Question

Imagine that for the purpose of some experiment you wished to create a spherical volume (radius R) within which there was a uniform magnetic field B . What current distribution on the surface of the sphere would be required? If you instead wanted a spherical volume of uniform electric field, how might you create that?

Answer 1

mmm thats intersting. First of all a magnetic field has a direction, so we hace to take that in account. a constant field has one direction all the time, so the current cant be symetric.also if we take little loops that are not parllel they wiil meet and cancel each other in some places. I can think of one nice idea. Think of a firt loop of current as the biggest one. the next two besides it parallel to it but with less current, and so on the whole sphere, it will be a little like a coil. a coil is a good example of an almost uniform magnetic field (it is uniform if the coil infinite). What do u say about it? The other example of a uniform magnetic field is the field from an infinite sheet of currnt. mmm for the second question a uniform electrical field exists betwenn the plates of a capacitor. but in a equally charged sphere there is no field. I have an idea I will post it if u like.
Enjoy! tnx for the nice question :)

Answer 2

I like how you think! The answer that I came up with is the surface current equivalent to a rotating sphere of radius R with static surface charge (\sigma), rotating at (\omega).

Then the magnetic field from such a configuration is,

\[ \mathbf{B} =\frac{\mu_0}{4\pi}\frac{8\pi}{3}\mathbf{M}\] where M is the magnatization given by,
\[ \mathbf{M}=\frac{\mathbf{m}}{4/3\pi R^3}=R\sigma\omega \sin\theta\hat{\phi}\]

Would you also try the case of the electric field?

Answer 3

Nice! a great solution! how did u calculate it? I don't have strong skills with some calculations :)
We had basiclly the same Idea :) because if u look at a rotating sphere its the same as saying there are a buch if currnt loops in the same way and I goes down like 1/r !

about conatant electric field... I had an Idea to do something similar with loops but it turned out the wrong way, but I think it got me to the answer!!
the thing is that I thought about differnt distributions of charge and than It occured to me that it cant be (uniform field) if all the charge is with the same sign all over the sphere.

So u need negative and positive surface charge.
My thought is of a sphere- half of it charged with positive surface charge the second half negative.
I think this will give a uniform field, because of the similar example of the uniform charged sphere in which the field is zero everywhere. also the symtery is good u can tell if ur up or down but not if the sphere has spun around the axis.

What do u say?
Tnx again for the nice question!