Magnetic Field of Rotating Cylinder w/ Linear Polarization

[thatmaceguy]

Homework Statement

Long dielectric cylinder of radius R carries a built-in electrostatic polarization P
that is linearly proportional to the distance to the axis, P=$$\alpha$$r, P is directed along
the radius-vector r. Cylinder is being rotated around the axis with angular velocity
$$\omega$$. Find the magnetic field B on axis.

Homework Equations

The Attempt at a Solution

Conceptually I think I understand the problem, I'm just having a hard time (as usual) setting up the math.

Basically, I should be able to solve for the bound surface and volume charge, then find the bound surface and volume current from those bound charges and the angular velocity. After that I think I should able to find the magnetic vector potential (A) and finally B since it is equal to the curl of A.

I've been referencing Example 5.11 and problem 5.13 and from Griffiths but as I've implied, I have a really hard time assembling the actual math.

Thanks for any help/guidance you can give.

 

[gabbagabbahey]

Basically, I should be able to solve for the bound surface and volume charge, then find the bound surface and volume current from those bound charges and the angular velocity. After that I think I should able to find the magnetic vector potential (A) and finally B since it is equal to the curl of A.

That sound like as good a plan as any...start with the bound surface and volume charges. What are the equations relating them to the polarization? Calculate them from those equations (show your steps if you get stuck!).

 

[thatmaceguy]

Hmm, yeah I meant to include that I had already done that in the original post. My apologies.

Here's what I have for that.

Surface Bound Charge = $$\alpha$$R (Alpha x R)

Volume Bound Charge = -2$$\alpha$$ (-2 x Alpha)

(sorry, not terribly familiar with latex)

I am not certain though how to move forward from here.

 

[gabbagabbahey]

Hmm, yeah I meant to include that I had already done that in the original post. My apologies.

Here's what I have for that.

Surface Bound Charge = $$\alpha$$R (Alpha x R)

Volume Bound Charge = -2$$\alpha$$ (-2 x Alpha)

Good.

I am not certain though how to move forward from here.

Now you want to calulate the volume and surface current densities... $\textbf{J}_b=\rho_b\textbf{v}(\textbf{r})$ and $\textbf{K}_b=\sigma_b\textbf{v}(\textbf{r})$...so, what is the velocity of any given point in the cylinder if it is rotating with angular speed $\omega$?

 

[paranormal]

I would suggest breaking down the problem into smaller steps and using known equations to solve for the magnetic field. First, you can use the given information to calculate the bound surface and volume charge, as well as the bound surface and volume current. Then, using the equation for the magnetic vector potential (A), you can solve for A. Finally, taking the curl of A will give you the magnetic field B. It may also be helpful to reference examples and problems from textbooks or online resources, and to seek guidance from a professor or colleague if needed.