I have been admiring the demonstration of the spherical shell in a magnetic field produced by Michael Trott:
I would like to be able to do produce the same results, but for a rectangular/square shell. A colleague of mine has been able to solve the spherical case using separation of variables, but I don't think this is possible for the rectangular shell. This leads me to believe that NDsolve might be the best tool for this problem.
The difficulty with using NDsolve is in setting the boundary conditions. As I see it, there are three separate regions
Each of these regions has it's own solution to the laplace equation. The boundary conditions between these solutions are that the magnetic flux density is continuous through each boundary.
Would it be possible to solve the laplace equation in each of these three regions and to couple the boundary conditions?
I would recommend checking out these blog posts:
And generally blog posts by Michael Trott.