Ohm's law — microscopic origin, resistivity: Visual walkthrough

We only need three plain ideas to start, so let's name them in pictures first.

Step 1 — Put one electron in the field

WHAT. Drop a single electron into a region where the field $E$ points to the right. It feels a force.

WHY this step. Everything in physics that moves starts with a force. We cannot talk about "current" (a crowd) until we know what one electron does. So we begin with the simplest possible honest question: what force acts here?

PICTURE. Look at the figure. The orange arrows are the field $E$ pointing right. The electron (teal dot) feels a force $F$ pointing left — opposite to the arrows — because its charge is negative.

$F = charge (negative) - e the push E$

$- e$ : the electron's charge. The minus is why the force points backward relative to $E$ .
$E$ : the field arrows. Bigger $E$ → bigger force. Simple proportionality.

This is just the rule "force = charge × field", the same rule from Electric field inside a conductor.

Step 2 — Force becomes acceleration

WHAT. Turn that force into a change in motion using Newton's second law.

WHY this step. A force alone tells us nothing about speed until we know the mass it acts on. Newton's law $F = m a$ is the bridge from "push" to "picking up speed."

PICTURE. Same electron, now with a green acceleration arrow. It points the same way as the force (left) and its length is the force divided by the electron's mass $m$ .

$a = \frac{F}{m} = how strongly it responds - \frac{e}{m} E$

$m$ : mass of the electron, $9.1 \times 1 0^{- 31}$ kg. A tiny mass, so the acceleration is huge.
$a$ : the rate the velocity grows. If nothing stopped it, the electron would go faster and faster forever.

Step 3 — The collision brake

WHAT. Between crashes the electron speeds up; then it slams into a lattice atom and its velocity is scrambled to random — on average, back to zero drift.

WHY this step. This is the heart of the whole story. The field keeps trying to build up velocity; the crashes keep wiping it out. Without this brake there is no steady state and no Ohm's law.

PICTURE. The zig-zag path. Each straight segment is a "free flight" where the field bends the path slightly (curving left). Each red dot is a collision that resets the direction. The average time of one free flight is called $τ$ — the relaxation time.

Step 4 — Average the drift gained in one flight

WHAT. Starting from zero drift right after a collision, the electron accelerates for a time $τ$ before the next crash. Its typical extra velocity is $a$ times $τ$ .

WHY this step. We want the steady average velocity of the crowd, not the wiggling instant-by-instant speed. Averaging "gain over one flight, then reset" gives one clean number that does not change with time — the definition of steady state.

PICTURE. A velocity-vs-time sawtooth: velocity climbs along the field during each flight, drops to zero at each crash. The dashed horizontal line is the average — that is $v_{d}$ .

$v_{d} = a τ = drift per unit field - \frac{e τ}{m} E$

$v_{d}$ : drift velocity — the small steady average of the whole electron sea (see Drift and diffusion of carriers).
$τ$ : longer time between crashes → more time to speed up → bigger drift.
Notice: $v_{d}$ is constant in time. That is why current is steady, not runaway.

This drift rides on top of fast random thermal jiggling, but the jiggling averages to zero and carries no net current — only $v_{d}$ does. This is the full logic of the Drude model of conduction.

Step 5 — Count the charge that crosses (drift → current)

WHAT. Turn "each electron drifts at $v_{d}$ " into "how much charge passes a cross-section each second."

WHY this step. Current is charge per second through a surface. We must count carriers, not track one. So we slice the wire and count everyone who sweeps through.

PICTURE. A cylinder of the wire, cross-section area $A$ . In a time $d t$ every electron within a distance $v_{d} d t$ of the face will cross it. That's a slab of volume $A v_{d} d t$ .

$I = \frac{d Q}{d t}, d Q = electrons per m^{3} n slab volume (A v_{d} d t) charge each e$

Dividing charge by time and by area:

$J \equiv \frac{I}{A} = n e v_{d}$

$n$ : how many free electrons sit in one cubic metre (for copper, $8.5 \times 1 0^{28}$ ).
$J$ : current density — current per unit area, the local, geometry-free version of current (see Electric current and current density).

Step 6 — Substitute and reveal Ohm's law

WHAT. Feed the drift from Step 4 into the current density from Step 5.

WHY this step. Steps 4 and 5 are two separate facts: one says how fast electrons drift, one says how drift makes current. Combining them removes $v_{d}$ and leaves current density in terms of the field alone.

PICTURE. The two boxed results snapping together like puzzle pieces, out pops a straight line: $J$ against $E$ .

Using magnitudes, put $v_{d} = \frac{e τ}{m} E$ into $J = n e v_{d}$ :

$J = n e \cdot \frac{e τ}{m} E = call this σ \frac{n e ^{2} τ}{m} E$

$J = σ E, σ = \frac{n e ^{2} τ}{m}$

Step 7 — Zoom out: from $J = σ E$ to $V = I R$

WHAT. Wrap the local law over a whole wire of length $L$ and area $A$ .

WHY this step. $J = σ E$ is a statement at each point. To use it in a circuit we need the whole-wire quantities voltage $V$ and total current $I$ .

PICTURE. The wire as a straight pipe. Across its length $L$ the voltage $V$ makes a uniform field $E = V / L$ ; the total current is $I = J A$ .

$E = \frac{V}{L}, I = J A = σ E A = σ \frac{V}{L} A$

Rearrange for $V$ :

$V = R \frac{L}{σ A} I = \frac{ρ L}{A} I \Rightarrow V = I R, R = ρ \frac{L}{A}$

$L$ : longer wire → more collisions in series → bigger $R$ .
$A$ : fatter wire → more parallel lanes → smaller $R$ .

This geometry factor is what stacks up in Resistors in series and parallel.

Step 8 — The edge & degenerate cases

We must never leave the reader in a scenario we didn't draw.

$E = 0$ (field off). Then $a = 0$ , so $v_{d} = 0$ , so $J = 0$ . No push, no drift, no current — only the random thermal jiggle that averages to nothing. The figure shows a pure random walk with zero net displacement.
$τ \to 0$ (crashes non-stop). Drift is wiped out instantly: $v_{d} \to 0$ , so $σ = n e^{2} τ / m \to 0$ and $ρ \to \infty$ . A perfect insulator-like limit — never enough free flight to build speed.
$τ \to \infty$ (no crashes). Then $σ \to \infty$ , $ρ \to 0$ : a perfect conductor. But also $v_{d} = a τ \to \infty$ — the runaway from Step 2 returns. This confirms collisions are exactly what tames the runaway.
Heat it up. More lattice vibration → more crashes → $τ$ drops → $ρ = m / (n e^{2} τ)$ rises. That is Temperature dependence of resistance. (In semiconductors rising $T$ boosts $n$ so much that $ρ$ can fall instead.)
Non-ohmic materials. If $τ$ or $n$ themselves change with $E$ (filaments, diodes, plasmas), then $σ$ is no longer constant and the straight line bends. Ohm's law was only ever the constant- $σ$ special case.

The one-picture summary

Recall Feynman retelling — the whole walkthrough in plain words

Picture pushing a shopping cart down a hallway packed with people. You lean on it steadily — that's the field. The cart starts to speed up (that's acceleration), but every couple of steps you thump into someone and stop dead, then start pushing again. Because you keep getting reset, you never actually go fast — you settle into a slow, steady walk. That steady walk is the drift velocity, and how far apart the people are sets how long you get to speed up between bumps — that's $τ$ . Now imagine a whole crowd of you, all pushing carts in the same direction. Count how many carts pass a doorway each second — that's the current. Since each of you walks at the same steady drift speed, more people per square metre or a wider doorway means more carts pass: that's $J = n e v_{d}$ . Put the two facts together — "your steady speed depends on the push" and "current depends on your speed" — and you get: current is proportional to push. Bigger voltage, more current, in a straight line. That straight line is Ohm's law. And if you heat the hallway so everyone wiggles and bumps you more often, you get reset sooner, walk slower, and it's harder to push through: resistance goes up.

Active Recall

Which single fact prevents the current from running away to infinity?

Collisions reset the drift velocity every relaxation time

τ

, so drift stays finite and steady.

In Step 4, why does the average drift equal

a τ

and not keep growing?

Because each collision returns the drift to zero, so only one free-flight's worth of speed-up ever accumulates.

What makes

J = σ E

deserve the name "Ohm's law"?

σ = n e^{2} τ / m

has no

E

in it, so

J \propto E

— a straight proportional line.

In the

τ \to \infty

limit, what happens and why does it matter?

σ \to \infty

ρ \to 0

(perfect conductor) but

v_{d} \to \infty

— the runaway returns, proving collisions are what tame it.

When

E = 0

, why is there still no current despite thermal motion?

Thermal velocities are random and average to zero, carrying no net charge across any surface.

Connections

Ohm's law — microscopic origin, resistivity (index 1.8.16)
Electric current and current density
Electric field inside a conductor
Drude model of conduction
Drift and diffusion of carriers
Resistors in series and parallel
Temperature dependence of resistance
Semiconductors vs metals
Joule heating — power dissipation