Question

A spherical capacitor has outer sphere of radius $5cm$ and inner sphere of radius $2cm$. When the inner sphere is earthed, its capacity is $C_1$ and when the outer sphere is earthed its capacity is $C_2$. Then $\frac{C_1}{C_2}$ is

• A. $\frac{5}{2}$
• B. $\frac{2}{5}$
• C. $\frac{7}{3}$
• D. $\frac{3}{7}$

Answer 1

Answer: (A)

If a spherical capacitor has outer radius $R_2$ and inner radius $R_1.$ Then the capacitance,
(a) when outer shell is earthed,
$C=4\pi {\epsilon }_0\frac{R_1{R}_2}{R_2-R_1}$
(b) When inner shell is earthed,
$C^{\prime }=C+4\pi {\epsilon }_0{R}_2$
Given, radius of outer sphere of capacitor,
$R_1=2cm$ and Radius of inner sphere of capacitor, $R_2=5cm$
$\therefore C_1=4\pi {\epsilon }_0\left[\frac{R_1{R}_2}{R_2-R_1}+R_2\right]$
$=4\pi {\epsilon }_0\left[\frac{2\times 5}{5-2}+5\right]$
$C_1=4\pi {\epsilon }_0\frac{25}{3}$ Farad
and $C_2=4\pi {\epsilon }_0\frac{R_1{R}_2}{R_2-R_1}=4\pi {\epsilon }_0\frac{2\times 5}{5-2}=4\pi {\epsilon }_0\frac{10}{3}$
Hence, $\frac{C_1}{C_2}=\frac{\frac{25}{3}}{\frac{10}{3}}=\frac{25}{10}=\frac{5}{2}$