Griffith's Electrodynamics 6.8

[monkeykoder]

Homework Statement

A long cylinder of radius R carries a magnetization $$\vec{M}=Ks^{2}\hat{\phi}$$ where k is a constant, s is the distance from the axis, and $$\hat{\phi}$$ is the usual azimuthal unit vector. Find the magnetic field due to $$\vec{M}$$ for points inside and outside the cylinder.

Homework Equations

$$\vec{J}_{b}=\nabla\times\vec{M}$$
$$\vec{K}_{b}=\vec{M}\times\hat{n}$$
Formula for the Vector potential That I can't seem to get to work in tex

The Attempt at a Solution

$$\vec{J}_{b}=3k\sqrt{x^{2}+y^{2}}\hat{z}$$
$$\vec{K}_{b}=3k(x^{2}+y^{2})\hat{z}$$

Truthfully my biggest annoyance right now is formulating the separation vector (I know I'm lame) then a little plug and chug and I'm done but I wouldn't mind someone checking my work (probably wrong...)

 

[jdwood983]

You're told that it is a cylinder, why are you using Cartesian coordinates?

$$\vec{J}_b=\vec{\nabla}\times\vec{M}=\frac{1}{s}\,\frac{\partial}{\partial s}\left(s\,ks^2\right)\hat{z}=3ks\hat{z}$$

Similarly for $\vec{K}_b$:

$$\vec{K}_b=\vec{M}\times\hat{n}=-kR^2\hat{z}$$

You can then use Ampere's Law to find the magnetic field.

 

[nickjer]

You might want to double check your bound surface current. Also, try to keep everything in cylindrical coordinates. Also, Ampere's law will be real useful for this problem.

EDIT: Seems jdwood983 beat me to it :(

 

[monkeykoder]

You might want to double check your bound surface current. Also, try to keep everything in cylindrical coordinates. Also, Ampere's law will be real useful for this problem.

EDIT: Seems jdwood983 beat me to it :(

Don't have Ampere's law to work with and not allowed to use cylindrical coordinates for some reason.

 

[nickjer]

Does it specifically say you can't use either of those? And what kind of question uses cylindrical coords and asks you afterwards not to use them.

 

[monkeykoder]

Ampere's law for polarized objects is about 3 sections ahead of where the problem is located in the book (not that that would usually stop me...).

 

[nickjer]

I have griffith's and Ampere's law can be found in section 5.3.3. Long before your problem.

 

[nickjer]

This problem is very similar to a long straight current carrying wire. Except the current density is radially dependent and you have a negative surface current.

 

[monkeykoder]

If I'm not mistaken there is some reason not to use that statement of ampere's law in cases of magnetization.

 

[nickjer]

If you are determined not to use Ampere's law (which I advise against) then you will have to use the Biot-Savart law. But you will just end up with the same answer.

 

[monkeykoder]

I'm pretty sure if I were allowed to use it in cases of magnetization there wouldn't be a section "6.3.1 Ampere's Law in Magnetized Materials"

 

[nickjer]

Well you aren't using H when solving for Ampere's law in this problem. When applying Ampere's law for this problem you only worry about bound currents and treat it as the total current. But as I said before you can always use the Biot-Savart law, just be very careful when solving it.

 

[nickjer]

Well I see what you mean since there is a problem in 6.3 that is roughly the same and asks you to solve it using the method in 6.2 for part (a) and using Ampere's law for part (b). I seriously don't think you will be deducted points for using ampere's law early. But you can always solve the magnetic fields using integrals just in case. Just be very careful :)