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Slightly philosophical question about magnetism

  1. Feb 2, 2008 #1
    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 [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!
  2. jcsd
  3. Feb 5, 2008 #2


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    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.
  4. Feb 8, 2008 #3
    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!
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