- #1

pterid

- 8

- 1

I have a slightly philosophical question about magnetism. It seems to be very simple, but I am struggling to find myself a convincing answer. Here it is:

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If you place a piece of magnetically susceptible material (for example, an ideal isotropic linear diamagnet or paramagnet) an an uniform external magnetic field, it will acquire a magnetisation [tex]\textbf{M}[/tex], which can be identified with the induced magnetic moment per unit volume. Say the susceptibility, [tex]\chi[/tex], is uniform across the sample. Must the magnetisation also be uniform across the sample?

Is it possible to find a solution in which [tex]\textbf{M}[/tex] varies? Or can we show that there are no such solutions? Are there geometries for which [tex]\textbf{M}[/tex] must be non-uniform?

Standard texts on magnetism that I have seen usually tend to approach a particular geometry (usually an ellipsoid),

*assume*uniform magnetisation, and then show that a solution for the magnetic fields [tex]\textbf{B}[/tex], [tex]\textbf{H}[/tex], [tex]\textbf{M}[/tex] exists. However, ellipsoids are a bit of a special case, as the [tex]\textbf{B}[/tex] & [tex]\textbf{H}[/tex] fields also turn out to be uniform in the sample, and I have reason to think that a uniform-[tex]\textbf{M}[/tex] solution might not work for other geometries e.g. a bar (cuboid).

I have some thoughts on this, but I fear trying to explain them would only confuse the issue. So I would appreciate your thoughts - cheers!