# General capacitance

1. May 26, 2017

### Pushoam

1. The problem statement, all variables and given/known data
Consider a system of N isolated conductors,with arbitrary shape and position.
We specify the charge Qi on the ith conductor for each i.
The capacitance Cij is:
Qi= ΣCijVj, sum over j.
To prove:
Cij = Cji
Hint: Consider how much energy is needed to start with the system uncharged, then add charge Qi to conductor i, and then add charge Qj 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 Qi to conductor i. Then think about how to use your answers to prove the desired result.]
2. Relevant equations
W=QV

3. 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?

2. May 28, 2017

### haruspex

That cannot be true. It will have some potential as soon as you start putting charge on it.
Similarly, the potential will change as you add charge.