Re: Magnetic field of a stationary particle

From: Tom Roberts (tjroberts_at_lucent.com)
Date: 10/14/04 

Date: Thu, 14 Oct 2004 18:50:43 GMT

Daveman750 wrote:
> I have read several explanations of why a magnetic field must exist as
> a consequence of relativity, and there is just one thing left that
> baffles me: why does the particle need to be moving in the lab frame?

First: you need to keep your theoretical context clear.

In SR, particles are generally considered to be pointlike without
internal structure, so a charged particle at rest in inertial frame S
generates no magnetic field in S. But because the electromagnetic field
is a two-form (a rank 2 antisymmetric tensor), in frames moving wrt S
that particle will generate a (time varying) magnetic field. Note,
please, that the actual field is the two-form field, and the traditional
magnetic and electric field are merely components of that two-form
field; that's why E and B vary from frame to frame (tensor fields do not).

In quantum theories like QED, however, particles with nonzero spin can
have a nonzero magnetic moment, which means that such a particle at rest
in inertial frame S will generate a magnetic field in S. Indeed, this is
observed for many/most elementary particles. I'm ignoring this, because
your context seems to be SR, not QED; such magnetic fields are quite
small in practice, and usually get reduced to essentially zero because
of cancellation due to summation over zillions of particles.

> If the negative charge is moving in the lab frame, wouldn't the
> distance between the negative charges become length contracted,

Let me guess you are thinking of a wire at rest in frame S (the lab
frame). When carrying a current, indeed the conduction electrons
(negative charges) move wrt S, and on average the distance between them
in S is not the same as the distance between them in their average rest
frame. The conduction electrons will redistribute themselves so the wire
remains neutral in S, and this is essential to remain consistent with
the way the E and B fields transform between frames.

> allowing for a magnetic field to affect a charged particle that is
> stationary in the lab frame?

While the current-carrying wire generates a B field in S, a charged
particle at rest in S is unaffected by the B field. As the current
redistributes itself so the wire remains neutral in S, the wire and its
current have no effect at all on a charge at rest in S.

> The charge densities still wouldn't be
> balanced, at least the way I see it.

They are, basically because when starting the current, the source of the
current pushes slightly more electrons into the wire than it removes.
This imbalance is related to the initial E field in the wire that starts
the current up, and the solid-state physics of the conduction electrons.
Once a steady-state current is achieved, the E field in the wire is
zero, and the rate of electrons entering the wire from the source is the
same as the rate of those leaving the wire back to the source.

Tom Roberts tjroberts@lucent.com

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