[electrodynamics] rotating hollow sphere

[Herbststurm]

Hi,

a homogeneous charged hollow sphere with the radius R and the complete charge Q is rotating with constant angular velocity omega around a fixed axle.

How to calculate the magnetic field at the north pole?

I have this hints:

The ring current I with the radius small r excites on the ring axle with the distance s from the center a magnetic field of

B = 2 \pi I \frac{r^{2}}{\sqrt{(r^{2}+s^{2})^{3}}}

okay, now I have some questions about that:

1.) "ring axle with the distance s from the center" I can't imagine this. Could you give me some hints how to draw a sketch please?

2.) How to calculate this magnetic field at the north pole? Please only hints. I will do it on myself.

3.) Is it important that the sprehe is rotating? I guess it could be just wrote to confuse people because nobody asked here about gyromagnetic relation.

thanks

greetings

 

[Thaakisfox]

Yes it is important that the sphere is rotating, otherwise there wouldn't be any magnetic field, since the whole configuration would be stationary, with no charges moving. But since it is rotating the charges are moving (in our frame), hence they represent a current. So you have to see how much current does a very little segment of the sphere represents. That is you have to break up the sphere into "stripes" and calculate the charge of one stripe. Now you know that this dQ charge goes once in every period. Hence the current:

dI=\frac{dQ}{T}=\frac{\omega}{2\pi}dQ

Draw a circle of radius r. Now draw a line perpendicular to the plane of the circle, through its center. The distance s is measured on this line. So that s=0 means the center of the circle itself.

So after you have calculated the dQ charge on one "stripe" of the sphere you have the current.

Now plug this into get the amount this stripe contributes to the magnetic field at the north pole (the formula you gave is not fully correct):

dB=\frac{\mu_0}{2}\frac{r^2 dI}{(r^2+s^2)^{3/2}}

s and r and dQ can be expressed with the Radius R and the polar angle.

Now integrate this and you are done :D

 

[Herbststurm]

Hello,

thanks for your help. I have two questions about that:

1.) Where does this formula of the magnetic field come from?

2.) Why is the given formula wrong and your formula is correct? What is the reason that the given formula should be wrong?

Thanks

Greetings

 

[Thaakisfox]

This formula can be derived from the Biot-Savart law.

Ok, the formula you gave, has the correct functional relationship, so basically its ok, just if we are working in SI, then the constant factor, should be what I wrote..

 

[Herbststurm]

Now it is clear. I forgot Biot-Savarts law but we discussed it in lecture.

We are using the cgs system from Gauß.

I will calculate it and post my solution.

thanks and greetings :)