.

$C_1=4\pi {ϵ}_0{R}_1$

Let $C_2$ be the capacitance of sphere when it is enclosed by an earthed concentric sphere.

$C_2=4\pi {ϵ}_0\left(\frac{R_1{R}_2}{R_2-R_1}\right)$

Given that $C_2=n{C}_1$

$\therefore \frac{R_1{R}_2}{R_2-R_1}=n{R}_1$

$=\frac{\frac{R_2}{R_1}}{\frac{R_2}{R_1}-1}=n$

$=\frac{R_2}{R_1}=\frac{n}{n-1}$

Hence, option (b) is the correct answer.