A sphere of radius a carries an electric charge $Q$ uniformly distributed over its surface. Calculate the electric field outside and inside the sphere. Also calculate the electrostatic potential outside and inside the sphere, assuming it vanishes at infinity. State the integral formula for the energy $U$ of the electric field and use it to evaluate $U$ as a function of $Q$

Relate $\frac{d U}{d Q}$ to the potential on the surface of the sphere and explain briefly the physical interpretation of the relation.