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

 

[kerimek]

Any help would be greatly appreciated.

Thank you for your question. It seems like you are on the right track by treating the concentric spheres as a parallel plate capacitor. To start, let's calculate the electric field between the spheres. As you mentioned, we can use the equation E = (Q x d) / (A x ε0), where Q is the charge on the inner sphere, d is the distance between the spheres (0.5cm in this case), A is the area of the plates, and ε0 is the permittivity of air.

To find the area of the plates, we can use the formula for the surface area of a sphere, A = 4πr^2. For the inner sphere, this would be A1 = 4π(10cm)^2 and for the outer sphere, it would be A2 = 4π(10.5cm)^2. Plugging these values into the equation for E, we get E = (5nC x 0.5cm) / (4π(10cm)^2 x ε0) = 1.25 x 10^4 V/m.

Now, to calculate the energy density of the electric field, we can use the equation u = 0.5 x ε0 x E^2. Plugging in our calculated value for E, we get u = 0.5 x (8.85 x 10^-12 C^2/Nm^2) x (1.25 x 10^4 V/m)^2 = 6.57 x 10^-5 J/m^3.

Finally, to find the total energy stored in the electric field between the spheres, we can use the equation U = u x V, where V is the volume of the space between the spheres. Since the space between the spheres is a thin layer of 0.5cm, we can approximate the volume as a cylinder with a height of 0.5cm and a radius of 10.5cm. This gives us V = π(10.5cm)^2 x 0.5cm = 55.12 cm^3 = 5.51 x 10^-5 m^3.

Plugging in our calculated values for u and V, we get U = (6.57 x 10^-5 J/m^3) x (5.51 x 10^-5 m^