Undergrad Magnetization of a material with linear susceptibility

[hylander4]

Fairly simple question:

If a highly permeable material (of any shape) is placed in a uniform magnetic field, will the material's magnetization density always line up with the uniform magnetic field, or do I have to take into account the fields that are created as the material becomes magnetized?

The material is paramagnetic.

I've wasted so much time re-reading my E+M textbook to figure this out, but it never seems to tell us how to compute the magnetization created by magnetic fields. We're generally just given objects that are already magnetized.

 

[chrisbaird]

If the magnetic material is linear, then yes the magnetization will line up with the applied field. That is the definition of linear. For linear materials, M, B, and H are trivially related. So if you find one, then you find all of them. You would solve for the fields separately inside and outside the material as if the material were not even there, but being careful to use the susceptibility of the material inside and the susceptibility of free space outside, then match up the two regions using boundary conditions.

Conceptually, you can think of the applied field as inducing bound net electric currents in the materials that give rise to the magnetization field.

Here is a http://faculty2.uml.edu/cbaird/all_homework_solutions/5magnetic_cylinder.pdf" of an originally unmagnetized cylinder being placed in an originally uniform external magnetic field.

 

[tiny-tim]

but isn't https://www.physicsforums.com/library.php?do=view_item&itemid=110", still linear but different for different directions within the material?

so the material's magnetization density lines up with the magnetic field only if that is parallel to one of the principal axes of the material?

 

[hylander4]

That makes sense. The field inside the paramagnetic material will follow the applied field and thus change the direction/magnitude of the field outside the paramagnetic material.

For some reason I was thinking that a uniform field on the inside of the material would require the field outside the material to stay pointing in the same direction (which I knew was impossible, since I'm actually designing magnetic shielding...and magnetic shielding wouldn't shield if it didn't change field line directions).

Thanks a lot for your help!

 

[chrisbaird]

Yes, I meant the simplest textbook case of a linear, isotropic, homogenous material. But that is a mouthful, so unless otherwise stated, "linear" often implies all of this.

 

[hylander4]

I've hit a snag. Using magnetic scalar potential, I calculated the B-field inside a hollowed cylinder filled with linear/isotropic/homogenous paramagnetic metal (there is a shell of known thickness filled with paramagnetic material, and then an empty inner cylinder). The magnetization lines up with the calculated B-field within the material (it has to), but the B-field within the material doesn't line up with the applied uniform magnetic field.

The cylinder inside the paramagnetic shield, which is a vacuum, is the only area where the resulting magnetic field lines up with the applied magnetic field.

This is annoying, because it means that I have to solve the Laplace equation for the exotic shape of the shield I'm actually using, and I'm pretty sure than I can only solve it numerically.

 

[chrisbaird]

Solving for magnetostatic fields numerically is actually not that bad. If using the magnetic scalar potential, you can use a http://en.wikipedia.org/wiki/Relaxation_%28iterative_method%29" type approach which is fairly easy to implement.