Question

A spherical capacitor consists of an inner sphere of radius $12\text{cm}$ and the outer sphere of radius $36\text{cm}$. The capacitance is $C_1$ when the inner sphere is charged and the outer sphere is earthed and $C_2$ when the inner sphere is earthed and outer sphere is charged. The ratio $\frac{C_1}{C_2}$ is

• A. $3$
• B. $\frac{1}{3}$
• C. $\frac{1}{6}$
• D. $\frac{1}{9}$

Answer 1

The correct option is B $\frac{1}{3}$

Let $R_1=12\text{cm}$ and $R_2=36\text{cm}$

When outer space is earthed

$C_1=\frac{4\pi {ϵ}_0{R}_1{R}_2}{R_2-R_1}.......\left(1\right)$

When inner space is earthed

$C_2=\frac{4\pi {ϵ}_0{R}_1{R}_2}{R_2-R_1}+4\pi {ϵ}_0{R}_2$

$\Rightarrow C_2=\frac{4\pi {ϵ}_0{R}_2^2}{R_2-R_1}.......\left(2\right)$

Using eqs. $\left(1\right)$ and $\left(2\right)$,

$\frac{C_1}{C_2}=\frac{R_1}{R_2}$

$\Rightarrow \frac{C_1}{C_2}=\frac{12}{36}=\frac{1}{3}$

Hence, option $\left(b\right)$ is correct.