Re: magnetic circuits

["John S. Denker" <jsd@MONMOUTH.COM>]

> Jack Uretsky wrote:
>
> > What makes you believe that the pole strengths of interacting
> > bar magnets is independent of distance?

Then at 03:42 PM 2/18/01 -0500, Ludwik Kowalski wrote:

> My high school textbook, I suppose. Thanks for a  good hint.
> Let me follow it? As one end of the bar magnet comes closer
> to another it induces "poles of the opposite kind at the surface.

That's not a good way to think about it.

> It is similar to what happens in many dielectric materials. The
> net pole becomes stronger at each bar magnet and the attractive
> force between them changes more rapidly with the distance
> than the 1/r^2 law would suggest. That is why magnets come
> toward each other suddenly.

What 1/r^2 law is that?  The law for magnetic monopoles?

The corresponding 1/r^2 law for electric charges does not produce a 1/gap^2
force for capacitor plates at constant charge;  in fact it produces a force
that goes like gap^0 (i.e. independent of gap) in that case.

So please, before we use analogies to explain facts, let's make sure the
facts are true, and draw analogies to things that really happen.

We are not trying to explain why the force goes faster than 1/gap^2;  we
are trying to explain why it goes faster than gap^0.

If the bar magnet really did have constant pole strength on its end, the
force law would be a nonsingular (indeed exceedingly gentle) function of
gap, and there would be no trouble establishing equilibrium between the
magnet force and the spring force.

======================

The sensible way to think about this is as a magnetic circuit.  Haul out
the usual formalism of flux, magnetomotive force, and reluctance.  The EB
writeup is not too bad:
  http://www.britannica.com/bcom/eb/article/printable/4/0,5722,51234,00.html

I don't know of a really good discussion of this, perhaps because it is a
somewhat flaky technique, but it would be nice to at least get a decent
discussion of the limits of validity.  Somehow I prefer controlled
approximations to uncontrolled approximations.

Anyway, to simplify the calculation, rather than using bar magnets, let's
suppose we have a horseshoe magnet, and we are putting the keeper on
it.  We measure the force as a function of the gap.  In this case,
practically all of the reluctance is in the gap, the flux goes like 1/gap
(assuming constant MMF and using the magnetic equivalent of Ohm's law), the
field energy density goes like 1/gap^2, the extensive field energy goes
like 1/gap, and the force goes like -1/gap^2 by PVW (principle of virtual
work).

(I reiterate that electrically charged spheres have a force as a function
of gap that doesn't look anything like this.)

> How come that Coulomb was able to determine the 1/r^2
> dependence using a torsion spring balance?

I suppose he was clever enough to use sources that were small compared to
the gap between them.

I also suppose he was clever enough to use stiffer springs to measure the
larger forces.

> I suppose he was dealing with repulsive forces only.

I strongly doubt he restricted himself to repulsive forces only.