# Electric and Magnetic field of an electron.

## Homework Statement

Picture the electron as a uniformly charged spherical shell, with charge e and radius R, spinning at angular velocity $$\omega$$.

(a) Calculate the total energy contained in the electromagnetic fields.

## Homework Equations

$$\oint \vec{B} \bullet d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \int \frac{d\vec{E}}{dt} \bullet d\vec{a}$$

$$U_{em} = \frac{1}{2} \int \epsilon_0 E^{2} + \frac{1}{\mu_0} B^{2} d\tau$$

## The Attempt at a Solution

So the solution should just be a matter of plugging in to the second equation. My dilemma is actually in finding the fields.
E(r<R) = 0 by symmetry,
$$E(r>R) = \frac{1}{4 \pi \epsilon_0} \frac{e}{r^{2}} \hat{r}$$
Per the maxwell amperian loop I get B(r<R) = 0 but this is apparently incorrect and I don't understand why.
And B(r>R) = dE/dt (4pi r^2)/(2pi r).

My problems seem to be with determining the magnetic field. Does anyone see what I'm missing here?

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at first glance:

So the solution should just be a matter of plugging in to the second equation
Not exactly: the electric field, for example, is not constant everywhere so to find its size (squared) you need to integrate all the space it reaches.

Per the maxwell amperian loop I get B(r<R) = 0 but this is apparently incorrect and I don't understand why.
I don't think there is an easy amperian loop here, you don't have the right symmetry.
can you show me your loop?

To find the magnetic field you probably need to integrate all the surface and use, for example, Biot-Savart. (by symmetry the magnetic filed will be in the direction of the spin axis)

P.S. I think your electric field are correct.