Re: Terminal velocity - which formula?

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
to others about how to get after a question. I've added comments here and
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 
started out thinking about what the average text-book says about electron 
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.