Drift velocity, mobility, conductivity: Visual walkthrough

We will build, in order: the random swarm → the nudge → one electron's zig-zag → the average creep $v_{d}$ → counting electrons through a slice → current $I$ → current density $J$ → conductivity $σ$ and mobility $μ$ → the temperature edge case.

Step 1 — The swarm with no battery (the starting picture)

WHAT. A metal is a fixed lattice of positive ions with a gas of free electrons — electrons no longer tied to one atom, free to roam the whole wire. With no battery, each electron flies in a straight line, hits an ion, ricochets in a random new direction, flies again. This is called thermal motion, and it is fast: about $1 0^{5}$ metres every second.

WHY start here. Because the surprise of this whole topic is that current is not electrons starting to move — they already move like crazy. Current is a bias laid on top. To see the bias we must first see the chaos it rides on.

PICTURE. Below, each grey dot is an ion; each coloured arrow is one electron's velocity. The arrows point everywhere. Add them tip-to-tail (the dashed black arrow) and you land back where you started: the vector sum is zero.

Figure — Drift velocity, mobility, conductivity

Step 2 — Switch on the field: a universal nudge

WHAT. Connect a battery. This sets up an electric field $E$ pointing along the wire. A field is "force-per-charge waiting to happen": drop a charge $q$ in it and it feels $F = q E$ . Our charge is the electron, $q = - e$ , where $e = 1.6 \times 1 0^{- 19}$ coulombs is the (positive) size of the electron's charge.

WHY the minus sign matters. Because $q = - e$ is negative, the force $F = - e E$ points opposite to $E$ . The field points one way (our chosen positive direction); electrons get shoved the other way. Hold this thought — it is exactly the minus sign that later cancels against "current runs with $E$ ", leaving the magnitude formulas clean.

PICTURE. The field is the faint lavender arrows filling the wire (same everywhere → "uniform"). On one electron we draw the force in coral: it points against the field.

Step 3 — Zoom in on ONE electron: the zig-zag

WHAT. Follow a single electron. Between two collisions it does two things at once: it keeps its old random thermal velocity and it slowly bends because of the constant acceleration $a$ . So its path is a slightly curved dart. Then — bang — it hits an ion, and the collision randomises its direction again, wiping out whatever sideways speed $a$ had built up.

WHY collisions are the whole trick. Without collisions, $a$ would speed the electron up forever ( $v = a t$ with no ceiling) and current would grow without bound. Collisions act like a reset button: each one throws away the field-gained velocity, so on average the electron only ever accumulates the gain from one free flight. That is what keeps drift small and steady.

PICTURE. The zig-zag: straight-ish darts (each bending gently toward the force direction) punctuated by collision points (coral dots). Notice each segment leans a little more downfield than pure randomness would — that lean is the seed of drift.

Step 4 — Average the darts: the drift velocity is born

WHAT. Average the field-gained velocity over one flight. It ramps linearly from $0$ to $a τ$ , so its average across many electrons at random points in their flights is one factor of $τ$ times $a$ : $v_{d} = a τ .$

WHY only one $τ$ . Because each collision throws the counter back to zero. The electron never gets to build up more than "one flight's worth" of field velocity — the mean flight lasts $τ$ . That single, humble $τ$ is the entire memory the metal keeps of the field.

PICTURE. Same swarm as Step 1, but now the arrows are no longer perfectly random — every one has a tiny extra component pushed against $E$ . Sum them tip-to-tail: they almost cancel, but the leftover (the thick mint arrow) no longer returns to the start. That small leftover, divided by $N$ , is $v_{d}$ .

Step 5 — Count the electrons crossing a line: current

WHAT. Pick a flat slice across the wire, area $A$ . Ask: how much charge crosses it in a time $Δ t$ ? Any electron within a distance $v_{d} Δ t$ behind the slice will reach it in that time (it creeps forward at $v_{d}$ ). So the electrons that cross fill an imaginary cylinder of length $v_{d} Δ t$ and base $A$ .

WHY a cylinder. It is just "distance = speed × time" turned into a volume. Length $v_{d} Δ t$ is how far the drift carries an electron in $Δ t$ ; multiply by the slice area $A$ to get the volume that empties through. (We use $v_{d}$ as a positive magnitude here — the speed of the creep; the direction was fixed in Step 4.)

PICTURE. The wire, the slice (mint), and the shaded cylinder of length $v_{d} Δ t$ behind it. Every dot inside the cylinder crosses; dots outside do not.

Count and charge: $electrons crossing N_{cross} = per m^{3} n \times cylinder volume A v_{d} Δ t, Δ Q = e N_{cross} = e n A v_{d} Δ t .$ Here $e$ is the positive charge magnitude and $Δ Q$ is the positive amount of charge delivered forward in $Δ t$ .

Step 6 — Assemble the master current relation

WHAT. Current is charge-per-time. Divide $Δ Q$ by $Δ t$ and the time cancels: $I = \frac{Δ Q}{Δ t} = \frac{e n A v _{d} Δ t}{Δ t} .$

WHY the $Δ t$ cancels. Because current does not depend on how long we watch — a steady stream delivers charge at a fixed rate. Wait twice as long, cross twice as much charge; the ratio is fixed.

PICTURE. A "term ledger": each factor of $I = n e A v_{d}$ drawn as a knob you can turn, showing what physically changes.

Strip away the wire's thickness by dividing by $A$ to get the current density $J$ (current per unit area): $J = \frac{I}{A} = n e v_{d} .$ This is a property of the material and field, not of how thick the wire happens to be. As a full vector, $J = n e v_{d} = - n e v_{d} \hat{E}$ points along $E$ because $v_{d}$ itself points against it — again the two signs cancel to give $J$ along $E$ .

Step 7 — Substitute drift into $J$ : conductivity appears

WHAT. We have two facts (both as positive magnitudes in the $+ E$ direction): $J = n e v_{d} (Step 6) and v_{d} = \frac{e E τ}{m} (Step 4) .$ Put the second into the first: $J = n e \cdot \frac{e E τ}{m} = (\frac{n e ^{2} τ}{m}) E .$

WHY this is the payoff. Everything in the bracket — $n, e, τ, m$ — is a fixed property of the metal, independent of $E$ (that was the assumption flagged in Step 3). So $J$ is simply proportional to $E$ . That proportionality is exactly what Ohm's law claims. We didn't assume Ohm's law; it emerged.

PICTURE. The substitution shown as $v_{d}$ (mint) being slotted into the $J$ expression, with the constant lump getting circled and renamed $σ$ .

Step 8 — Mobility: the same physics, re-packaged

WHAT. Define mobility $μ$ as "how much drift you get per unit field": $μ = \frac{v _{d}}{E} .$ Since $v_{d} = e E τ / m$ , the $E$ cancels: $μ = \frac{e τ}{m} .$

WHY bother, when we already have $σ$ ? Because $μ$ isolates the responsiveness of one carrier, separate from how many carriers there are. In semiconductors, where $n$ can change by huge factors, splitting "how many" ( $n$ ) from "how nimble" ( $μ$ ) is the natural language.

PICTURE. A straight line: $v_{d}$ on the vertical axis, $E$ on the horizontal. Its slope is $μ$ . Steeper line = more mobile carrier.

Step 9 — The temperature edge case (and what happens near absolute zero)

WHAT. Heat the wire. The ions vibrate harder, so a moving electron meets them more often — the average free flight $τ$ gets shorter. Nothing else in $σ = n e^{2} τ / m$ changes much. So $τ ↓\Rightarrow σ ↓\Rightarrow ρ ↑$ .

WHY this is a trap worth its own step. Thermal speed rises with temperature ( $\propto T$ ), which tempts you to think drift rises too. But drift depends on $τ$ , and $τ$ falls with heat. Bigger thermal jiggle → more obstacles → smaller $τ$ → smaller drift for the same field. The two speeds move in opposite directions with temperature.

WHY $σ$ does NOT blow up at $T \to 0$ . Cooling makes ion vibration (called phonon scattering) fade away, which alone would send $τ \to \infty$ and $σ \to \infty$ — unphysical. Real metals dodge this because electrons also scatter off frozen-in impurities and defects, which do not care about temperature. This leftover scattering fixes a floor on collisions, so at $T \to 0$ the resistivity settles at a nonzero residual resistivity $ρ_{0}$ , and $σ$ stays finite. (This split — vibration part + impurity part — is Matthiessen's rule.)

PICTURE. Two panels: cold lattice (ions still, long free flights) vs hot lattice (ions blurred with vibration, short chopped-up flights).

The one-picture summary — the nontrivial takeaway

The nine steps chain up, but the insight worth carrying away is subtler than the chain itself: one microscopic constant, $τ$ , secretly controls the entire macroscopic story. Change $τ$ (by heating, by adding impurities) and $v_{d}$ , $I$ , $J$ , $σ$ , $ρ$ , and $μ$ all move in lockstep — every observable in this topic is really $τ$ in disguise. The figure below makes that single lever visible: pull $τ$ and watch the whole column respond.

Recall Feynman retelling — the whole walkthrough in plain words

Picture a jar of bees, each zipping around super fast, going nowhere on average (Step 1). Now tilt the jar ever so slightly — that tilt is the electric field's push (Step 2). Each bee still zips randomly, but every flight bends a hair downhill, and every time a bee bumps a wall it forgets the bend and starts fresh (Step 3). Average over all the bees and their bumps and you find a tiny, steady drift downhill (Step 4) — set by one average time-between-bumps, $τ$ . To count how many bees pass a doorway per second, sweep out the little cylinder they can reach in that time (Step 5): that gives $I = n e A v_{d}$ (Step 6). Strip the doorway's size and you get $J = n e v_{d}$ ; plug in the drift and the number of bees, their charge, their bump-time and weight all lump together into one constant — the conductivity $σ = n e^{2} τ / m$ , which is just Ohm's law wearing a microscope (Step 7). Re-label the "nimbleness of one bee" as mobility $μ$ , and $σ = n e μ$ (Step 8). Finally, heat the jar: the walls shake, bumps come faster, $τ$ drops, and the whole thing conducts worse — but even in a perfectly cold jar the bees still bump the odd stuck crumb (an impurity), so it never conducts perfectly (Step 9). One knob, $τ$ , quietly runs the whole show.

Recall Quick self-test

Where does the single factor of $τ$ come from? ::: The field-gained velocity ramps from 0 to $a τ$ over one flight; averaging leaves one $τ$ . Why does $Δ t$ vanish from $I = n e A v_{d}$ ? ::: Current is a rate; the cylinder's charge and the time both scale with $Δ t$ , so their ratio is fixed. Why are $I = n e A v_{d}$ and $J = n e v_{d}$ sign-clean despite the electron's $- e$ ? ::: The electron's negative charge and its backward drift are two minus signs that cancel, so current runs with $E$ and the magnitudes are all positive. Split $σ$ into "how many" and "how nimble". ::: $σ = (n e) \times μ$ — carrier count times mobility. Why does heating raise resistance if thermal speed rises? ::: Drift depends on $τ$ , not thermal speed; hotter ions collide more, so $τ$ and hence $σ$ fall. Why doesn't $σ \to \infty$ as $T \to 0$ ? ::: Impurity/defect scattering survives, leaving a residual resistivity $ρ_{0}$ ; $σ$ stays finite.

Connections

Ohm's Law — Step 7's $J = σ E$ is $V = I R$ under a microscope.
Current density — Step 6's $J = n e v_{d}$ .
Free electron model of metals — supplies $n$ and the random swarm of Step 1.
Electric field inside conductors — the $E$ of Step 2.
Resistivity and temperature dependence — Step 9's $τ (T)$ and residual resistivity.
Semiconductors — where the $n$ vs $μ$ split of Step 8 earns its keep.