Proving C12=C21: General Capacitance

[Pushoam]

Homework Statement

Consider a system of N isolated conductors,with arbitrary shape and position.
We specify the charge Q_i on the i_th conductor for each i.
The capacitance C_ij is:
Q_i= ΣC_ijV_j, sum over j.
To prove:
C_ij = C_ji
Hint: Consider how much energy is needed to start with the system uncharged, then add charge Q_i to conductor i, and then add charge Q_j to conductor j. Then consider starting again with the system uncharged, and performing these operations in the opposite order. That is, add charge _jj to conductor j, and then Q_i to conductor i. Then think about how to use your answers to prove the desired result.]

Homework Equations

W=QV

The Attempt at a Solution

For N=2,
To show: C₁₂ = C₂₁
No energy is needed to put Q₁ on the conductor no.1.
This charge Q₁ creates a surface charge density and potential V ₂ on the conductor no.2.
To put charge Q₂ on the conductor no.2 takes energy Q₂V ₂.
Similarly, putting Q₁ on the conductor no.1 after putting Q₂ on the conductor no.2 takes energy Q₁V ₁.
Is this correct till now? What to do next?

 

[haruspex]

No energy is needed to put Q1 on the conductor no.1.

That cannot be true. It will have some potential as soon as you start putting charge on it.

To put charge Q2 on the conductor no.2 takes energy Q2V 2.

Similarly, the potential will change as you add charge.