Re: [Phys-l] Force on a charged particle from a magnetic field

[John Denker <jsd@av8n.com>]

On 11/28/2006 04:14 PM, Bob LaMontagne wrote:

> That's why I carefully defined the magnetic field to be uniform, so there is
> no change in flux with time (classically). Specifically, how do you get the
> time varying flux that ultimately produces an electric field to move the
> proton? (Again, without an appeal to relativity.)

Without an appeal to relativity?  Wow.  I think it is
possible -- but very tricky -- to make that requirement
meaningful.

We have only one universe, and the laws of relativity apply
in our universe.  It is very hard for me to pretend that I
don't know that time is the fourth dimension.  The best I
can do is to consider various low-order expansions.

For obvious reasons, things that are first-order in v/c
were known long before there was a detailed understanding
of relativity.  The so-called nonrelativistic limit doesn't
mean v=0;  to me it means v/c small enough that we need
only look at the leading-order term.
 -- For a lot of classical things, like the 3-velocity itself,
  the leading term is first-order in v.
 -- For some classical things such as kinetic energy, the
  leading term is second-order in v (namely 0.5 m v^2).
  Is this kinetic energy a classical thing?  Or is it a
  relativistic thing approximated to second order?  To me,
  that's a distinction without a difference.
 -- For the problem at hand, the E/B ratio is first order
  in v.  So in some sense it ought to be as classical as
  classical can be.  Or is it a relativistic thing, valid
  to first order?  To me, that's a distinction without a
  difference.  To put the point more strongly, there is
  no non-relativistic limit that will make the E/B effect
  go away, except for the trivial case of v exactly zero,
  in which case there is nothing that needs explaining.

To me it seems reasonable to work the original question as
follows:  We know momentum is conserved.  That means the
force must be the same in both frames.  That allows us
to ascertain how strong the E field must be, valid to (at
least) first order in v/c.  If we only require the answer
to be valid to first order in v, we can get it right without
even needing to fuss over "relativistic" corrections such
as x/x' or t/t' or t/tau, since those ratios are all unity,
to second order.  The key to this approach is a completely
classical conservation of momentum argument.

If you're not interested in details, stop reading here.
----------------------------------------------------------

Now let's ask what happens if we try to get the exact answer,
not just the first-order answer.

At this point I may need to partially retract some things I
said earlier today.  It's still true that the Maxwell equations
are not _by themselves_ sufficient to solve all the world's
problems.  For sure the Lorentz force law is also needed, and
/technically/ it's not one of the Maxwell equations.  But I
won't stoop to such technicalities.  I have to agree with most
(maybe all) of the spirit of what John M. said:  A very great
deal of special relativity is "built in" to electrodynamics,
and it will foist itself on you whether you want it or not.

In particular, contrast this with the 0.5 m v^2 expression
for kinetic energy.  That is valid in the classical limit,
i.e. small v/c, but without additional assumptions we have
no clue as to what the next-order term is.  The next term
involves higher powers of v/c, but we don't have a good way
of knowing that c has anything to do with classical mechanics.
Classical mechanics doesn't tell us how far we can push it
before the next-order terms show up.  We can start to say
something similar for electrostatics and magnetostatics.
However (!) the Maxwell equations do have a built-in value
for c^2, so we *do* know how far we can push electrostatics
and magnetostatics before the next-order terms show up.

I'm not saying /all/ of special relativity will drop into
your lap, but I think you can get the correct-to-all-orders
expression for what a purely magnetic field (in the lab frame)
looks like in other frames, just by considering a succession
of very small boosts.  Some mild assumptions are required,
but only verrry mild, such as
 -- t = t' = tau to first order
 -- x = x' to first order
 -- fields add linearly
 -- B = B' to first order
 -- E = E' to first order
I'm thinking of following the "bootstrap" procedure used in
  http://www.av8n.com/physics/spacetime-trig.pdf
but with even fewer assumptions required.

I'm not 100% sure of the second half of this note, because
I usually play by the rules that I can use everything I know;
I'm not good at pretending I don't know that our universe has
a Minkowski spacetime geometry.  I may be tacitly assuming
something I shouldn't.