Scott Aaronson is explaining “Physics for Doofuses,” and has started with electricity. He’s got a nice breakdown of the basic quantities that you need to keep track of to understand electricity, leading up to Ohm’s Law. He asks for a little help on this point, though:

Well, as it turns out, the identities don’t always hold. That they do in most cases of interest is just an empirical fact, called Ohm’s Law. I suspect that much confusion could be eliminated in freshman physics classes, were it made clear that there’s nothing obvious about this “Law”: a new physical assumption is being introduced. (Challenge for commenters: can you give me a handwaving argument for why Ohm’s Law should hold? The rule is that your argument has to be grounded in terms of what the actual electrons in a wire are doing.)

As it turns out, this is the topic of tomorrow’s lecture for my class, and one of the things I need to do today is to dust off my slides for that lecture. Scott’s post provides a nice way to turn that productive activity into bloggy procrastination, and help me avoid grading papers while jet-lagged…

So, Ohm’s Law in terms of the motion of electrons. The key to the whole thing is a miocroscopic picture of a solid as a regular array of atoms, with electrons moving about inside. In the absence of any external forces, the electrons will sort of noddle around aimlessly, always in motion, but never really going anywhere. Any given electron is as likely to be headed to the right as to the left, and every so often, they’ll hit an atom, and change direction more or less at random. The **average** velocity of all the electrons in the material is zero, even though the average speed of an individual electron is pretty high.

This all changes if you put a voltage across the material, though.

When you put a voltage across the material, you’re essentially establishing an electric field inside the solid. Any electron in that field will experience a force that pushes it in the opposite direction from the field lines (since electrons have negative charge). A force, as we all know, produces an acceleration, so the electrons will accelerate in that direction. If you apply an electric field pointing from right to left, the electrons will feel a force accelerating them from left to right. An electron headed to the right will start moving faster, while an electron headed to the left will slow down, and eventually turn around.

Of course, there are still the ions making up the solid lattice to worry about, and the electrons will still collide with them from time to time. After a collision, the electron velocity will be redirected more or less at random, so the effect is really to limit the amount of time that an electron spends accelerating in the field. On average, an electron will travel for a time *t _{avg}* before hitting something, so the average velocity for

**all**the electrons can be written as:

v= – (_{avg}e/m)E t_{avg}

where *e* is the electron charge, *m* the electron mass, and *E* the magnitude of the electric field. The electric field is related to the voltage *V* by *V = E/L* (where *L* is the length of material in the direction of the voltage), so we have a relationship between the velocity of the electrons and the voltage across the material, which is a good start.

Now, we need to relate that velocity to the current. Current is, as Scott notes, basically a measure of the number of electrons that pass a given point in a given amount of time. You can easily relate this to the velocity of the electrons (making the simple approximation that every electron in the material is moving at the average velocity found above). The number of electrons passing a given point in any time interval *Δ t* is equal to:

N=n A v_{avg}Δ t

where *A* is the cross-sectional area of the material, and *n* is the density of electrons in the material.

You can understand this by thinking of the electrons like water in a pipe. If you want to figure out how many liters of water will come out the end of a pipe in a given amount of time, you determine the length of pipe that can be emptied in that amount of time, which is just the velocity of the water in the pipe multiplied by the time. The volume contained in that length of pipe is just the length multiplied by the area of the pipe. The only thing that changes when we’re talking about electrons is that the electrons, unlike the water, do not completely fill the volume, so we need the extra density factor.

The current through the material is the **charge** per second passing through, but we can easily account for that by multplying the number by the charge, so:

I=e N/Δ t=e n A v_{avg}

Combining this with our earlier result, we get:

I=e n A (e/m) (V/L) t_{avg}

A bunch of these things are just properties of the material: the charge density, the average time between collisions, the effective charge and mass of the electrons. We can save a bunch of writing by grouping this together into a quatity called the “conductivity” * σ *:

σ=n q^{2}t_{avg}/ m

and this lets us relate current and voltage by:

I=σ (A/L) V

or, putting it in the usual form for Ohm’s Law:

V=I (L/A σ)

The resistance, then, is the stuff after I: the length of the material that the electrons flow through divded by the cross-sectional area of the material and the conductivity (1/* &sigma* is often given the symbol * ρ*, and called the resistivity of the material, so you might see it written that way).

What does this tell us? Well, it explains a bunch of basic rules about resistors. Thin wire has higher resistance than thick wire, for example, because the cross-sectional area is smaller. It also tells you that longer wires have higher resistance. So, if you want to make a heater, use a long piece of thin wire. If you want to connect two electronic devices and not lose any voltage along the way, use a short piece of thick wire.

It also explains the rules for adding resistances in series and parallel. If you’re adding resistors in series, you’re essentially increasing the length of the material that the electrons have to flow through, so the resistances just add. If you’re connecting resistors in parallel, you’re effectively increasing the cross-sectional area, so the total resistance should decrease.

And all of that stuff comes from looking at the microscopic behavior of electrons in a solid responding to an external voltage across that solid.

(I should note that this treatment is largely lifted from the Six Ideas That Shaped Physics book, and also Principles of Physics by Serway and Jewett, which is the textbook we actually use for the class…)