Consider two charged metallic spheres S1 and S2 of radii R1 and R2 respectively . The electric field E1 (on S1) and E2 (on S2) …..

Q: Consider two charged metallic spheres S1 and S2 of radii R1 and R2 respectively . The electric field E1 (on S1) and E2 (on S2) on their surfaces are such that $\frac{E_1}{E_2} = \frac{R_1}{R_2}$ . Then the ratio V1 (on S1)/V2(on S2) 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 $

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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} $