Understanding dissipation of energy in a resistor through the Drude model

[zenterix]

TL;DR Summary

There is a snippet in a book explaining why there is dissipation of energy in a resistor by explaining what happens at the level of charge carriers. I'd like to understand this snippet.

In section 4.8 entitled "Energy Dissipation in Current Flow" of Purcell and Morin's Electricity and Magnetism, there is the following snippet

The flow of current in a resistor involves dissipation of energy. If it takes a force $\vec{F}$ to push a carrier along with average velocity $\vec{v}$, any agency that accomplishes this must do work at the rate $\vec{F}\cdot\vec{V}$. If an electric field $\vec{E}$ is driving the ion of charge $q$, then $\vec{F}=q\vec{E}$, and the rate at which work is done is $q\vec{E}\cdot\vec{V}$. The energy thus expended shows up eventually as heat. In our model of ionic conduction, the way this comes about is quite clear. The ion acquires some extra kinetic energy, as well as momentum, between collisions. A collision, or at most a few collisions, redirects its momentum at random but does not necessarily restore the kinetic energy to normal. For that to happen the ion has to transfer kinetic energy to the obstacle that deflects it. Suppose the charge carrier has a considerably smaller mass than the neutral atom it collides with. The average transfer of kinetic energy is small when a billiard ball collides with a bowling ball. Therefore the ion (billiard ball will continue to accumulate extra energy until its average kinetic energy is so high that its average loss of energy in a collision equals the amount gained between collisions. In this way, by first “heating up” the charge carriers themselves, the work done by the electrical force driving the charge carriers is eventually passed onto the rest of the medium as random kinetic energy, or heat.

The model in question is the Drude model I believe (though the book does not seem to give the model any name).

There are some paragraphs like this in the book that I have a difficult time understanding, and this difficulty does not arise because I didn't understand the material that came before. It seems to be because the language used isn't clear.

For example, what does "restore kinetic energy back to normal" mean? In particular, it is not clear what "normal" means here.

Here is my interpretation
- We have electrons that are modeled as moving randomly and colliding with each other and with protons, the latter with much larger mass. The average thermal speed of these electrons can be estimated with a theory such as the kinetic theory of gases.
- Then we add an electric field. The model predicts that this gives a drift velocity to electrons of, say, a wire.
- An electron collides with a proton and loses part of the newly acquired momentum due to the electric field. Then it gains more momentum than it had before due to the electric field.
- This keeps happening until the electrons are moving fast enough such that when they lose momentum in a collision, they gain the exact amount in between collisions due to the electric field.

So in this intepretation, "normal" means the thermal speed of electrons. "Heating" up the electrons means increasing their speed to a particular level at which the loss of momentum in collision equals gain of momentum between collisions. Since speed of electrons is related to temperature, this means we have increased the temperature.

I am not sure about the distinction between temperature and heat at this point.

 

[lightgrav]

This is essentially the Drude model.
In your interpretation, line 1: the electrons don't need to collide with each other if they collide with the lattice. The lattice is not protons (except in Jupiter's core), but rather copper or other metal ion cores.

In the "standard intro" Drude model [say, in Sears, Halliday, or Tipler], the electron loses _all_ the extra momentum it got from the Electric Field. So after any collision it always starts out with randomly-oriented momentum at the speed which has ½mv^2 = ½kT ... keeping the Temperature of the electrons the same as the Temperature of the wire ions.

Because momentum is a conserved vector, the electron momentum loss in a collision must equal the lattice momentum gain in that collision. Purcell hints that the Energy transfer might go as the ratio of the masses. This would mean that a Cu ion only gets 1/115000 of the Electric Field's Work that was done to the electron - so the electron must do 114999 collisions before it gets back to thermal equilibrium with the lattice. If it does this many, on average, while it travels a "drift velocity * collision time" distance, it stays in thermal equilibrium.
(This holds for Electric fields less than about 0.04 V/m in copper).

So Purcell says that if you push them hard enough, the electrons can get hotter than the lattice.

He wants to embellish the "intro" Drude model, because Drude finds that the good metals should have their resistance proportional to the square-root of the Temperature ... not linear with T (oops).

"Heat" is a verb where some other Energy form is transformed into Thermal Energy (kT)