A spherical capacitor has outer sphere of radius $5 \mathrm{~cm}$ and inner sphere of radius $2 \mathrm{~cm}$. 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

Question

A spherical capacitor has outer sphere of radius $5 \mathrm{~cm}$ and inner sphere of radius $2 \mathrm{~cm}$. 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, $\frac{5}{2}$, $\frac{2}{5}$, $\frac{7}{3}$, $\frac{3}{7}$

MARKS App · Accepted Answer

Key Idea 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, $C^{\prime}=C+4 \pi \varepsilon_0 R_2$ Given, radius of outer sphere of capacitor, $R_1=2 \mathrm{~cm}$ and Radius of inner sphere of capacitor, $R_2=5 \mathrm{~cm}$ $\begin{aligned} \therefore \quad C_1 & =4 \pi \varepsilon_0\left[\frac{R_1 R_2}{R_2-R_1}+R_2\right]=4 \pi \varepsilon_0\left[\frac{2 \times 5}{5-2}+5\right] \ C_1 & =4 \pi \varepsilon_0 \frac{25}{3} \text { Farad } \end{aligned}$ and $C_2=4 \pi \varepsilon_0 \frac{R_1 R_2}{R_2-R_1}=4 \pi \varepsilon_0 \frac{2 \times 5}{5-2}=4 \pi \varepsilon_0 \frac{10}{3}$ $\frac{C_1}{C_2}=\frac{\frac{25}{3}}{\frac{10}{3}}=\frac{25}{10}=\frac{5}{2}$ Hence, the correct option is (a).

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