MadSci Network: Physics

Re: How do you calculate the force between to opposing magnets?

Date: Thu Jul 8 10:02:31 1999
Posted By: Georg Hager, Grad student, Theoretical Particle Physics
Area of science: Physics
ID: 930542556.Ph

Dear Ian!

This is a good question. The reason why the problem you are referring to is usually not discussed in tutorials and exercises is that this kind of calculation is not easy to carry out for nontrivial situations.

Let us consider two magnetized objects with some distance r between them. What we like to know is the force resulting from their interaction as a function of r. As a further simplification we assume that the magnetic field of one object does not alter the magnetization of the other, no matter how close they are together (the material is `magnetically hard'). I think we can accept the notion that the force is zero if r is infinitely large. Bringing the objects closer to each other, we will feel some force, and let us assume that it is repulsive. So we have to invest some energy to decrease the distance, energy which was e.g. previously stored in our muscles. Where does this energy go? Well, the only `entity' that is able to store this energy in our setup is the combined magnetic field of the two objects! If the force is attractive, we must obviously have `drained' the magnetic energy to do some work (pulling the objects together against our muscles).

So all we have to do to calculate the force is to compute the overall energy that is stored in the magnetic field at each distance r. This gives us some function W(r), a so-called potential function. To get the force out of such a potential we have to differentiate -W(r) with respect to the coordinate r: F(r)=-dW(r)/dr. The problem is solved.

Of course, the problem is not yet solved. How do we get the field energy in the first place? From electrostatics we know that the field energy of an electromagnetic field can be calculated by integrating the sum of the squares of the electric and the magnetic fields over all space. In our case there is no electric field, and with the correct prefactors the expression for the energy is

           1   |  2  3
   W(r) = ---- | B  d r
          4 Pi |
This expression depends, of course, on the distance r between the objects, as well as their shape and magnetization. The magnetic field is just the vectorial sum of the fields of each object. It is the calculation of W(r) which is the real difficulty in this problem, but if you know the magnetic field of each object, you could do the integral numerically, of course. In some simple cases like identical cuboid magnets on a common axis there might even be an analytical solution (not sure about that, though). If you've understood the physics, it all boils down to doing integrals :-)

Hope that helps,

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