|MadSci Network: Physics|
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.
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