MadSci Network: Physics |

Posted By:

Area of science:

ID:

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)=-d*W*(*r*)/d*r*. 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

Hope that helps,

Georg.

Try the links in the MadSci Library for more information on Physics.

MadSci Network, webadmin@www.madsci.org

© 1995-1999. All rights reserved.