- #1
jfy4
- 649
- 3
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
[tex]
\vec{E}=\frac{q}{4\pi\epsilon_0 r^2}\hat{r}
[/tex]
between the conductors, but when I calculate the energy, should I only integrate between the spheres?
[tex]
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)
[/tex]
Then to express it in terms of the original charges and the potential difference,
[tex]
\Delta\phi=\frac{q}{4\pi\epsilon_0}\left(\frac{1}{a}-\frac{1}{b}\right)
[/tex]
then
[tex]
W=\frac{q}{2}\Delta\phi
[/tex]
but how would I write this in terms of the charges? Does it want me to split it up like
[tex]
q=\frac{1}{2}(q-(-q))
[/tex]
and put this in the above equation?
Thanks in advance,