Express electrostatic energy in terms of both charges

[jfy4]

Homework Statement

For two concentric conducting spheres (radius a and b, b>a) that form a capacitor with charge q on the inner sphere and -q on the outer sphere, express the electrostatic energy in terms of q and -q and the potential difference between them.

Homework Equations

Gauss's Law, the equation for electrostatic potential, the equation for the energy stored in a static electric field.

The Attempt at a Solution

I have the field
$$\vec{E}=\frac{q}{4\pi\epsilon_0 r^2}\hat{r}$$
between the conductors, but when I calculate the energy, should I only integrate between the spheres?
$$W=\frac{\epsilon_0}{2}\int E^2 d\tau=\frac{q^2}{8\pi\epsilon_0}\int_{a}^{b}\frac{1}{r^2}dr=\frac{q^2}{8\pi\epsilon_0}\left( \frac{1}{a}-\frac{1}{b} \right)$$

Then to express it in terms of the original charges and the potential difference,
$$\Delta\phi=\frac{q}{4\pi\epsilon_0}\left(\frac{1}{a}-\frac{1}{b}\right)$$
then
$$W=\frac{q}{2}\Delta\phi$$
but how would I write this in terms of the charges? Does it want me to split it up like
$$q=\frac{1}{2}(q-(-q))$$
and put this in the above equation?

Thanks in advance,

 

[I like Serena]

between the conductors, but when I calculate the energy, should I only integrate between the spheres?

Isn't the electric field zero at every other point?

You have calculated a 1-dimensional integral here.
Shouldn't you integrate over 3-dimensional space?

but how would I write this in terms of the charges? Does it want me to split it up like
$$q=\frac{1}{2}(q-(-q))$$
and put this in the above equation?

Thanks in advance,

The phrasing of your problem implies that q and -q are equal and opposite.
It's a bit weird that the problem asks for you to use q and -q, since "just" q should suffice.

 

[jfy4]

Yeah that seems reasonable, from Gauss' law.

I integrated the phi and theta parts in the background, sorry I wasn't more explicit, the constants out front should already reflect those integrals being done.

yeah, I don't know how to interpret the question about the q and -q, that's why I brought it here lol.