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

when sphere’s are kept in contact, charge re-distribution takes place to get same potential .
Hence new charge densities will be 
$$\begin{gathered} \frac{\mathrm{K}(\mathrm{\sigma }4{\mathrm{\pi R}}^2+\mathrm{Q})}{\mathrm{R}}= \frac{\mathrm{K}(16{\mathrm{\sigma \pi R}}^2-\mathrm{Q})}{2\mathrm{R}} \\ Q=\frac{8\mathrm{\sigma \pi R}}{3} \\ {\mathrm{\sigma }}_1=\frac{(\mathrm{\sigma }4{\mathrm{\pi R}}^2+\mathrm{Q})}{4{\mathrm{\pi R}}^2}=\frac{5}{3}\mathrm{\sigma } \\ {\mathrm{\sigma }}_2=\frac{(\mathrm{\sigma }16{\mathrm{\pi R}}^2-\mathrm{Q})}{16{\mathrm{\pi R}}^2}=\frac{5}{6}\mathrm{\sigma } \\ {\mathrm{\sigma }}_1=\frac{5}{3}\mathrm{\sigma } , {\mathrm{\sigma }}_2=\frac{5}{6}\mathrm{\sigma } \end{gathered}$$