|MadSci Network: Physics|
I do not quite understand to which exact physical setup you are referring, but I think I can answer your question just by describing how to calculate the force between two dipole (bar) magnets if you place the poles close together:
+---------+ +---------+ + S | N + + S | N + +---------+ +---------+Let us assume that the magnets are homogeneously magnetized along their axis with some constant magnetization M (actually a vector parallel to the magnet's axis), and that M is constant even under the influence of an external magnetic field. Such a material is called a hard ferromagnet. Using the Maxwell Field Equations, one can see that due to the fact that the curl of the magnetic field outside the magnets is zero, the magnetic field H is a gradient field, i.e. there exists some scalar potential whose gradient is equal to H. How do we get this scalar potential? It turns out that the laplacian of the scalar potential is proportional to the divergence of the magnetization M. As M is constant inside and outside the material, the only contribution to the gradient comes from the surface of the magnets, and, in this case, especially from the abutting faces.
Now we have reduced the problem to an electrostatic one: It is totally equivalent to the problem of calculating the force between the plates of a capacitor! The (magnetic) `charge density' on the plates is equal to the normal component of the magnetization just below the surface. It should now be easy for you to solve the problem, using elementary electrostatics. The resulting force is proportional to the magnetization and the abutting surface area of the bar magnets, assuming of course that the poles are close enough together as to avoid significant `stray fields'.
This is one of the rare cases where one is able to establish a one-to-one correspondence of a magnetic to an electrostatic problem, by introducing a magnetic surface charge density. We know, of course, that magnetic charges (monopoles) are highly hypothetical, but that is of no concern here.
The treatment of hard ferromagnets is described in detail in many textbooks, e.g. in J.D. Jackson: Electrodynamics.
Hope that helps,
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