# Homework Help: Griffith's Electrodynamics 6.8

1. May 3, 2010

### monkeykoder

1. The problem statement, all variables and given/known data
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.

2. Relevant 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
3. 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...)

Last edited: May 3, 2010
2. May 3, 2010

### 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.

Last edited: May 3, 2010
3. May 3, 2010

### 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 :(

4. May 3, 2010

### monkeykoder

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

5. May 3, 2010

### 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.

6. May 3, 2010

### 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...).

7. May 3, 2010

### nickjer

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

8. May 3, 2010

### 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.

9. May 3, 2010

### monkeykoder

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

10. May 3, 2010

### 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.

11. May 3, 2010

### 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"

12. May 3, 2010

### 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.

13. May 3, 2010

### 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 :)