Question

A spherical capacitor has outer sphere of radius $5\, cm$ and inner sphere of radius $2\, cm$. 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,
$C=4 \pi \varepsilon_{0} \frac{R_{1} R_{2}}{R_{2}-R_{1}}$
(b) When inner shell is earthed,
$C'=C+4 \pi \varepsilon_{0} R_{2}$
Given, radius of outer sphere of capacitor,
$R_{1}=2\, cm$ and Radius of inner sphere of capacitor, $R_{2}=5 \,cm$
$\therefore \,C_{1}=4 \pi \varepsilon_{0}\left[\frac{R_{1} R_{2}}{R_{2}-R_{1}}+R_{2}\right]$
$=4 \pi \varepsilon_{0}\left[\frac{2 \times 5}{5-2}+5\right]$
$C_{1}=4 \pi \varepsilon_{0} \frac{25}{3}$ Farad
and $C_{2}=4 \pi \varepsilon_{0} \frac{R_{1} R_{2}}{R_{2}-R_{1}}=4 \pi \varepsilon_{0} \frac{2 \times 5}{5-2}=4 \pi \varepsilon_{0} \frac{10}{3} $
Hence, $\frac{C_{1}}{C_{2}}=\frac{\frac{25}{3}}{\frac{10}{3}}=\frac{25}{10}=\frac{5}{2}$