Interaction force between magnetic dipoles

[Akyu]

Greetings

In order to make a mathematical model of the forces that act on a mechanism, I need to use the formula for the force between two magnetic dipoles.

It would have to be a complete formula, that should take into consideration, aside magnetic moment and environment permeability:

• 
  the position of the two dipoles relative to an absolute coordinate system (of arbitrary origin)

• the orientation of the two dipoles, relative to the absolute coordinate system

To give a specific example: knowing the location and orientation vectors of two magnetic dipoles, I want to determine the components (on x, y and z) of the attractive/repelling force that they exercise on each other.

Oddly enough, as it should be a general knowledge imo, it is hard to find this specific formula. I have only found it on wikipedia (http://en.wikipedia.org/wiki/Force_between_magnets). But I would feel safer if someone with a background in this field could confirm it, or recommend a trusted source where it is eventually explained as well.

Thanks in advance.

 

[rude man]

If you insist on more data than dipole moments and environmental parameters, can one even talk about a 'force on a dipole'? It's going to be a force on each charge separately, is it not?

 

[Akyu]

Let me clarify the issue.

Two magnetic dipoles, in arbitrary locations and with arbitrary orientations in space, interact on each other with a force given by their magnetic field.

This force is either repelling or attractive, depending on the relative orientation of the magnets.

This force is a function of relative location and orientation of the dipoles, their magnetic moments (essentially the source of their field), and magnetic permeability of the environment.

I would like to know the exact formula for this force and hopefully have it explained.

 

[Leveret]

As far as I know, you wouldn't calculate the force on one dipole due to another, per se: you'd calculate the field created by one dipole, then find the force on the other dipole due to that field.

The first part is just the magnetic field due to a dipole, which you can look up or derive from a multipole expansion of the Biot-Savart Law. As for the second, you can derive that by imagining a dipole as some square loop of current in an inhomogeneous magnetic field. If you let the size of the square go to zero and add up the force due to the field on each side of the square, you should be able to rederive the equation in the link you posted:
\boldsymbol{F} = \nabla(\boldsymbol{m} \cdot \boldsymbol{B})