Re: copper rods

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Chuck Britton commented on William Beaty's vision of ubiquitous sound and
radio waves with:
> William Beaty wrote
> > Regardless, I'd better make my rods a million KM long, so that it takes a
> > few seconds for my "work" done on the generator shaft to appear on the
> > distant motor shaft. 
>
> The electrons themselves will move surprisingly slowly. A few mm/s for 
> several amps in common lab sized wires! But the electromagnetic 
> disturbance will move at a speed comparable to that of light.

to which William Beaty responded:

> Yep!  I recently found yet another way to express this.  If we could move
> electrons at a few cm per second, the wire would heat up a rate of
> kilowatts per cubic mm, it would glow white hot and melt, if not vaporize. 
> Or conversely, if we could grab the electron-stuff of a wire and shove it
> along, it would take kilo-newtons force to make it move at cm per second
> rate (and of course it would smoke and melt.)  The "electric fluid" within
> wires is not like water.  It behaves more like cold tar being pumped
> through a sponge!  Amazing that electrical devices work at all.

I think that there *might* be a misconception afoot here.  It is *not* true
that the conduction electrons in a current-carrying wire move at speeds
anywhere near the mm/s order of magnitude.  Rather, the order of magnitude of
the speeds of the electrons responsible for conduction is closer to around
(10^(-3) c to) 10^(-2) c (which is *much* faster than the speed of sound in
such materials).  The mean free path length for these conducting electrons is
large compared to the metal atomic lattice spacing and is determined by: the
impurity/lattice defect concentrations in the metal, the temperature which
determines the concentration of the thermally excited phonons (off of which
the conducting electrons scatter), and the intrinsic strength of the
interaction between electrons and phonons (scattering cross-section).  It is
true that the *average drift* speed of the conducting electrons in a current
carrying wire may be of the mm/s order of magnitude,  but that drift speed is
hardly representative of the motion of the electrons themselves.  The typical
speeds of the conduction electrons is governed by the band structure of the
metal in the vicinity of the Fermi surface for that metal.  Using the quasi-
classical approximation for a metal's responding electrons when it is driven
by an external force (such as a small electric field) which induces
transitions between various states with partial occupancy, we see that the 
electron velocity in a given band for a given crystal momentum is the
gradient of the band energy taken wrt the crystal momentum.  Since the Fermi
energy is typically a few eV, the speed of the corresponding electrons has
the order of magnitude typical of a particle with a few eV of kinetic energy
with a mass corresponding to the electron's effective mass in the metal.  The
drift velocity, OTOH, is determined by the quotient of the average current
flux density in a given region divided by the average effective charge
density of the thermally excited electrons (and holes) in that region in the
vicinity (energywise) of the Fermi surface which are responsible for
carrying the current.

Because of the large mean free path length and mostly single independent
particle-like nature of electron conduction, the image of tar as an analogous
fluid may be somewhat misleading.  But, because the responding fluid very
quickly reaches its terminal velocity due to interactions (i.e. collisions)
with defects, impurities and phonons, the picture of any friction-dominated
fluid such as cold tar flowing through a sponge *does* have something
going for it.  My picture is one of air flowing through a porous open-spongy
or fluffy insulating material such as attic insulation.  In this case the
typical speeds of the molecules in the air are orders of magnitude faster
than the wind speed of the light breeze flowing through the fluffy medium.
This picture also has its faults, as well.  Because electrons are *much* less
massive than the molecules of the gases found in air, their speeds would be
much faster, at a given temperature, than the speeds of the air molecules--
even if they (the electrons) were not, additionally, effectively sped up much
further (and reduced in concentration) by the indirect effects of the Fermi-
Dirac statistics.  I suppose that if we imagined, instead of air, the flow of
a vapor of a substance at a temperature and pressure below its triple point
through the insulation we might be able to *sort of* picture the effect of
the quantum statistics effectively freezing out most of the potential
carriers.  If we had, say, CO2 vapor flowing through the wadding and the
temperature had been cooled to a little below the sublimation temperature for
dry ice, then a frost of dry ice would have condensed (er, deposited) out
onto the fiberglass wadding surfaces.  In this case only the remaining vapor
molecules would be responsible for the flow.  The problem with cold CO2 is
that in this case the individual molecular speeds would have slowed down over
the warm air picture case, rather than have sped up, as would be needed for a
more accurate analogy.  It seems that any macroscopic classical analogous
picture is plagued by faults in the analogy.  The best picture, it seems, is
that provided by the metallic electrons themselves.

If I could find the time I would like to comment more fully on Bill Beaty's
epiphany relating sound to work.  I also would like to some day make a post
defending the use of the term 'thermal energy' and defending the distinction
between 'internal energy' and '(total) energy'.  But since I have little of
the needed time for these things, don't hold your breath.

David Bowman
dbowman@gtc.georgetown.ky.us