Re: ORIGINAL simple magnets question

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

At 10:42 AM 6/25/99 -0700, William Beaty wrote:
> I still think we are having communication problems.  My "original problem"
> was my prediction that a spinning disk-magnet would affect the trajectory
> of an electron differently than a non-spinning magnet, even if the b-field
> is not being varied by nonuniformities in the spinning magnet.
> Is this wrong?

Alas, that prediction is mostly wrong.   Specifically...
 (a) In the limit of rotational velocities much less than c, and
 (b) if we ignore junk effects (such as mechanical distortions of the magnet due to centrifugal forces) and
 (c) if the magnet is macroscopically electrically neutral, then ...
spinning the magnet has no significant effect.

First answer:  When all else fails, look at the Maxwell equations.  Trying to do electrodynamics without Maxwell equations would be like teaching a snake to walk.

The magnetic field has a source term for changing electrical fields, but by symmetry we don't have any of those in this situation.  The magnetic field has a source term from flowing charges, but by hypothesis (c) we don't have any (unbalanced) flowing charges, so we are left with zilch.  It's that simple.



Second answer:  In order to make this result more intuitive, please consider the particular case where the magnet is an electromagnet, consisting of a 1-amp transistorized constant-current source feeding a metal ring.  We build two of the things.  We rotate one of them, current source and all, in the plane of the ring.  In the (slightly non-inertial) frame of the rotating current source, it continues to put out one amp.  In the lab frame, the rotating magnet's electrons are circulating slightly faster, so the electron-current is larger.  On the other hand, the ions of that metal ring are circulating in the same direction, creating a countervailing current.  Therefore to first order in v/c, the current is the same, and the magnetic field is the same whether or not we spin the ring.

To increase the generality of the argument, consider the following arrangement of current loops, where each Q is a small loop;  the tail on the loop is meant to designate the constant-current source:


                       QQQQQ
                     QQQQQQQQQ
                    QQQQQQQQQQQ               (1)
                     QQQQQQQQQ
                       QQQQQ


The foregoing argument applies to each of the little loops separately.

Here's another way to come to the same conclusion:  By the usual Stokes-type arguments, this arrangement is identical to a large ring sort of like

                        ___
                       /   \
                     /       \
                    |         |               (2)
                     \       /
                       \___/

with one amp flowing around the edge, so figure (1) is just equivalent to the single ring we already analyzed.

A chunk of magnetic iron is just like figure (1), except that the little source-loops are *exceedingly* small and numerous.