Subject: 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.  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. 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.  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.  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?

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

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.

I 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!  
Generations of science students deluding themselves into thinking they 
understand current electricity when they dont. 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.   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.

Re: Terminal velocity - which formula?