The text derived Green’s reciprocity theorem for a set of conductors as a special case of a more general result. For conductors with charges and potentials (q_k, ϕ_K) and the same set of conductors with charges and potentials (˜q_k, ˜φ_k), the theorem reads

(a) Use the symmetry of the capacitance matrix to prove the theorem directly.

(b) Three identical conducting spheres are placed at the corners of an equilateral triangle. When the sphere potentials are (φ, 0, 0), their charges are (q, q₀, q₀). What is the charge q' on each sphere when the potential of each sphere is φ'?

(c) What is the potential on each sphere when their charges are (q'', 0, 0)?

N N = i=l i=l