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 Originally Posted by a500lbgorilla
These aren't proofs to me. How can a linear relationship like V=IR be a proof? I was thinking more of Algorithms and how you can prove the sort of 'time' they execute on. O(n) or O(ln(n)) time kinda of stuff. I haven't really studied it.
Did you just use 'big-O' notation up in here?
The end result of the proof is not the proof, no. The proof starts with axioms and applies logical operations to demonstrate the result is consistent with the axioms.
Here's a bit of a derivation based on classical mechanics which still shows the temperature dependence of resistivity.
This is short, but heavy in math symbols. I like it because it should have occurred to me sooner that the mean free path of conducting electrons would be parabolic trajectories between bounces. Of course, it's more accurately described by waves and QM, but that classical picture is cool.
The relationship V = IR holds in a certain domain, but it is a generalization (a linearization) of a more complicated relationship. It only even applies to "good conductors." In general, the resistivity of a material is a function of its temperature, atomic components and crystal lattice structure.
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The outermost electrons in an atom are shielded from the charge of the nucleus by the inner electrons. For conductors, the outermost electrons are weakly bound. This means that they may flow from one atom to another with a small amount of energy. For what we call conductors, it can be shown that the average thermal motions of the particles is above this energy threshold. This means that there is enough thermal energy to cause the electrons to randomly hop around their neighbor atoms' outer shells.
A look at the electric fields shows that the periodic nature of the metallic crystalline bonds gives rise to an interesting structure of allowed energies. This observed periodic potential becomes the axiom we add to QM to postulate an explanation for resistivity.
A free particle can have any energy, but a bound particle can only have specific energies. We call those specific energies "allowed" energies. The electrons in a material are bound, and therefore have specific energies available to them, while other energies are prohibited. The periodic potential gives rise to a banding of allowed energies. There will be many allowed energies with very close values, interspersed with wide gaps where there are no allowed energies. The nature of this banding determines the resistivity of the material.
If there is an allowed band near the thermal energy, then the electrons are allowed to flow freely. This is called a conductor.
If the thermal energy is near a "forbidden" band, then the electrons do not flow freely. This is called a dielectric (insulator).
For Semi-conductors, the material has a conducting band just above the thermal energy. A small electric field is enough to add the needed energy to turn the material from a dielectric into a conductor.
It is by this model, based on the axioms of QM, that we can prove that resistivity and Ohm's Law are implicated by the Standard Model.
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