Slightly philosophical question about magnetism

[pterid]

Hello. I am a new poster; I hope I am correctly observing the forum rules.

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 $$\textbf{M}$$, which can be identified with the induced magnetic moment per unit volume. Say the susceptibility, $$\chi$$, is uniform across the sample. Must the magnetisation also be uniform across the sample?

Is it possible to find a solution in which $$\textbf{M}$$ varies? Or can we show that there are no such solutions? Are there geometries for which $$\textbf{M}$$ 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 $$\textbf{B}$$, $$\textbf{H}$$, $$\textbf{M}$$ exists. However, ellipsoids are a bit of a special case, as the $$\textbf{B}$$ & $$\textbf{H}$$ fields also turn out to be uniform in the sample, and I have reason to think that a uniform-$$\textbf{M}$$ 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!

 

[pam]

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 $$\textbf{M}$$, which can be identified with the induced magnetic moment per unit volume. Say the susceptibility, $$\chi$$, is uniform across the sample. Must the magnetisation also be uniform across the sample?

The magnetization will NOT usually be uniform, because the shape of the object is important.
For a linear, isotropic, homogeneous susceptibility, the calculation of M is mathematically the same as that of P for a dielectric in an electric field. That case is treated in most EM texts as a boundary value problem. It turns out that, for a sphere, M is constant within the sphere, but for any other shape, it is r dependent.

 

[pterid]

The magnetization will NOT usually be uniform, because the shape of the object is important.
For a linear, isotropic, homogeneous susceptibility, the calculation of M is mathematically the same as that of P for a dielectric in an electric field. That case is treated in most EM texts as a boundary value problem. It turns out that, for a sphere, M is constant within the sphere, but for any other shape, it is r dependent.

Yes - for any ellipsoid (not just a sphere) the induced magnetisation is uniform; in other cases it cannot be. I think I've sorted that out now - thanks!