Week 8: Faraday’s Law and Induction 259
8.1: Magnetic Forces and Moving Conductors
Last week we saw that our study of the sources of the magnetic field, even before we reconsider
the forces produced by those fields, are starting to raise red flags concerning the consistency of
electromagnetic theory. As we begin this week, Newton’s Third Law istoast, directly violated by
magnetic forces between moving charged particles, and we ought to be very worried about things like
momentum conservation that we derived from Newton’s Third Law. There is some sort of hidden
problem with Ampere’s Law (that you may or may not have figured outon your own from the hints)
but it seems as though it might have something to do with dynamics andinvariant currents. Finally,
I noted that magnetic forces, by their very nature and defining equation, can do no work.
This week we willbegin by considering certain puzzles associated with this last incontestable
fact. Under some very easily constructible scenarios, it certainlylookslike magnetic fields do work.
However, when we take advantage of the freedom weshould have in physics tochange inertial
reference frames, we can do some surprising things (like make magnetic fields and forces acting on
moving charged particles vanish entirely). Changing frames ought not to alter the total force or
classical motion produced by the total force, but this means thatsomehowmagnetic fields must be
able to transform into electric fieldsand vice versa as we change frames. Electric fields can indeed
do work, so this might actually resolve our paradox!
B (in)
Fm
Fe
q v
L
Figure 91: A conducting rod of lengthLmoving through a uniform magnetic field into the page.
The fieldpolarizesthe free charge in the rod until a region of crossed fields is produced.
To see the nature of the difficulty, we begin with a very simple picture –a conducting rod of
lengthLmoving through a uniform magnetic field at right angles to the field as show in figure 91.
The rod is, of course, made up of many, many microscopic point charges, and as the rodmoves to the
right at velocity~vin the magnetic field, all of those charges experience a magnetic force (according
to the Lorentz Force Law that we learned two weeks ago). Because it is a conductor, it has an
“inexhaustible” supply of free charge that can move within the conductor under the influence of this
force while the equal and opposite charge of the presumed neutral conductor is pushed the other
way. We will assume that the free charges have magnitudeq(which might be positive or negative)
- none of what we work out will depend on its sign.
The magnetic force on any given carrier is thus:F~m=q(~v×B~) (558)which isupin 91. We therefore expect the magnetic field to push free charge up until it reaches the
end of the rod, where a surface potential holds it in (the vacuum beyond is basically an insulator, if
you like). Every charge that migrates to the upper end leaves behind a “hole” (ion of the opposite
charge in the lattice) and following the exact same reasoning we usedin our study of the Hall Effect,
we conclude that these negatively charged “holes” will migrate (via backfilling) until they are located
at the lower end, at which point there is no charge available to backfillthem.