George Holz wrote:
Stephen A. Lawrence wrote:
The thing you need to keep firmly in mind is that all motions considered
here are as seen by an observer stationary in the "laboratory frame" and
in that frame the motion of the charges is not simple rotation about the
ring's axis; it's got a translation added to it. The charges are
following something more akin to a cycloidal path than a circular path.
Here's a picture which may help; the red line, motion of the charges, is
supposed to be the sum of the green line (translation) and the black
curve (rotational velocity) (ok ok the lines are not drawn not to scale,
sorry):
http://www.physicsinsights.net/images/moving-ring.png
The B field is assumed to point UP through the page, so the Lorentz
force lies in the page and is perpendicular to the net motion of the
charges.
So with your model the ring would not slow down if the box with the dipole
field moved instead of the ring?
Certainly it would. All you've done is change your frame of reference,
which doesn't affect the outcome, and doesn't affect the 4-force which
causes that outcome.
If the box is in motion, then to find the field which will be observed
in the "laboratory" frame we need to transform the B field which is
observed in the box's frame into the lab frame. When we do that we find
there's now an E field in the lab frame. In this case the ring is
slowed down by the E field, which (in the ring's frame of reference)
_does_ do work.
By the way, whether work is done is frame-dependent, which should come
as no surprise, since kinetic energy is a frame-dependent quantity.
The E and B fields together form a rank 2 tensor (the "Faraday"
tensor). The application of that tensor to a charged particle's
4-velocity yields the 4-force on that particle due to the EM field.
Changing FoR changes the components of the Faraday tensor but doesn't
change the effect it has on a particular object.
George Holz
Varitronics Systems
