Consider two conducting spheres with radii R1 and R2 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 q1 + q2, where q1 represents the charge on the first sphere and q2 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 U = keq2/2R. (Do this on paper. Your instructor may ask you to turn in this work.)
(b) Find the total energy of the system of two spheres in terms of q1, the total charge Q, and the radii R1 and R2. (Use the following as necessary: R1, R2, Q, q1, q2, and ke.)
U =
(c) To minimize the energy, differentiate the result to part (b) with respect to q1 and set the derivative equal to zero. Solve for q1 in terms of Q and the radii. (Use the following as necessary: R1, R2, Q, and ke.)
q1 =
(d) From the result to part (c), find the charge q2. (Use the following as necessary: R1, R2, Q, and ke.)
q2 =
(e) Find the potential of each sphere. (Use the following as necessary: R1, R2, Q, and ke.)
V1=
V2=
(f) What is the potential difference between the spheres? (Use the following as necessary: R1, R2, Q, and ke.)
ΔV =