Explanation

(a) [image:1_image245.emf]
Let $Q_{1}$ and $Q_{2}$ are the charges on sphere of radii $R_{1}$ and $R_{2}$ respectively. Surface charge density $\sigma = \frac{Charge}{Area}$
According to given problem, $\sigma_{1} = \sigma_{2}$
$\frac{Q_{1}}{4\pi R_{1}^{2}} = \frac{Q_{2}}{4\pi R_{2}^{2}}$
$\therefore\frac{Q_{1}}{Q_{2}} = \frac{R_{1}^{2}}{R_{2}^{2}}$ …(i)
In case of a charged sphere, $V_{s} = \frac{1}{4\pi\varepsilon_{0}}\frac{Q}{R}$
$\therefore V_{1} = \frac{1}{4\pi\varepsilon_{0}}\frac{Q_{1}}{R_{1}},\ V_{2} = \frac{1}{4\pi\varepsilon_{0}}\frac{Q_{2}}{R_{2}}$
$\Rightarrow \frac{V_{1}}{V_{2}} = \frac{Q_{1}}{R_{1}} \times \frac{R_{2}}{Q_{2}} = \frac{Q_{1}}{Q_{2}} \times \frac{R_{2}}{R_{1}}$
$= \left( \frac{R_{1}}{R_{2}} \right)^{2} \times \left( \frac{R_{2}}{R_{1}} \right) = \frac{R_{1}}{R_{2}}$ …[Using(i)]