Electric and Magnetic field of an electron.

[Bhumble]

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 into 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?

 

[gomunkul51]

at first glance:

So the solution should just be a matter of plugging into 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.

may be helpful: www.demul.net/frits/images/ppt/B-field_homog_sphere.ppt[/URL]

 

[Bhumble]

So I found an example problem in the book that flat out gave the inner magnetic field and gave the auxiliary field for the outside so I took curl to get the magnetic field outside.
The part that confuses me (possibly because I haven't taken an amperian loop since we recently changed ampere's law with maxwell's correction) is that you could take an amperian loop anywhere to account for the magnetic field.
So I imagined the sphere as a series of rings each with a current flow and I took a square amperian loop through the center of the shell and closing outside of the shell. So the current enclosed should be the sum of the current through each ring. And like you were saying should be in the direction of the spin axis by the right hand rule. Then the only way that I could think to differentiate the inner and outer is by taking the boundary that an amperian loop inside the shell would have no current and since E = 0 then the total magnetic field should be 0. I'm still not sure where my logic is incorrect.