[Phys-l] relativity +- electromagnetism +- pedagogy

[John Denker <jsd@av8n.com>]

On 11/29/2006 09:21 AM, Bob LaMontagne wrote:

> ...... My concern is that so many introductory texts bring up this
> example in the end-of-the-chapter problem sets. They then leave the students
> hanging, with the implication that Maxwell's equations are incorrect. 

That's definitely a "concern"!

That's really bad pedagogy.

It is trivial to cobble up many questions such that the student can
understand the question but doesn't have the tools to understand the
answer (let alone discover the answer from scratch).

To me, the following are all in the same category (although perhaps
differing somewhat in degree):
 -- Asking students to rediscover the relativistic invariance of the
  Maxwell equations (before they've been taught anything about relativity).
 -- Asking students to rediscover the proof of the four-color map theorem.
 -- Asking students to rediscover the proof of Fermat's last theorem.
 -- Et cetera.

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

> On 11/29/2006 11:38 AM, John Mallinckrodt opined:
> > .... the moment we  
> > introduce the Lorentz force, we become almost obliged to point out  
> > the seeming absurdity of a force that depends on velocity--not  
> > *relative* velocity, just velocity--and that vanishes in the frame of  
> > the moving particle. 

That really hits the nail on the head.  It's almost a syllogism:
  *) Even the most introductory course emphasizes Galileo's
   principle of relativity.
  *) Galilean relativity + Lorentz force law ==> big problem.
  *) Solving the problem ==> special relativity.

I don't see any way to dumb down electricity & magnetism to the
point where we can avoid questions that demand SR answers.

So ... what are the options for dealing with this?
 -- At one extreme, you could put that "syllogism" on the board and
  tell the students "Until we get to special relativity, y'all are
  going to have to live with some unanswered questions."
 -- At the other extreme, you could introduced relativity in all
  its gory glory before doing electrostatics and magnetostatics.
 -- Or ... I think there is a third way, as we now discuss.

I am reminded of something  Prof. Charlie Peck said to his class,
with a big smile on his face:

   "I'm not going to teach you how to do Lorentz transformations.
    I'm going to teach you how to /avoid/ doing Lorentz transformations."

I'll always remember that, and I'll always smile when I remember it.

His point was that anything worth doing can be done in terms of
spacetime geometry; that is: spacetime diagrams, 4-vectors, invariant
length, proper time, invariant mass, et cetera.
  Doing it that way is IMHO vastly easier than doing it terms of
  Lorentz transformations, time dilated clocks, Lorentz contracted
  rulers, velocity-dependent mass, et cetera.

In Maxwell's original paper on electrodynamics, he wrote out the
equations component by component, not using vectors.  It was a
mess.  (I hope those of you who tout the "historical" approach
to physics aren't cruel enough to inflict this bit of historical
realism on your students.)  Now we all know that the laws of
electrodynamics are invariant w.r.t rotations.  This is not at
all self-evident when the equations are written out component by
component ... but it becomes self-evident when the equations are
rewritten in terms of vectors.  So nowadays everybody writes the
equations in terms of vectors.

The rule is, if you're not up-to-speed on vectors, you're not
ready to study the Maxwell equations.  This rule is the consensus,
and it makes sense.

My point is that the laws of electrodynamics are also invariant
w.r.t boosts.  This is not at all self-evident when the equations
are written in terms of E-fields and B-fields ... but it becomes
self-evident when the subject is reformulated in terms of the
electromagnetic bivector.

The "boosted magnetic field" problem will always be hard for
students so long as they think of the magnetic field as a
vector (or pseudovector).  There's no way to visualize the
right answer in terms of vectors.  The magnetic field is not
a 3-vector, nor a 4-vector, nor even the spacelike part of a
4-vector.  If you draw a picture in the lab frame, where the
electric field is zero and the magnetic field is nonzero,
the picture is well-behaved under arbitrary rotations, but
badly-behaved under almost all boosts.  The picture is wrong
... irreparably wrong, as long as you are stuck with vectors.

At this point some would say "you must do without the picture;
you must rely on the math".  Phooey, I say.  I want the picture.
Most of the students want the picture.
   Of course, I want the math, too.  I want the picture to
   guide me as to what math to do, and I want the math to
   guide me as to how to draw the picture.

A suitable picture is readily available!

It's easy:
 -- You do /not/ need to spend days and days studying Lorentz
  transformation, time dilatation, Lorentz contraction, and
  all that.
 -- You do /not/ need to teach them how to multiply vectors
  and bivectors.  Draw the picture first;  this will motivate
  them to learn the math later.
 ++ You need to say that time is the fourth dimension, and
  show them how to draw spacetime diagrams.
 ++ A boost mixes the X and T variables much as an ordinary
  spacelike rotation mixes the X and Y variables.
 ++ What was heretofore called Bz is really a bivector in the
  XY plane.  You can draw the picture, or (better) model it
  with a piece of cardboard.
 ++ Similarly what was heretofore called Ey is really a
  bivector in the YT plane.

Now draw some axes on the chalkboard:  X to the left, T upward,
and Y coming out of the board.  Hold the piece of cardboard
horizontally, representing the magnetic field as a bivector in
the XY plane.  Now rotate it a small amount in the XT plane,
i.e. raise the +X edge of it relative to the -X edge, like
this:
     http://www.av8n.com/physics/img48/magnetic-boosted.png
where the overall EM field is shown in red.  In the boosted
frame, it is purely magnetic.  In the lab frame, it has been
resolved into components:  the green, striped component is
purely magnetic in the lab frame, while the gray component
is purely electric in the lab frame.

The difference between the purely magnetic field in the moving
frame and the magnetic component in the lab frame is a small
small bivector component in the YT direction.  This is the correct
answer!  If you boost a magnetic field, it picks up a component
of electric field.

What was inexplicable in terms of vectors is obvious in terms
of bivectors.  The math is correct /and/ the picture of the
electromagnetic field (the red bivector) is correct in all
frames.  The decomposition into a so-called electric piece
and a so-called magnetic piece is frame-dependent ... but
the total field (i.e. the real electromagnetic field bivector)
is the same in all frames.

To repeat:
 *) You can make the math work for electric vectors and magnetic
  vectors.
 *) You can make the math *and* the picture work for electromagnetic
  bivectors.