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

[John Denker <jsd@av8n.com>]

On 11/28/2006 11:53 AM, Bob LaMontagne wrote:

> It is the classic question of whether or not a magnetic field can produce a
> force on a stationary charge. I pose the question in the following limited
> way. I have a UNIFORM magnetic field that is pointing out of the page (as we
> say in textbooks or exams). A proton at a particular instant has a velocity
> within the plane of the page pointing to the left. It experiences a force at
> that moment toward the top of the page.
>
> Many texts then ask the question, based on relativity, that if we consider
> the electron to be at rest and that the uniform magnetic field is produced
.     ^^^^^^^^ ???
> by a huge set of pole pieces or a Helmholtz coil that is at rest within a
> frame of reference that is approaching the proton from the left side of the
> page with a velocity to the right, is there still a force on the proton
> toward the top of the page?

     Note: I assume the "electron" is a typo for "proton".

The answer is easy:  In the lab frame, the field is purely magnetic.
In the frame comoving with the proton, the field is partly magnetic
and partly electric, and the electric field accounts for the observed
force.

If you define force the "right" way -- as d(momentum)/d(proper time) --
then the force is the same in both frames, as it must, by conservation
of momentum.  (Beware that other definitions of "force" can be found
in the literature.)

There is a nice way to get your head around this result.  This way is
simultaneously the easiest to understand, the easiest to depict, the
easiest to teach, the most sophisticated, and the most powerful.

The key idea is that the EM field is not a vector.  Never was, never
will be.  Rather, it is a bivector.  What we call the electric field
in a particular frame is a vector, but it is merely a /component/ of
the electromagnetic bivector.  Similarly the magnetic field vector is
merely a /component/ of the electromagnetic bivector.

As everybody knows, if you change from one basis to another, a physical
vector stays the same, but its representation in terms of components
changes.  The same goes for bivectors.  The EM field bivector (as a
physical, geometric object) is the same in both frames, but its
representation in terms of components changes.

I worked out the mirror-image problem, i.e. an EM field that is purely
electric in one frame, but partly magnetic in another.  Here are the
details, including a diagram of the bivectors:
  http://www.av8n.com/physics/magnet-relativity.htm