A "successful" exchange in a science group is, IMHO, one in which I
learn something.
This one's been successful ;-)
George Holz wrote:
Stephen A. Lawrence wrote:
[ snip ]
> Remember, E and B fields (apparently!) follow the law of
> superposition, which means overlapping fields themselves don't
> interact; they just sum.
They really don't just sum, it's more complex than that in real
geometries with real materials, mu and saturation. Thats why we need
finite element programs.
Um ... As I learned E&M ... and this is just quoting from Griffiths, I
sure can't claim to have proved this experimentally -- both E and B
fields obey superposition perfectly, and the "pure" form of Maxwell's
equations holds everywhere, inside matter as well as outside. The
_apparently_ different equations we get in matter -- with D vs E and H
vs B and with funky values for mu and strange saturation effects --
_just_ result from the superposition of E and B fields induced in the
matter as a result of the effect of the externally applied fields. They
can come from tiny current loops, or from rotated/stretched electric
dipoles, but either way it's actually an additional field associated
with the matter which is added to the applied field. Or so say the
textbooks I've read.
But as I said, I certainly can't claim to have _verified_ that! :-)
But in any case finite element programs are needed to understand the
unbelievable hair which grows even if you take the classical theory at
face value.
[ snip ]
The currents are almost entirely electron spin dipoles aligned
within in changing domains which reorient to aid the applied field.
Standard magnetic materials have only a small contribution from
orbital dipoles. Unusual materials with large spin-orbit coupling do
exist but even here the electron spin dipoles dominate.
Well, well.
Well, well, well.
Hmmm..
An electron in a B field has a magnetic dipole field, and unless
Wikipedia got it totally wrong, the strength of the electron's dipole is
independent of the strength of the external field.
If my mental picture is right, then a free electron's dipole must be
aligned with the external field (parallel or antiparallel).
But in that case an electron in a non-uniform B field must feel a force,
proportional to the gradient of the field ('cause that's what magnetic
dipoles _do_, and besides, if the electron sourced a dipole field but
didn't feel a force as a result of being immersed in somebody else's
dipole field we'd violate conservation of linear momentum which would be
unfortunate).
But then if we let the electron go in a non-uniform B field it'll
accelerate, which means something did work on it; it gained kinetic energy.
Where'd the energy come from? I have no idea. Since the force on the
electron depends on whether it's spin-up or spin-down there's certainly
no simple "potential-gradient" model one can use here, either.
Interesting. There must be something wrong with this picture, but I
don't know what. :-)
