Consider two conducting spheres with radii and separated by a distance much greater than either radius. A total charge Q is shared between the spheres. We wish to show that when the electric potential energy of the system has a minimum value, the potential difference between the spheres is zero. The total charge Q is equal to , where represents the charge on the first sphere and the charge on the second. Because the spheres are very far apart, you can assume the charge of each is uniformly distributed over its surface.
(a) Show that the energy associated with a single conducting sphere of radius R and charge q surrounded by a vacuum is .
(b) Find the total energy of the system of two spheres in terms of , total charge Q and the radii and .
(c) To minimize the energy, differentiate the result to part (b) with respect to and set the derivative equal to zero. Solve for in terms of Q and the radii.
(d) From the result to part (c), find the charge .
(e) Find the potential of each sphere.
(f) What is the potential difference between the spheres?
