(a) Suppose a charge distribution P₁ (r) produces a potential V₁ (r), and some other charge distribution p₂(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 P_l and 0₂ are not present at the same time; we are talking about two different problems, one in which only P_l is present, and another in which only ,0₂ is present.] Prove Green's reciprocity theorem:

(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).

Transcribed Image Text:

S all space Q a P1V₂ dt = S all space P2V1 dr. b