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I'm a maths student working on a project. I want to know, how I can find the aggregate 3D equations for a magnetic field induced by a series of bar magnets of various sizes placed at various positions on a plane. Obviously this is a very difficult problem, but can anyone suggest a couple sources where I can begin to learn how to calculate this field?

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  • $\begingroup$ No not silly. The field lines of a bar magnet are three dimensional. I want to find the equation for the magentic field at an arbitrary point above the magnets $\endgroup$ – robbentheking Sep 23 '16 at 11:22
  • $\begingroup$ You have probably read this, sorry, but even for two magnets in a plane, it looks like you have a lot of work to do, best of luck with it en.wikipedia.org/wiki/Force_between_magnets $\endgroup$ – user108787 Sep 23 '16 at 11:35
  • $\begingroup$ @CountTo10 I have, but thanks anayway. What I (or rather my boss) has in mind is far more irregular than this, so I'm not very sanguine, to say the least. $\endgroup$ – robbentheking Sep 23 '16 at 11:52
  • $\begingroup$ a permanent magnet is (grossly simplifying) a conglomerate of a lot of small dipoles. So in the end you "simply" have to integrate over the contributions from all these dipoles. $\endgroup$ – Bort Sep 23 '16 at 12:20
  • $\begingroup$ If you really want macroscopic bar magnets (non vanishing size) I assume that will be very hard. If you just want magnetic dipole moments (fields) I guess that should be possible but not too easy either. You could take the field of a magnetic dipole and superpose different ones. With Computer algebra this should be doable. $\endgroup$ – N0va Sep 23 '16 at 12:23
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I do not know what kind of magnets you want to model. If idealized dipole moments work for you it is not difficult: Just super pose their fields: $$\vec{B}(\vec{m},\vec{r})=\frac{\mu_0}{4\pi r^3}(3(\vec m \cdot\vec r)\frac {\vec r }{r}-\vec m)+\frac{2 \mu_0}{3}\vec m ~\delta^3(\vec r).$$

The superposed fields will get extremely messy but they are straight forward to construct. Here a plot of two different dipole moments in the x-y-plane: enter image description here

The formula gives you the field in all 3 dimensions but it is difficult to visualize that nicely so I only added that one 2D plot of the field.

If you really want to simulate real bar magnets with non vanishing size you would need to do magnetic FEM http://www.femm.info/wiki/HomePage to do real simulations. This will get complicated especially if you want to go up to 3 dimensions. To write a code for that completely from scratch will be challenging.

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