Two charged spherical conductors of radii $R_1$ and $R_2$ when connected by a conducting wire acquire charges $q_1$ and $q_2$ respectively. Find the ratio of their surface charge densities in terms of their radii.

Two parallel plate capacitors of capacitances $C_1$ and $C_2$ such that $C_1=2 C_2$ are connected across a battery $V$ volts as shown in figure below. Initially the key $\left(K\right)$ is kept closed to fully charge the capacitors. The key is now thrown open and a dielectric slab of dielectric constant $k$ is inserted in the two capacitors to completely fill the gap between the plates. Find the ratio of the net capacitance, before and after the introduction of the dielectric slab.

Two parallel plate capacitors of capacitances $C_1$ and $C_2$ such that $C_1=2 C_2$ are connected across a battery $V$ volts as shown in figure below. Initially the key $\left(K\right)$ is kept closed to fully charge the capacitors. The key is now thrown open and a dielectric slab of dielectric constant $k$ is inserted in the two capacitors to completely fill the gap between the plates. Find the ratio of the energies stored in the combination, before and after the introduction of the dielectric slab.