1. The problem statement, all variables and given/known data
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
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.
2. Relevant 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/ph...SepVarsCyl.pdf
on page two.
Summed from n = 1 to infinity
3. 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.