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## Homework Statement

a. Calculate the energy density of the electric field at a distance r from an electron (presumed to be a particle) at rest.

b. Assume now that the electron is not a point but a sphere of radius R over whose surface the electron charge is uniformly distributed. Determine the energy associated with the external electric field in vacuum of the electron as a function of R.

## Homework Equations

$$ u_e = 1/2\epsilon_0E^2 $$

## The Attempt at a Solution

a. The electric field of an electron can be assumed to be the same as a point charge, that is

$$ E = \frac{q}{4\pi\epsilon_0r^2} $$

Since

$$E^2 = \frac{q^2}{16\pi(\epsilon_0)^2r^4} $$,

$$u_e = \frac{q^2}{32\pi \epsilon_0 r^4} $$

b. We use Gauss's law to find the electric field of this sphere.

$$ EA = \frac{\sigma A}{\epsilon_0} $$

So that

$$ E = \frac{\sigma}{\epsilon_0}$$, where $$\sigma$$ is the charge per unit area.

So the energy density is

$$ u_e = \frac{1}{2}\epsilon_0\frac{\sigma}{(\epsilon_0)^2} = \frac{1}{2} \frac {\sigma^2}{\epsilon_0} $$

The total energy is therefore the energy density multiplied by the volume, so

$$ U = \frac{4\sigma^2\pi R^3}{6\epsilon_0} $$

Is this correct?