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hylander4

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**"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.

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