Charged concentric metal spheres

[ChrisBaker8]

Homework Statement

Two concentric metal spheres have radii R1 =10cm and R2=10.5cm. The inner sphere has a charge of Q=5 nC spread uniformly over its surface, and the outer sphere has charge −Q spread uniformly over its surface.

Calculate the total energy stored in the electric field between the spheres. (Hint : the spheres can be treated as flat parallel slabs separated by 0.5cm)

Homework Equations

U = 0.5 x C x (V^2)
=(Q x V)/2

Energy Density=1/2 x epsilon_0 x E^2

The Attempt at a Solution

None, unless confused scribbles count. I know I can treat this as a parallel plate capacitor (from the hint), but that doesn't seem to help me.

I've been looking though my textbooks for hours but I can't find a clear way to work this out. I tried using (Q x d) / (A x Permittivity of air) to work out the electric field strength (E), but I didn't know which area to use for A.

If I can work out the energy density of the field, I can multiply it by the volume of the space between the spheres to find the energy stored, but again, I can't work out E.

 

[Andrew Mason]

Use Gauss' law or Coulomb's law to calculate the field between the spheres:

$$\int \vec{E}\cdot dA = \frac{q_{encl}}{\epsilon_0}$$

$$E = Q/4\pi \epsilon_0r^2$$

It is not quite, but approximately equal over the .5 cm distance between spheres.

The potential difference between the spheres is V = Ed (in volts or joules/coulomb).

Since potential difference is the energy in joules per coulomb of charge: U = QV

AM