A sphere of radius R has charge q.

a. What is the infinitesimal increase in electric potential energy dU if an infinitesimal amount of charge dq is brought from infinity to the surface of the sphere?

b. An uncharged sphere can acquire total charge Q by the transfer of charge dq over and over and over. Use your answer to part a to find an expression for the potential energy of a sphere of radius R with total charge Q.

c. Your answer to part b is the amount of energy needed to assemble a charged sphere. It is often called the self-energy of the sphere. What is the self-energy of a proton, assuming it to be a charged sphere with a diameter of 1.0 × 10^-15 m?