Physics - Classical Theory of Conduction

The following is a brief overview of the classical theory of conduction.

Fluid Analogy
First I would like to invoke the fluid analogy to electricity. Basically there are many similarities between different phenomena for fluid mechanics and electronics. The reason why we have this analogy is simply that most people are used to dealing with fluids, especially those of us with access to plumbing and running water. However, most of us don't even think about electricity, we just put a plug in an outlet and stuff works.

Now how does a battery work? Its no mystery though Bill O'Reilly might disagree. We can see on a battery's label a + and a - sign. We can think of this as a lake at the top of a mountain and a lake on the bottom of the mountain. They have different potential energy due to gravity. When we plug in a battery its like digging a trench to connect the two lakes and putting a water mill in between it to drive our machine.

Relation to Conduction
We can then relate two other phenomena. Current in a wire is very similar to trying to get water to move through a pipe. The smaller our pipe, the less current can get through, which is similar to a wire with high resistivity. In electronics we can think of a wire as a lattice of of atoms. Our electrical current is then though of as electrons traveling through this lattice. An electrical field caused by our difference in potential drives electrons through this lattice but the lattice is stationary and stops/causes ricochets when an electron hits it. We can describe this with the following two equations:

I =ΔQ/Δt = neAv_d and
λ = <v> τ =1/n_aπr² as

Where for the first equation I is the current described by n the number of electrons, with charge e, passing through an area A with electron drift velocity v_d. The second equation gives us λ is the mean free path, or the average distance a particle travels before a collision with <v> the average velocity multiplied against τ the average time between collisions. Through this we can figure out resistivity and conductivity and can derive Ohm's law. 

V = IR

Defects

This is a great approximation but there are significant errors when scaled to different temperatures and other quantum effects.

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