Slightly philosophical question about magnetism

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In summary, the conversation discusses the question of whether the magnetisation of a magnetically susceptible material placed in a uniform external magnetic field will also be uniform. It is noted that for a linear, isotropic, and homogeneous susceptibility, the calculation of magnetisation is similar to that of polarization in an electric field. However, it is mentioned that for shapes other than a sphere, the magnetisation will be dependent on the radius. The conversation concludes that for ellipsoids, the magnetisation will be uniform, but for other shapes, it will not be.
  • #1
pterid
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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!
 
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  • #2
pterid said:
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?
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.
 
  • #3
pam said:
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!
 

Related to Slightly philosophical question about magnetism

1. What exactly is magnetism?

Magnetism is a fundamental force of nature that causes objects with magnetic properties to either attract or repel each other. It is caused by the alignment of particles within a material, known as magnetic domains, which create a magnetic field.

2. Why are some materials magnetic and others are not?

Materials become magnetic when their atoms have unpaired electrons, which can align and create a magnetic field. These materials are known as ferromagnetic. Other materials, such as copper and aluminum, do not have unpaired electrons and therefore cannot create a magnetic field.

3. How does magnetism relate to electricity?

Magnetism and electricity are closely related and are both part of the electromagnetic force. Moving electric charges, such as electrons, create a magnetic field. Similarly, a changing magnetic field can cause an electric current to flow.

4. Can magnetism be used for anything other than attracting or repelling objects?

Yes, magnetism has many practical applications in our daily lives. It is used in generators to produce electricity, in motors to convert electrical energy into mechanical energy, and in medical devices such as MRI machines to create images of the body.

5. Is magnetism a renewable resource?

No, magnetism is not a resource that can be depleted or used up. It is a natural force that exists in the universe. However, the materials used to create magnets, such as iron and cobalt, are limited resources and must be mined from the earth.

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