Current of Spinning sphere of constant voltage

[Z90E532]

Homework Statement

Find the magnetic dipole moment of a spinning sphere of voltage $V$ and radius $R$ with angular frequency $\omega$.

Homework Equations

The Attempt at a Solution

To find the dipole moment, we need to do $I \int d \vec{a}$, which would be $I 4 \pi R^2 \hat{r}$, but I need to find the current knowing only the voltage and angular velocity. I'm not sure how to proceed.

Edit: Thought about this again and noticed a mistake. What I need to do is integrate to add up all the current rings on the sphere. So I want to be able to do something like (the sphere is spinning along the z axis):

$$\vec{m} = \sigma R^3 \omega \int ^{2 \pi} _{0} \int ^{\pi} _{0} \sin^3 \theta d \theta \phi \hat{z}$$ .

Where $I = \sigma R \omega \sin \theta$ and $\vec{A} = R^2 \sin^2 theta$

 

[Orodruin]

Your expression for the dipole moment is the dipole moment of a conductor carrying a current I. You have a more general current density. It is also unclear what you mean by $\hat r$.

Hint: You know the potential so you should be able to find the charge density on the sphere. From the charge density and rotation of the sphere, you should be able to deduce the current density.

 

[Z90E532]

Your expression for the dipole moment is the dipole moment of a conductor carrying a current I. You have a more general current density. It is also unclear what you mean by $\hat r$.

Hint: You know the potential so you should be able to find the charge density on the sphere. From the charge density and rotation of the sphere, you should be able to deduce the current density.

I corrected my mistake I think, but I still am not sure how to find the surface charge density from only the voltage. I think my brain stopped working.

 

[Orodruin]

I corrected my mistake I think, but I still am not sure how to find the surface charge density from only the voltage. I think my brain stopped working.

Did you perhaps do electrostatics before magnetostatics?

 

[Z90E532]

Did you perhaps do electrostatics before magnetostatics?

Yeah, I think I was just having a brain malfunction... it's pretty late here.

I think this is the right approach: For a surface charged sphere: $\vec{E} = \frac{R^2 \sigma}{\epsilon _{0}r^2}$ then we have $V = - \int ^{R} _{\infty} \frac{R^2 \sigma}{\epsilon _{0}r^2} dr =\frac{R^2 \sigma}{\epsilon _{0}R}$ which gives $\sigma = \frac{V \epsilon _{0}}{R}$.

Edit: Just heard from a friend this is the correct solution, although my answer is missing a factor of R somehow. I have $I = \sigma \omega \sin \theta R$ and $da = \pi R^2 \sin ^2 \theta$, putting them together I get $\vec{m} = \sigma \omega R^3 \int ^{\pi} _{0} \sin ^3 \theta d\theta$

 

[Orodruin]

although my answer is missing a factor of R somehow

What is the width of the circular loop between $\theta$ and $\theta + d\theta$?

Edit: Dimensional analysis is also a powerful tool to figure out where a factor of $R$ is missing ...