A spherical capacitor has outer sphere of radius (5 ~cm ) and inner...

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).