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:

- - -

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!