Re: What is the electric field near a rotating wire, and why doesn't it follow logic of moving individual sections?
- From: Lorentz <drosen0000@xxxxxxxxx>
- Date: Wed, 18 Jul 2007 19:14:33 +0000 (UTC)
On Jun 29, 2:15 am, "Neil Bates" <neil_del...@xxxxxxxxxxxxxxx> wrote:
Hi, I'm not getting enough response elsewhere:I do not think this is a reasonable assumption because the
What is the electric field near a rotating wire section rotating like a
baton? (Let's say, close to the relevant side and axis of a slowly rotating
rectangular current loop.) I am wondering why the use of some valid field
rules conflicts with what we expect from this apparently reasonable
assumption: to pretend that each piece of the wire "projected" the effect
it's magnetic field would have, if the segment is considered separate and
moving in a straight line at its velocity.
electric charges that comprise the wire are accelerating. If you
really break the wire into its component parts, consider one electron
in the wire. The electron is not moving in a straight line at a
constant velocity. In addition to the Law of Biot and Savart that is
valid for a nonaccelerating charge, one has to consider that an
orbiting electron emits synchrotron radiation. The elimination of
energy flux obviously occurs by some type of mutual interference
effect caused by the radiation emitted from other electric charges.
Otherwise, wiping any neutral wire would generate radiation.
Obviously mutual interference isn't an accident. When some
physical effect "happens" to cancel out, one looks for a hidden
symmetry that causes the cancellation. it is something caused by the
Lorentz invariance of the electromagnetic field. So to really analyze
the problem effectively, you have to include (maybe in an indirect
way) special relativity in the problem. Consider the fact that the
electric charges are no moving in a straight line and at a constant
velocity, and that they are distributed at significant distances from
each other in the wire. There is also a speed-of-light time delay
between the arrival of the magnetic field of each segment, which is
called a retardation.
Doing the problem this way is really too hard for me. I just
want to convince you that ignoring the acceleration of an
electromagnetic charge is usually not always reasonable. One easy
trick to avoid analyzing accelerations is include the geometric
symmetries of the complete system from the beginning, before one
mentally dissects the system into segments, and that way force the
acceleration terms to cancel out early in the problem. Before they
confuse you.
In your case, you plucked a segment of wire out of the system
early in the problem. By doing so, you hid the cylindrical symmetry
implicit in the problem. The rotating wire is moving in a circle. If
you want to do the problem right from this perspective, you have to
sum up all the contributions from all the segments of wire. Then, you
have to reorder the summation so that things like synchrotron
radiation disappear.
The Maxwell equation curl E =I note that the A field of a segment can not by its nature be
- @B/@t doesn't give the E field directly. We can use the A field (magnetic
vector potential) instead, which is parallel to the current producing it and
decays as 1/r. The electric field projected from sources can be given as:
parallel to the segment. The segment is infinitesmally short.
Therefore, there are regions far in front of the segment where the A
field will not be parallel to the wire segment. What you unconsciously
did was impose the cylindrical symmetry of the wire on the
infinitesmal segment of wire. This is a no-no. The reason the A field
is parallel to the wire with electric current is that perpendicular to
the wire cancel out, but only under conditions where that electric
current is constant. However, if the electric charges in the different
segments were accelerating at different rates, these perpendicular
components couldn't possibly cancel out.
That 1/r dependence looks like a quantity derived using the
cylindrical symmetry of the wire. This is a pretty good assumption if
one is looking at a long length of wire. But it probably doesn't work
for an infinitesmal segment. At the very least, you need a retardation
delay to modify the 1/r dependence. I am guessing, as I am not sure
exactly how you came up with these formulas. It would be helpful if
you describe exactly which electromagnetic gauge you are working in.
To summarize: Your analysis is working for your potential
calculation rather than your field calculation because in the
potential calculations you are using the symmetry of the total system
to cancel out the acceleration terms. I suspect that you can probably
do that in a field calculation provided you somehow include special
relativity in it.
.
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