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Two metal spheres, one of radius R and the other of radius 2R respectively have the same surface charge density $\sigma$. They are brought in contact and separated. What will be the new surface charge densities on them?
Detailed Solution
$Q_1=\sigma4\pi R^2$, $Q_2=\sigma4\pi(2R)^2$, so $Q_{net}=20\pi\sigma R^2$. After contact the potentials are equal, so $\frac{Q_1'}{Q_2'}=\frac{R}{2R}=\frac{1}{2}$: $Q_1'=\frac{20}{3}\pi\sigma R^2$, $Q_2'=\frac{40}{3}\pi\sigma R^2$. $\sigma_1=\frac{Q_1'}{4\pi R^2}=\frac{5}{3}\sigma$, $\sigma_2=\frac{Q_2'}{4\pi(2R)^2}=\frac{5}{6}\sigma$.