Postby **Zamfir** » Fri Mar 03, 2017 1:34 pm UTC

It's often useful to model a (uniformly magnetized) permanent magnet as a current sheet on its surface. That is, the magnetization of the magnet consists of microscopic current loops in the magnet that cancel each other out everywhere, except at the boundary of the magnet. In addition, you model the interior of the magnet as a permeable material (which can simply have the permeability of vacuum, for example in rare earth magnets).

So you basically replace a cylindrical magnet with a very thin-wired solenoid with the same length and radius, and with its turns around the cylinder. The coercivity of the permanent magnet becomes the current density (in Ampere-turns per axial meter of length) in the current sheet.

The above is enough information to accurately model the magnet in a program like FEMM, and find the flux density at any point around it. A closed form will be complicated, though perhaps it's doable if you;re only interested in the axis. I am not sure.

You can do a simplification: model the magnet as a current loop with the same radius but no length, and a current I = Hc*L. That gives a simple expression that will be more exact the further away you get from the magnet.

B(z) is flux density on the axis of the current loop, at distance z from the middle of the loop.

B(z) = muzero / (4*pi) * (2 * pi * R^2 * I)/ (z^2 + R^2)^(3/2)

Now, if z is much larger than R, then that formula simplifies to

B(z) = muzero / (4*pi) * (2 * pi * R^2 * I)/ (z^3)

Which is is the same as your formula for an infinitely small magnet, if we take magnetic moment equal to pi * R^2 * I = pi R^2 * Hc * L = Hc * volume

So at large distances in vacuum (or air), a permanent magnet behaves as a magnetic moment equal to its volume times its coercivity. But close to the magnet (like the distances you mention), the field behaves different depending on the shape of the magnet.

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EDIT: I realized I skipped a complication. When we simplify the magnet to a current loop, we should base that on the total magnetisation of the magnet. That's the coercivity, plus the extra magnetisation induced by the flux if the material has a permeability above vacuum. So the magnetic moment is magnetization times volume, not just coercivity times volume.

Low grade neodymium magnets are easy, they have basically constant magnetisation. So magnetization is roughly coercivity, and remanence is coercivity times muzero. The magnetisation of higher grade neo magnets increases a bit with flux, but they saturate quickly and operate mostly in the saturated area all the way to the remanence point. So the magnetisation is still mostly constant for practical purposes, and can be calculated from the remanence as remanence/muzero. It will be a tad higher than only coercivity.

Some magnets, like Alnico magnets, are unsaturated when you keep them in free space ("open circuit"). Then you need to find out the operating point in free space first, before you can calculate the magnetization.