Re: boosted electromagnetic fields (was: motional emf)

[John Denker <jsd@AV8N.COM>]

Hugh Logan wrote:
> .... The very elementary treatments hardly use the results of special
> relativity. .... I think the difficulty is more conceptual than
> mathematical.

1) That's an excellent point.

2) Of course the conceptual and pedagogical issues are not
to be sneezed at.  For us as trained physicists, the thought
of E&M without relativity sounds like a logical impossibility,
... but students are not born knowing relativity, and commonly
students are exposed to E&M (including motional "emf") before
they are exposed to enough relativity to be useful.

*Something* has to come first; either students see intro
E&M before intro relativity or vice versa.  So concepts like
this, which involve both topics, pose a problem (at first)
as well as an opportunity (later).
-- At first it is a scheduling problem.  Students will ask
questions during the E&M discussion that cannot be properly
answered until later, during the relativity discussion.
-- Later it is an opportunity to show the unity and elegance
of physics.  Relativity can be applied to the electromagnetic
field (not just to positions and velocities).  This refreshes
and reinforces what the students previously learned about E&M.

Most textbooks don't have nearly enough cross-topic discussions
or cross-topic exercises.  Grrrrr.

> ... I don't think this is exactly true, but it is OK if v/c is small.

A) Yup.  The exact result is sinh(v/c).  Hint: E^2 - B^2 is
a relativistic invariant scalar.  [This is closely analogous
to t^2 - x^2 being an invariant scalar.]

B) If you think of the electromagnetic field as a _bivector_,
all guesswork is removed.  The bivector approach allows you
to figure out the correct transformation law just by looking
at the geometry of the situation in a spacetime diagram.

For example, the case of a field that is purely electical in
the lab frame is diagrammed at:
http://www.av8n.com/physics/magnet-relativity.htm#fig-wire-bivector

(I realize that the problem that started this thread is
purely magnetic, not electric, in the lab frame, but the
concepts are the same.  Just rotate the diagram 90 degrees
and relabel the t,x axes to convert one problem into the
other.)

I consider the electromagnetic field bivector (F) to be
primary and fundamental.  In any particular frame, the
projection of F onto the spacelike directions is called
the magnetic field bivector.  The rest of F is called
the electric field bivector (and has one spacelike edge
and one timelike edge).

[This is closely analogous to considering the momentum
4-vector to be primary and fundamental.  In any given
frame, the projection in the timelike direction is called
energy, while the spacelike part is called the 3-momentum.]

In a different frame, the same physical object (F) will
have different components.  In the figure, let the red
bivector be the actual physical electromagnetic field.
It is entirely electric in the primed frame (t',z').
This red bivector can be represented as the sum of the
green bivector plus the dark-gray bivector ... which
are respectively the electric and magnetic components
in the unprimed frame (t,z).

========

For presenting this in class, beware that it takes time
and effort to draw the diagram with sufficient accuracy
and clarity.  Options include:
-- pre-drawing the diagram, perhaps on a viewgraph
that can be projected when needed, and/or
-- using props, i.e. three slabs of card-stock or
poster-board taped together.


*** Philosophical/metaphysical remark #1:

For me, physics is simultaneously
 -- part intuition, and
 -- part calculation.

Each part reinforces and guides the other.

Spacetime diagrams are a tremendous example of how this
works.  A rough diagram tells me what calculations to do,
and the calculations tell me how to refine the diagram.


*** Philosophical/metaphysical remark #2:

I'd rather not get into a discussion of whether electric
field lines are "real" or magnetic field lines are "real",
because people often have trouble agreeing on what the
word "real" really means.  Instead, a better question is
to what extent our conception of E and B is a faithful
representation of the way nature really works.

It is entirely clear that F is a more faithful representation
than E or B.  E and/or B exist only in a particular reference
frame, whereas F has a "reality" (i.e. is a faithful
representation) that transcends the choice of frame.  That is,
F is a well-behaved geometric object, whereas E and B are not.