Foundations — Drift velocity, mobility, conductivity

This page assumes nothing. Before you meet $v_d$, $\tau$, $\sigma$, or $\mu$ on the parent topic, we build every letter, arrow and picture it stands on. Read top to bottom; each symbol is earned before the next uses it.

0. What is a "free electron"? (the actors)

A metal is a lattice of positive ions — atoms that gave up their outermost electron and stayed frozen in a grid. Those handed-over electrons roam the whole block freely; we call them free electrons. They are the only things that actually move to carry current.

Look at the figure: the violet dots pinned in a grid are ions; the magenta dots zig-zagging between them are free electrons. Every symbol below describes these electrons. See Free electron model of metals for why this simple picture works so well.

1. Charge $e$ — the size of one electron's "electric weight"

Picture: think of $e$ as the "electric weight" each electron carries — the same for every electron, a fixed pebble of charge.

Why the topic needs it: current is charge flowing per second. To count how much charge crosses a line, we multiply the number of electrons by the charge each one carries, $e$. Without $e$ we could count electrons but never convert them to amperes.

Why the minus sign matters: because the electron's charge is $-e$, an electric field pushes it backwards relative to the field. We'll keep track of this in §5.

2. Electric field $E$ — the "sideways nudge"

Picture: an arrow at every point in the wire, all pointing the same way when a battery is connected — a steady wind blowing through the metal. The little arrow over $\vec{E}$ (and $\vec{F}$) means "this quantity has a direction", not just a size.

Why the topic needs it: with no field the crowd goes nowhere. The field is the only reason a net drift appears. It is the cause; drift is the effect. Where does this field come from and how deep into the wire does it reach? See Electric field inside conductors.

3. Force $F$, mass $m$, acceleration $a$ — Newton enters

Why we bring in $F=ma$ and not something fancier: we want to know how the field speeds up an electron. The most basic law of "force makes things accelerate" is exactly Newton's — no relativity, no quantum machinery needed, because drift speeds are absurdly small. So the right tool is the simplest one.

Combine §2 and §3: an electron in the field feels $\vec{a}=\frac{\vec{F}}{m}=\frac{-e\vec{E}}{m}.$ This is the very first line of the parent's derivation — now every symbol in it is defined.

4. Thermal motion vs. drift — the two speeds

Here is the subtle part the whole topic hinges on. An electron has two motions at once, and confusing them is the classic error.

Look at the two panels. Left: no field — every red arrow points a different way; add them tip-to-tail and you return to the start (net = zero). Right: field on — the same wild zig-zag, but each segment sags slightly leftward (electron, negative, drifts against $\vec{E}$); add them up and you land a little to the left. That tiny leftover displacement per second is $v_d$.

5. Relaxation time $\tau$ — why the drift stays small

If the field keeps accelerating electrons, why don't they get faster and faster forever? Because they keep colliding with the vibrating ions, and each collision randomizes direction — wiping out the speed the field had just built up.

Read the sawtooth: between collisions the field builds drift speed linearly (slope $=a$); at each collision (the vertical drops) the gained drift is knocked back toward zero. The electron never runs away — it gains for a time $\tau$, loses it, gains again. So the average extra velocity is roughly $a\tau$: $v_d\approx a\tau =\frac{eE}{m}\tau .$

Why $\tau$ matters for temperature: heat the metal → ions vibrate harder → electrons hit them sooner → $\tau$ shrinks → drift shrinks → resistance rises. This is the seed of Resistivity and temperature dependence.

6. Number density $n$ — how crowded the wire is

Picture: the density of dots in the metal block of §0. More dots per box = larger $n$.

Why the topic needs it: current counts electrons crossing a line. To know how many cross, you must know how densely they're packed. Even a snail-slow drift moves a giant current when $n$ is this huge — that's the resolution of "slow electrons, big current."

7. Area $A$ and the counting cylinder — building $I=neA{v}_d$

The counting trick (why a cylinder?): in a time $\Delta t$, every electron within a distance $v_d\Delta t$ behind the slice will just manage to cross it. That's a cylinder of length $v_d\Delta t$ and face area $A$. Its volume is $A{v}_d\Delta t$; it holds $n\times (A{v}_d\Delta t)$ electrons; they carry charge $\Delta Q=enA{v}_d\Delta t$. Divide by $\Delta t$: $I=\frac{\Delta Q}{\Delta t}=neA{v}_d.$

Dividing both sides by $A$ gives current density $J=I/A=ne{v}_d$ — the current per unit area. See Current density.

8. Conductivity $\sigma$ and mobility $\mu$ — the two "how easily" numbers

Two final symbols summarise everything by asking "how easily does this metal conduct?"

Why two symbols? $\mu$ describes one electron's agility; $\sigma$ folds in how many electrons ($n$) and their charge ($e$) to describe the whole material. The bridge $\sigma =ne\mu$ links the single-particle picture to the bulk material — and $J=\sigma E$ is exactly Ohm's Law wearing a microscope. In Semiconductors, both $n$ and $\mu$ change dramatically, which is why that link is so useful.

Prerequisite map

Equipment checklist

Connections

• Free electron model of metals — where $n$ and the random motion come from.
• Electric field inside conductors — the source of $\vec{E}$ that drives drift.
• Current density — $J=ne{v}_d$, built in §7.
• Ohm's Law — $J=\sigma E$ is its microscopic face.
• Resistivity and temperature dependence — how $\tau$ shrinks with heat.
• Semiconductors — where $n$ and $\mu$ both vary strongly.