(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that for two arbitrary conductors the coeﬃcients of capacitance C12 and C21 are always equal.

Hint: Consider a two-conductor system in which the two conductors have been charged so

that their potentials are φ1f and φ2f respectively (f for ”ﬁnal”). This condition might have been brought about, starting from a state with all charges and potentials zero, in diﬀerent ways. Two possible ways are of particular interest:

a) Keep φ2 at zero while raising φ1 from zero to φ1f , then raise φ2 from zero to φ2f while

holding φ1 constant at φ1f .

b) Carry out a similar program with the roles of 1 and 2 exchanged.

Compute the total work done by external agencies for each of the two charging programs. Then complete the argument

3. The attempt at a solution

I actually finally gave up and found the solution online. And it's a good thing I did, because even looking at it I still don't understand it. There's this one bit that I can't see where they got it from.

For For φ1 : 0 → φ1f and φ2 = 0 the work done by external agencies is:

[itex]\int[/itex][itex]^{\phi_{1f}}_{0}[/itex]C[itex]_{11}[/itex][itex]\phi^{'}_{1}[/itex]d[itex]\phi[/itex][itex]^{'}_{1}[/itex] + [itex]\int[/itex][itex]^{\phi_{1f}}_{0}[/itex]C[itex]_{21}[/itex][itex]\phi[/itex][itex]_{2}[/itex]d[itex]\phi[/itex][itex]^{'}_{1}[/itex]

Now the first term there makes perfect sense to me, but I have no idea why that second integral is there. Any help understanding what that represents would be appreciated.

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# Homework Help: Proof that Capacitance Coefficient C21=C12

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