Two metallic solid spheres of radii R and 2R are charge such that both of them have same charge density $\sigma .$ If the spheres are located far away from each other and connected by thin conducting wire, the new charge density on the bigger drop is __

Total charge $Q=Q_1+Q_2=\sigma 4\pi R^2+\sigma 4\pi 4R^2$

$Q=20\pi R^2\sigma ,$ when sphere are connected V is same.

$V=\frac{Q_1}{4\pi R}=\frac{Q^1}{4\pi .2R}\Rightarrow Q_2^1=2Q_1^1,$ but total charge = constant

$Q_1^1+Q_2^1=Q_1+Q_2\Rightarrow \frac{Q_2^1}{2}+Q_2^1=20\pi R^2\sigma$

$\frac{3}{2}20\pi R^2\sigma \Rightarrow Q_2^1=\frac{40}{3}\pi R^2\sigma$

New charge density on bigger drop ${\sigma }^1=\frac{Q_2^1}{4\pi .4R^2}=\frac{40\pi R^2\sigma }{3\times 16\pi R^2}=\frac{5}{3\times 2}\sigma =\frac{5}{6}\sigma$