Magnetic dipole interaction

tris_d

Two permanent magnets are on the table some distance apart and having some arbitrary orientation relative to each other. When we let go of them they will rotate and translate until they stick together. I am looking for suitable equations to model this interaction on a computer. Also, if anyone knows some software that can already do this please let me know.

AJ Bentley

I would suggest to treat each magnet as a dumbbell of fixed length with all the mass concentrated in the poles.
It's then possible to model all of the forces on each pole and iterate the motion.

Some possible motions could be chaotic though.

tris_d

You mean to treat each pole separately? I don't think there is such equation since magnetic monopoles are not supposed to exist, but that does seem to be the right direction to approach this problem.


So far I only found this equation:
http://en.wikipedia.org/wiki/Magnetic_dipole_moment



The problem is that only tells me how much will they attract, but not how much will they rotate and how much will they translate. You suggestion might solve this problem as instead of one force I would have two, and that would hopefully model rotation and thus possibly solve the whole problem. All I need now is some equation for it. Do we have equation for magnetic monopoles, or can it be derived from the equation above?

AJ Bentley

The basic equation of magnetic force (the inverse square law) is based on the concept of a monopole.

tris_d

You mean Biot-Savart law, magnetic field due to moving charge? That's cylindrical magnetic field rather than "spherical", and is defined by the velocity vector. Right in front and behind magnetic field goes to zero as it gets aligned with the velocity vector (doughnut), so I don't think that would work as I don't think that's how individual poles of a magnetic dipole look like.

AJ Bentley

No - back before the dawn of time, when the unit of magnetisation was the Oersted. The world was a simpler place.

tris_d

What do you mean "no"? What equation are you talking about?

AJ Bentley

Seems like the knowledge has been lost. lol!

The magnetic field from a monopole follows the coulomb law just like the electric field. Originally these two subjects were treated identically. You can substitute a pair of electric charges for your magnetic dipole and except for a few constants the mathematics are identical.

tris_d

I see. But the geometry should be different, field lines of an electric field are straight and radial, but magnetic field lines are circular. Still, the force lines might come up to be about the same, so it actually might work. It certainly is the best solution I have so far. Cheers!

AJ Bentley

The geometry is identical. The fields are identical. Maxwell's equations are symmetric (when monopoles are allowed) . Electric filed lines are not necessarily straight (Curl E ≠ 0). Everything is exactly the same except for the values of a few constants.

tris_d

That's wonderful.

Ok, so if I wanted to replace the two permanent magnets with two electrons and their intrinsic dipole magnetic moment, how would I get the value I need to use for my monopoles?

AJ Bentley

Well, the only change is that you use μ (permeability) in place of ε (permittivity)

You can use ε.μ = 1/c² so it's just a case of sticking in a factor of c²

tris_d

That's a bit too much for me. I expected some actual number as electron's dipole moment is constant value. I think magnetic dipole moment drops with inverse cube not inverse square law, and both poles are taken into account, so I expected it would be more complex to obtain the value for each pole separately.