Proof that Capacitance Coefficient C21=C12

[Opus_723]

Homework Statement

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

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.

 

[rude man]

The reason no one has responded is - what the heck is "coefficient of capacitance"?
We're all familiar with mutual inductance being M₁₂ or M₂₁, but I never heard of an equivalent electrostatic "mutuality" between two conductors. There is capacitance between two conductors, is all.

 

[Opus_723]

Well, that's okay. I managed to figure it out on my own. The second term is the work that must be done in order to hold one conductor at a particular voltage while you're collecting charge on the other. I was just confused because it turns out to be zero for this first step, but they included the term anyway. It only matters once one conductor is charged and you start to increase the voltage on the other.

In my book, they're using coefficient of capacitance to relate the charges on conductors to their potentials, or to the potentials on other conductors. So that C11 = Q1/V1, C12 = Q1/V2, C21 = Q2/V1, and C22 = Q2/V2, like you're starting from boundary values. The point was to prove that C12 and C21 are always the same, so that there is just one capacitance between any two conductors, as you said.