Two metal spheres, one of radius R and the other of radius 2R respectively have the same surface charge density σ. They are brought in contact and separated. What will be the new surface charge densities on them ?

Two metal spheres, one of radius R and the other of radius 2R respectively have the same surface charge density σ. They are brought in contact and separated. What will be the new surface charge densities on them ?

(a) \( \sigma_1 = \frac{5}{3} \sigma,\quad \sigma_2 = \frac{5}{6} \sigma \)

(b) \( \sigma_1 = \frac{5}{6} \sigma,\quad \sigma_2 = \frac{5}{2} \sigma \)

(c) \( \sigma_1 = \frac{5}{2} \sigma,\quad \sigma_2 = \frac{5}{6} \sigma \)

(d) \( \sigma_1 = \frac{5}{2} \sigma,\quad \sigma_2 = \frac{5}{3} \sigma \)

Answer :  (a) \( \sigma_1 = \frac{5}{3} \sigma,\quad \sigma_2 = \frac{5}{6} \sigma \)

\( \sigma = \frac{q_1}{4 \pi R^2} = \sigma_2 = \frac{q_2}{4 \pi (2R)^2} \)

\( q_1 = \frac{q_2}{4} \)

After touching:

\( V_1 = V_2 \)

\( \frac{k q_1'}{R} = \frac{k q_2'}{2R} \)

\( q_1' = \frac{q_2'}{2} \)

\( q_1 + q_2 = q_1' + q_2' \)

\( \frac{q_2}{4} + q_2 = \frac{q_2'}{2} + q_2' \)

\( \frac{5}{4} q_2 = \frac{3}{2} q_2' \)

\( \sigma = \frac{q_2}{4 \pi (2R)^2} \)

\( \sigma_2 = \frac{q_1'}{4 \pi (2R)^2} = \frac{5}{4} \cdot \frac{2}{3} \cdot \frac{q_2}{4 \pi (2R)^2} = \frac{5}{6} \sigma \)

\( \sigma = \frac{q_1}{4 \pi R^2} \)

\( \sigma_1 = \frac{q_1'}{4 \pi R^2} = \frac{q_1' / 2}{4 \pi R^2} = \frac{5}{4} \cdot \frac{2}{3} \cdot \frac{1}{2} \cdot \frac{q_2}{4 \pi R^2} \)

\( \sigma_1 = \frac{5}{3} \sigma \)