The force between two conductors (Duffin)

[Smith987]

Homework Statement

I have been reading electricity and magnetism by Duffin in order to understand my first year EM course. So far it has been straightforward- however, I've hit a wall when it derives the force between 2 conductors (p.117 section 5.7 if you have the book!). The book says:

Homework Equations

Force along x = - dU/dx with Q held constant, where U=electrical energy stored(I accept this)

Elementary increment dU= 0.5QdV (i accept)
=0.5 Q^2 * d(1/c) (I accept)
=-0.5 V^2 * dC.

This I do not accept, as in the first equality, it implies V is changing (dV) yet in the last equality V is constant? I'm also unsure as to where the minus sign cam from but I assume it was a misprint.

The Attempt at a Solution

The only answer I can come up with is that the equation was meant to be
dU= 0.5 d[ V^2 * C]. But if I use this for the example of a parallel plate capacitor I get a sign error.

Please help!

 

[Smith987]

I think I solved the problem- when you consider dU as 0.5 *d(V^2 * C), this is equal to

0.5 * (V + dV)^2 * (C + dC) - 0.5*V^2 * C, which by using dv/dc= -Q/(C^2) leaads to the answer given. And pleasingly it gave me that annoying negative sign :)

I would still like a pat on the back though.