Q: Consider two charged metallic spheres S_{1} and S_{2} of radii R_{1} and R_{2} respectively . The electric field E_{1} (on S_{1}) and E_{2} (on S_{2}) on their surfaces are such that $\frac{E_1}{E_2} = \frac{R_1}{R_2}$ . Then the ratio V_{1} (on S_{1})/V_{2}(on S_{2}) of the electrostatic potentials on each sphere is

(a) $\displaystyle (\frac{R_1}{R_2})^3$

(b) $\displaystyle \frac{R_2}{R_1}$

(c) $\displaystyle \frac{R_1}{R_2}$

(d) $\displaystyle (\frac{R_1}{R_2})^2 $

Ans: (d)

Sol: $\displaystyle \frac{E_1}{E_2} = \frac{R_1}{R_2} $ (given) …(i)

$\displaystyle E_1 = K\frac{Q_1}{R_1^2}$

$\displaystyle E_2 = K\frac{Q_2}{R_2^2}$

$\displaystyle \frac{E_1}{E_2} = \frac{Q_1}{Q_2} \times \frac{R_2^2}{R_1^2}$

$\displaystyle \frac{R_1}{R_2} = \frac{Q_1}{Q_2} \times \frac{R_2^2}{R_1^2}$ (from (i))

$\displaystyle \frac{Q_1}{Q_2} = \frac{R_1^3}{R_2^3} $ …(ii)

$\displaystyle V_1 = K\frac{Q_1}{R_1}$

$\displaystyle V_2 = K\frac{Q_2}{R_2}$

$\displaystyle \frac{V_1}{V_2} = \frac{Q_1}{Q_2} \times \frac{R_2}{R_1}$

$\displaystyle \frac{V_1}{V_2} = \frac{R_1^3}{R_2^3} \times \frac{R_2}{R_1}$ (from (ii))

$\displaystyle \frac{V_1}{V_2} = \frac{R_1^2}{R_2^2} $