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Re: Park City Paradox ?



At 15:36 -0700 1/5/02, Ludwik Kowalski wrote:

Is this a paradox or not? I was skiing down
on the left side of another skier this morning.
We were moving in the same direction with the
same speed. And I was speculating:

1) Suppose each of us was charged negatively.
Being at rest with respect to each other we
would repel each other by Coulomb's law.

2) But electrons drifting along two parallel
wires in the same direction produce an
attractive force between the wires.

This is true, but it is relatively easily shown that this is a
relativistic effect (even at the very low drift velocities of the
electrons). The key of the analysis is that, although the net charge
on a wire carrying a steady state current is zero, the wire (less the
negative charge of the electrons that comprise the current) is
positively charged. The net attractive force between the wires is due
to the apparent increased charge density of the oppositely charged
objects in the other wire (this is the relativistic effect).


3) This triggered my imagination. Suppose
each of us is skiing as specified above,
but intide a glass tube. Will our motion
produce an attractive force between the
two tubes? By analogy this should happen,
right? But this would contradict our
mutual repulsion, as in #1 above.

If you and the other skier are both negatively charged, then the
force between you will still be repulsive. Only if the glass tubes
contain a charge opposite to you two (and there are loots of skiers,
all carrying negative charges, so that the whole thing *looks*
static, will there be an attractive force between the glass tubes.


Is it a paradox or not? The experiment is
of the gedanken type, allow any Q and any
v<c. Looks like a paradox to me. Note that
in this example attraction and repulsion are
perpendicular to the common v of two skiers.
In other words cosines of angle are equal
to unity for both moving and stationary
observers. So how will special relativity
help to resolve the paradox?

I think there is no paradox. Two electron beams moving parallel to
each other and in the same direction and at the same speed will feel
a mutual repulsion, just as you and the other skier will. In fact,
the electrons in a wire feel such a mutual repulsion, but it is
mostly screened by all the other charge present in the wire, and so,
at least at low frequencies the current-carrying charge is more or
less uniformly distributed throughout the wire (after all, the drift
velocities of the current-carrying electrons is very tiny compared to
their thermal velocities, so they behave just like the electrons in
an isolated conductor, distributing themselves about uniformly within
the swire so as to keep the E-field inside the wire to zero). Take
away the supporting wire, so that all you have is a beam of electrons
and the beam will become increasingly spread out in space due to
their mutual repulsion. This creates all sorts of problems with low
energy electron beams--they tend to become "defocused" fairly
quickly. High energy beams have the same problem but its magnitude is
normally less because, due to their high speed they get where they
are going quickly enough that there isn't much time for the space
charge effect to defocus the beam, unless the desired focal point is
very small. Designers of TV tubes and other CRTs have to worry about
this a lot. How the space charge effect plays out in an electron beam
is usually worked out in some detail in any text dealing with
electron beam technology, and is done also in Morse & Feshbach
(sorry, I don't have my copies handy so I can't give a specific
reference).

If one does the relativistic calculation for the force between the
charges of two parallel current-carrying wires, one gets the same
result as is predicted by assuming that the interaction between the
two wires is due to their magnetic fields. I have worked out the
arithmetic, and if anyone would like to see it (I urge you to try it
yourself--it is not a difficult relativistic calculation), let me
know. I can send it to you as a MS word attachment, off-list.

It's an interesting problem, Ludwik, but two charged skiers, even if
they are skiing inside parallel glass tubes are not analogous to a
current-carrying wire.

Hugh
--

Hugh Haskell
<mailto://haskell@ncssm.edu>
<mailto://hhaskell@mindspring.com>

(919) 467-7610

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