Re: Non-Coulomb forces (was: Batteries)

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

At 14:42 2002/02/13, Joel Rauber wrote:

> page 559 ...
> "The magnetic force evB is a non-Coulomb force ...
> The work done by this force in moving an electron from one end of the
> bar to the other is FNC L = evBL. . . ."
...
> IMO this is a blunder of a statement.  magnetic forces of the ev X B variety
> do no mechanical work!

Joel's got an excellent point.
Obviously dx dot F = 0 when F = v cross B.

I don't have the book in question, so I can't say
exactly what the mistake is, but my best guess is
that the book gets the numerically-correct answer
by a conceptually-incorrect argument.

In particular:

1) The velocity v is presumably the velocity of the rod
(relative to the lab frame).  This is _not_ the velocity
of the electrons, since they are moving relative to the
rod.

2) The electrons are constrained to remain inside the
wire.  This force of constraint is essentially Coulombic.
Without this constraint, the apparatus wouldn't work.
So in some sense, Coulombic forces are an essential
part of the physics.

Specifically:  The non-Coulombic v cross B force is
opposed by a Coulombic constraint.  The latter does
the work.  But since it is numerically equal to the
non-Coulombic force, one gets the numerically-correct
answer by multiplying the non-Coulombic force by the
distance. Numerically correct, but conceptually incorrect.

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

Tangentially related remarks:

A) The velocity of the electrons differs from the
velocity of the rod by a very small amount.  There
is a factor of "carrier density" in the denominator,
and that's huge.  So you might think that blurring
the definition of v introduces only a negligible
error.  But...

OTOH when calculating the total force of constraint,
the carrier density enters in the numerator.  So
the total force required to "straighten out" the
path of the electrons is a leading-order term.

B) In the Lorentz force law, one should not speak of
the v cross B term as "The" non-Colombic term.  The
other term, q E, also contains non-Coulombic contributions
when there is a changing magnetic field.

I doubt this is directly relevant to Joel's message,
but it is something to keep in mind.  Neglecting the
dB/dt term is an all-too-common mistake.

C) This whole discussion gives me the creeps.  Who
cares whether a given E field is Coulombic or non-
Coulombic?  The electron doesn't care!

It is easy to set up a situation where a force that
is 100% Coulombic in one frame is 100% non-Coulombic
in another frame.

The classic example is a wire carrying a current.
You observe that the wire is electrically neutral,
so there is no Coulombic force in your frame.  An
electron moves parallel to the wire, in the direction
that produces a force away from the wire.  This force
is 100% non-Coulombic, due solely to the magnetic
field. .......  Now re-analyze this in the frame
comoving with the electron.  Obviously there are
no magnetic forces in this frame.  No v cross B, and
no dB/dt.  But there is still a force;  everybody agrees
that the electron is accelerating away from the wire.
Homework:  How do you explain that?

So the pedagogical question is:  Why did the book
bring up the concept of "non-Coulombic" at all?  Why
pollute the students' brains with a notion that (even
if correctly presented) would be not very useful?