Fun Magnetic Scalar Potential Problem

[hylander4]

"Fun" Magnetic Scalar Potential Problem

Homework Statement

An infinite cylindrical shell of radius b is placed inside a constant field B which points along the upwards z-axis. A second cylindrical shell of radius a<b is placed inside the first cylindrical shell, and the volume from b>r>a is filled with a paramagnetic material of permeability u. Find the magnetic field everywhere.

Homework Equations

H = B/(u_0) + M

H = -grad W

Laplacian W = -grad M

W is continuous over all boundaries.
The change in dW/dr over a boundary is equal to the negative change in Magnetization over the boundary.

Cylindrical laplace equation solution (From my undergraduate E+M notebook)

W(r, phi) = D_0 + A_0*(a+b*phi) +
$\EPSILON$ [r^n + (A_n*r^-n)]*[B_n*cos(n*phi)+C_n*sin(n*phi)]

Another version of this equation can be found here http://www.cord.edu/faculty/gealy/physics315/SepVarsCyl.pdf" on page two.

Summed from n = 1 to infinity

The Attempt at a Solution

Since there's no free current in this situation, I tried using magnetic scalar potential to solve this problem. Unfortunately, I end up with too many variable in the proposed Laplace equation solutions that I need to create to use the boundary conditions.

In my main attempt I had four boundary conditions and seven types of variables.

My main problem is that I need to find the H field to find the B-field, but in order to find the H-field, I also need a function for the paramagnetic material's Magnetization density M.

The forum wants me to post my attempts, but I've already filled three pages of notebook paper with failed algebra and I doubt that that would be constructive. I'm not really looking for an exact solution, I really just need general guidance.

 

[grey_earl]

Usually you have B = μ0 (H + M) = μ0 μ H, which determines M to be (μ-1) H. In vacuum, of course, μ = 1.