### Re: Terminal velocity - which formula?

Date: Wed Jan 27 00:23:54 1999
Area of science: Physics
ID: 916421050.Ph
Message:
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You seem to have basically answered your main question, although there are
some aspects of it that perhaps could use a little more explanation. I've
included your exposition here...it can serve, I think, as a good example
there (indicated by --->).

Terminal velocity - which formula?

I have been trying to derive Ohms law from first principles lately so as to
satisfy myself that I really understand current electricity. At first I
drift velocities and mean time between collisions. I thought that if I
applied the equations of motion to an electron and I knew the mean time
between interactions (collisions), then by knowing the length of a
conductor I could calculate the time it took for an electron to move
through the conductor, and from this I would then know the current flowing.

---> Electrons in a metal are a little more complicated than that. They
consitute an almost-free electron gas in thermal equilibrium with
the lattice of metal ions.

However, this line of thought didnt work at all, and I found that if I
doubled the voltage, i.e - doubled the force on the electron, then rather
than moving through the material twice as fast, the electron actually moved
through the material sqrt(2) times as fast. This was obviously at odds with
Ohms law so I knew I was doing something wrong.

---> Such behavior does exist, but when the carrier velocities approach
what is called the scatter-limit velocity. Ohm's Law does not hold
for very high fields. See below, too.

Anyway, it took me a long time to realise, no thanks to any text-book, that
I could not use the equations of motion for a single particle because what
I was actually dealing with was a body of electrons moving under the
influence of an electric field, as a whole, due to their strong
electrostatic interactions.

---> The other major consideration is that the induced velocity due to the
field is actually rather low compared to the random thermal motion of
the electrons. In other words, they are bouncing around in there
pretty darned fast without any field present. This means the added
velocity from the field is really a small correction, and is readily
transferred to the lattice as the electrons remain in thermal
equilibrium with it.

And that this body of charge was trying to move through a resistive
material that offered a resistive force. This led me to realise that in
effect, what is happening is the same as a body moving through a viscous
fluid, and that the same equations must be applied to derive Ohms law from
first principles.

---> Insofar as viscosity essentially reflects momemtum transfer in a
fluid, this is a reasonable analogy to the electron-electron and
electron-lattice momentum transfer.  You must remember, however, that
viscosity is more of a phenomenological rather than a first-principles
parameter. That is, there is no particular "fundamental reason" why
the relationship between force and velocity has to be linear. Indeed,
for many materials it is not; these are called non-Newtonian fluids.
The emphasis you find in books, on a linear relationship between force
and velocity arises because the Navier-Stokes equations are not very
solvable (without computers, which are a recent phenomenon in the
history of fluid mechanics) if viscosity is not held to be a constant.

The current that is drawn by a certain resistance is equivalent to the
terminal velocity of an object falling through a viscous fluid. But there
are 2 seperate equations for terminal velocity in the text books and I'm
not sure which one I should be using. One is for a body falling under
gravity through air. The other is a body falling through a viscous fluid.
Both of these are describing essentially the same phenomena so why are they
so different?

--->Fluids behave very differently in the laminar (or creeping) flow regime
and when the flow is turbulent. In the latter case, viscous effects are
relatively secondary. I suspect your two formulae (not having seen them
or the books you got them from, it is hard to be positive here) reflect
this. In turbulent cases, force is proportional to the square of the
velocity. For creeping flow, it is linear (for a Newtonian fluid, i.e.
the viscosity is a constant).

For example, in the free-fall in air equation, the resistive force is
proportional to the velocity squared. But in the viscous fluid, the
resistive force is proportional to the velocity. Why the difference? I
assume it's got something to do with the fact that air is compressible
while the fluid is not.

--->See above.

Therefore, you dont need as much velocity for the same force because the
resistance in the viscous fluid is greater.

--->I don't follow this line of reasoning.

Anyway, I've been using the viscous fluid equation for terminal velocity
and it explains current electricity beautifully, and fully allows you to
derive Ohms law from 1st principles.

--->What this says is that a viscous type approach...treating the electrons
as essentially a viscous fluid inside the wire...is a good model for
certain regimes of electric field. This model does not hold under all
conditions in all materials. In semiconductors, we recognize three
regimes. For low fields, velocity of the carriers (and, hence, the
current) is proportional to the applied electric field, i.e. Ohmic
behavior, and the proportionality constant is called the mobility of
the carriers in the semiconductor. However, if you go to higher fields,
your carriers approach what is called the scatter-limit velocity and
you get a carrier velocity that is proportional to the square-root of
electric field. This behavior is asymptotic to the third, very high
field, regime wherein the carriers are at scatter-limit velocity and
the velocity of the carriers, and the current, no longer increases with
increases in electric field.

feel however that the standard texts do nothing to really explain
electricity and simply quote Ohms Law as though it's obvious. It's not
obvious!

--->True. Ohm's Law is a phenomenological consequence of the fact that
field-induced carrier motion is more or less negiligible with respect
to thermal carrier motion in the material. This is not necessarily a
trivial or obvious concept.

Generations of science students deluding themselves into thinking they
understand current electricity when they don't.

---> There are levels of understanding. They may have understood at a
rudimentary level, but not grasped the full difficulties. This tends
to be true for all of us, at whatever level of comprehension we have.
People, including scientists and engineers, tend to model physical
systems with equations. These models are often very useful, but they
are also often incomplete and/or even incorrect in some ways. Many
students make the mistake of believing the model is "truth". I have
seen engineers come into the semiconductor industry who got very
flustered when real-world transistors did not "fit" the models they
had been taught in university. They were certain the devices were
"misprocessed" when in actual fact they did not understand all the
subtleties the real-world devices can exhibit. Models must always be
fitted to actual physical systems, never the reverse.

I can honestly say that only now, years after graduating do I finally
understand current electricity. When I was at University I thought I did,
but I now realise that I really didn't, and that I am by no means alone.

--->See long-winded comment above.

Do you know of any books that actually go into detail at the microscopic
level and explain current electricity from 1st principles. Even though I've
finally worked it out for myself, I'd still like to own such a book.

--->I'd suggest that you start looking into books on solid-state physics,
e.g.Kittel. These much more complicated texts tend to go much more
deeply into what's "really happening" (i.e. more sophiticated models)
inside the materials.

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