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I'm working on a project which uses cylindrical permanent magnets, and I'm trying to determine the expected magnetic field due to these magnets. One problem I run into is that these magnets are usually not completely horizontal above the point at which I want to calculate the expected magnetic field. Assuming the point is on the axis of the cylinder, it's easy enough to calculate the expected magnetic field using the following equation: $$B = \frac{B_r}{2}\left[\frac{D + z}{\sqrt{R^2+(D+z)^2}} - \frac{z}{\sqrt{R^2+z^2}}\right]$$

where $B_r$ is the remanence field (found in a magnet's data sheet), $z$ is the distance from a face of the cylinder on the axis, $D$ is the thickness of the magnet, and $R$ is the radius of the cylinder. However, this equation only works on the axis of the cylinder itself. How can I calculate the off-axis magnetic field for a cylindrical permanent magnet?

Note: I don't particularly need the derivation for the off-axis equation for now, although it would definitely be helpful.

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  • $\begingroup$ I don't have time for a complete answer right now, but how to work this kind of thing out is covered in problem 5.4 of Jackson's Classical Electrodynamics (third edition). (Problems 5.3 and 5.5 are also related). $\endgroup$
    – Buzz
    Nov 21 '20 at 2:27
  • $\begingroup$ I'm not sure but this research of multipole cylindrical magnets magnetic field strength expansion off-axis should help. $\endgroup$ Apr 9 at 9:26
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It looks like there are no way avoiding solving the elliptical integral equations. More details you can find in this article (check chapter 3 and mentioned references): https://webspace.science.uu.nl/~kuipe103/Publications/JournalOfMagnetism2018.pdf [https://doi.org/10.1016/j.jmmm.2018.02.003]

Two articles below are about the solenoids, but since the cylindrical (bar) magnet are equivalent to the solenoid in terms of the field lines, they still can be useful:

Hope this may help a bit.

UPDATE 2021.09.04: there is a free Python package Magpylib which may help you to solve magnetic field calculation problems just in a couple of lines of code (https://doi.org/10.1016/j.softx.2020.100466)

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