(a) Suppose a charge distribution ρ1(r) produces a potential V1(r), and some other charg...

(a) Suppose a charge distribution ρ1(r) produces a potential V₁(r), and some other charge distribution ρ₂(r) produces a potential V₂(r). [The two situations may have nothing in common, for all I care—perhaps number 1 is a uniformly charged sphere and number 2 is a parallel-plate capacitor. Please understand that ρ₁ and ρ₂ are not present at the same time; we are talking about two different problems, one in which only ρ1 is present, and another in which only ρ₂ is present.] Prove Green’s reciprocity theorem:²⁵

[Hint: Evaluate ? E₁ . E₂ dτ two ways, first writing E₁ = −∇V1 and using integration by parts to transfer the derivative to E₂, then writing E₂ = −∇V₂ and transferring the derivative to E₁.]

(b) Suppose now that you have two separated conductors (Fig. 3.41). If you charge up conductor a by amount Q (leaving b uncharged), the resulting potential of b is, say, V_ab. On the other hand, if you put that same charge Q on conductor b (leaving a uncharged), the potential of a would be V_ba . Use Green’s reciprocity theorem to show that V_ab = V_ba (an astonishing result, since we assumed nothing about the shapes or placement of the conductors).