# What is resistance, if electrons do not hit atoms?

> From measured numbers: an electron in copper passes about 150 atoms before one deflection. Change metal and temperature to see where resistance comes from.

Canonical URL: https://learn.chabulab.com/nature/what-is-electrical-resistance. Facts checked: 2026-11-04.

- Resistance does not come from electrons hitting every atom. In copper at room temperature an electron flies about 39 nm, about 150 atoms, between two deflections.
- Electrons are deflected where the lattice loses its order: atoms shaking with heat, impurities and defects. That is why a metal's resistance falls as it cools.
- A metal's resistance usually does not reach zero when cold: copper stops at about 2 × 10⁻¹¹ Ω·m because the impurities are still there.
- Aluminium has twice as many free electrons as copper and still conducts worse, because its electrons are deflected about 3.4 times as often.

## What is resistance?

Resistance says how much something opposes an electric current, and the energy lost to it turns into heat. That is why a charging cable gets warm after a while.

The resistance of a wire is `R = ρ × L / A`. A longer wire resists more and a thicker one resists less. The number `ρ`, called **resistivity**, belongs to the material. Copper at 20 °C has `ρ` of about 1.68 × 10⁻⁸ Ω·m, so one metre of 1.5 mm² wire has about 0.011 Ω. At 10 A, that metre gives off about 1.12 W.

The formula says how much, not why. The more interesting question is where `ρ` comes from.

## Do electrons hit every atom?

No. In copper at room temperature an electron flies about 39 nm between two deflections, while neighbouring atoms are only about 0.256 nm apart. So each flight passes about **150 atoms**.

This number comes from measurements, with no assumption about collisions:

1. Time between deflections: `τ = m / (n × e² × ρ)`. For copper, `τ` is about 2.5 × 10⁻¹⁴ s.
2. Free path: `ℓ = Fermi speed × τ`, with copper's Fermi speed of 1.57 × 10⁶ m/s. The result is about 39 nm, close to the 40 nm that [a study of metal wires](https://www2.avs.org/symposium2014/Papers/Paper_EM-TuM12.html) gives for bulk copper.
3. Atom spacing from the lattice constant of 361.5 pm: `361.5 / √2` gives about 255.6 pm.

[School lessons](https://loigiaihay.com/ly-thuyet-dien-tro-vat-li-11-canh-dieu-a150864.html) often say that electrons collide with each other and with the ions in the lattice. That picture is enough to remember that heat raises resistance, but it cannot explain how an electron passes a hundred atoms without being deflected.

A more accurate way to see it: electrons behave as waves, and a wave travels a long way through an evenly spaced lattice. Light passes through a sheet of glass in much the same way, even though the glass is full of atoms. [The free path is not the result of electron and ion collisions](https://en.wikipedia.org/wiki/Free_electron_model). It comes from imperfections in the material: defects, impurities and thermal vibrations.

![An illustration of a copper wire cut open, showing even rows of copper atoms. An electron path runs straight past many atoms and bends only at one atom out of its row.](/art/lesson-lattice.webp)

*The bench: pick a metal and a temperature and read the resistivity on the gauge. Each bright ring in the viewport is one electron being deflected.*

Try them in this order:

1. **Copper, 293 K.** The viewport is 14 atoms wide, so an electron usually crosses all of it without being deflected.
2. **Lead.** The free path falls to about 6 atoms, and rings appear all the time.
3. **Copper, 77 K and then 4 K.** The atoms nearly stop and the rings nearly vanish, but the resistivity never reaches 0.

## Why does a cold metal resist less, but not zero?

As it cools, atoms shake less, so electrons are deflected less. But impurities and defects are still there, so the curve stops at a floor.

The table takes pure copper's resistivity from [Matula's tables](https://srd.nist.gov/jpcrdreprint/1.555614.pdf). The 77 K row is interpolated between the 70 K and 80 K rows. The last two columns come from the same formulas as above.

| Temperature | Resistivity (Ω·m) | Free path | Atoms passed |
| --- | --- | --- | --- |
| 600 K | 3.79 × 10⁻⁸ | 17 nm | about 68 |
| 293 K | 1.68 × 10⁻⁸ | 39 nm | about 153 |
| 77 K | 1.95 × 10⁻⁹ | 337 nm | about 1,300 |
| 20 K | 2.80 × 10⁻¹¹ | 23 µm | about 92,000 |
| 4 K | 2.00 × 10⁻¹¹ | 33 µm | about 129,000 |

From 293 K to 77 K, resistivity falls about 8.6 times. From 20 K to 4 K it falls only 1.4 times more, because it has reached the floor. The floor depends on the sample's purity: 2 × 10⁻¹¹ Ω·m is what Matula chose for copper, and gold in the same paper has 2.2 × 10⁻¹⁰ Ω·m. In some other materials the resistance drops straight to 0 at a low enough temperature. That is superconductivity, the subject of the next lesson.

## Why does aluminium conduct worse with more electrons than copper?

Because resistivity depends on the number of electrons and on how often they are deflected, and aluminium loses on the second. It has about 2.1 times copper's free electrons, but only about 1/3.4 of the time between deflections. In the table, ρ is resistivity, time is the time between deflections, and atoms is how many an electron passes.

| Metal | Electrons (10²⁸/m³) | ρ (10⁻⁸ Ω·m) | Time (fs) | Atoms |
| --- | --- | --- | --- | --- |
| Silver | 5.86 | 1.59 | 38.2 | 184 |
| Copper | 8.47 | 1.68 | 25.0 | 153 |
| Gold | 5.90 | 2.21 | 27.2 | 132 |
| Aluminium | 18.1 | 2.65 | 7.4 | 52 |
| Iron | 17.0 | 9.71 | 2.2 | 17 |
| Lead | 13.2 | 22.0 | 1.2 | 6 |

The table shows that the number of free electrons does not set the order: silver comes first with the fewest electrons. It only shows the result. Why each metal has its time needs a calculation from its electron structure and its vibrations, which this lesson does not attempt. Read iron's numbers for their size only, because the free-electron model describes iron poorly.

## Do electrons move fast or slowly?

Both, depending on which speed you mean. Electrons fly in every direction at about 1,570 km/s, but on average they drift along the wire very slowly. [HyperPhysics](https://hyperphysics.gsu.edu/hbase/electric/ohmmic.html) gives the same two scales: millimetres a second and a million metres a second.

In a 1.5 mm² copper wire at 10 A, the drift is `I / (n × e × A)`, about 0.49 mm/s, about 3.2 billion times slower than the flight. A lamp still lights almost at once, because electrons already fill the wire and the electric field spreads very fast, so electrons along the whole wire start to drift almost together.

## Which number matters in which situation?

| Situation | Number to look at | What happens |
| --- | --- | --- |
| Swapping copper wire for aluminium of the same size | Resistivity | Heat rises about 1.58 times, so it needs about that much more cross-section to stay the same |
| Same material, twice the cross-section | Cross-section | Heat per metre halves, because `P = I² × ρ / A` |
| Cooling copper wire to 77 K | Resistivity at that temperature | Resistance falls about 8.6 times |
| Copper wire near 39 nm across, as in a chip | Free path | [Surfaces and grain boundaries deflect electrons too](https://www2.avs.org/symposium2014/Papers/Paper_EM-TuM12.html), so resistivity rises above bulk copper |

This table is for understanding the mechanism, not for choosing real wiring. For wiring in a building, follow the electrical code and use a qualified electrician.

## How can I work these numbers out myself?

The code below only needs Node.js, with nothing else to install. It ran on Node 26.11.1 on macOS on 4 November 2026. The install commands for Windows and Ubuntu below have not been run.

**Install Node.js on Windows, macOS or Ubuntu**

Check first with `node --version`. If it prints a version number you already have Node and can skip this.

- **Windows:** open PowerShell, run `winget install OpenJS.NodeJS.LTS`, then reopen PowerShell.
- **macOS:** install [Homebrew](https://brew.sh) if you do not have it, then run `brew install node`.
- **Ubuntu:** run `sudo apt update && sudo apt install -y nodejs`.



Create a file named `dien-tro.mjs` with this content:

```js
// Electrons in a metal, from three measured numbers per metal.
const m = 9.109e-31; // electron mass, kg
const e = 1.602e-19; // electron charge, C

// rho: resistivity at about 20 °C (ohm metres). n: free electrons per m³.
// v: Fermi speed (m/s). d: distance between neighbouring atoms (m).
const metals = {
  silver: { rho: 1.587e-8, n: 5.86e28, v: 1.39e6, d: 2.889e-10 },
  copper: { rho: 1.678e-8, n: 8.47e28, v: 1.57e6, d: 2.556e-10 },
  gold: { rho: 2.214e-8, n: 5.9e28, v: 1.4e6, d: 2.884e-10 },
  aluminium: { rho: 2.65e-8, n: 18.1e28, v: 2.03e6, d: 2.863e-10 },
  iron: { rho: 9.71e-8, n: 17e28, v: 1.98e6, d: 2.482e-10 },
  lead: { rho: 22e-8, n: 13.2e28, v: 1.83e6, d: 3.5e-10 },
};

const amps = 10;
const area = 1.5e-6; // a 1.5 mm² household wire, in m²

console.log("metal      time (fs)  path (nm)  atoms  drift (mm/s)  heat (W/m)");
for (const [name, { rho, n, v, d }] of Object.entries(metals)) {
  const time = m / (n * e * e * rho);
  const path = v * time;
  const drift = amps / (n * e * area);
  const heat = (amps * amps * rho) / area;
  console.log(
    name.padEnd(10),
    (time * 1e15).toFixed(1).padStart(9),
    (path * 1e9).toFixed(1).padStart(10),
    (path / d).toFixed(0).padStart(6),
    (drift * 1000).toFixed(2).padStart(13),
    heat.toFixed(2).padStart(11),
  );
}
```

Run it with `node dien-tro.mjs`. The output must be exactly this:

```text
metal      time (fs)  path (nm)  atoms  drift (mm/s)  heat (W/m)
silver          38.2       53.1    184          0.71        1.06
copper          25.0       39.2    153          0.49        1.12
gold            27.2       38.0    132          0.71        1.48
aluminium        7.4       15.0     52          0.23        1.77
iron             2.2        4.3     17          0.24        6.47
lead             1.2        2.2      6          0.32       14.67
```

Common errors: `command not found: node` or `'node' is not recognized` means Node is not installed or the terminal needs reopening. `Cannot find module` means you are not in the folder that holds the file.

## What does this lesson measure, and what not?

- **Published measurements:** resistivity, electron density, Fermi speed and lattice constants. The lesson does not measure these itself.
- **Worked out from them:** time between deflections, free path, atoms passed, drift speed and heat.
- **The model:** free electrons, which treats a metal's conducting electrons as a gas. It fits copper, silver and gold fairly well and iron poorly.
- **Missing:** temperature curves for aluminium, iron and lead, so the simulation locks these three at 20 °C.

## Check what you learned

1. In copper at room temperature, about how many atoms does an electron pass between two deflections?
   Show the answer: About 150 atoms. Copper's free path is about 39 nm and neighbouring copper atoms sit about 0.256 nm apart. Dividing gives about 150.
2. Why does very cold copper still have resistance?
   Show the answer: Impurities and defects do not go away when cold. When cold, the shaking almost stops, so the curve goes flat at a floor. That floor comes from impurities and defects, so a purer sample has a lower floor.
3. Aluminium has more free electrons than copper. Why is its resistivity still higher?
   Show the answer: Its electrons are deflected much more often. Resistivity depends on the number of electrons and on the time between deflections. Aluminium has about 2.1 times the electrons but only about 1/3.4 of that time.
4. A 1.5 mm² copper wire carries 10 A. About how fast do electrons move along it on average?
   Show the answer: 0.5 mm/s. About 1,570 km/s is how fast electrons fly, in every direction. Their drift along the wire is only about 0.49 mm/s. A lamp lights almost at once because the electric field spreads very fast and electrons already fill the wire.

## Sources

- [Matula (1979): Electrical resistivity of copper, gold, palladium, and silver](https://srd.nist.gov/jpcrdreprint/1.555614.pdf)
- [HyperPhysics: Table of resistivity](https://hyperphysics.gsu.edu/hbase/Tables/rstiv.html)
- [HyperPhysics: Fermi energies, Fermi speeds and free electron densities](https://hyperphysics.gsu.edu/hbase/Tables/fermi.html)
- [HyperPhysics: Microscopic view of Ohm's law](https://hyperphysics.gsu.edu/hbase/electric/ohmmic.html)
- [Wikipedia: Free electron model](https://en.wikipedia.org/wiki/Free_electron_model)
- [Wikipedia: Electron mobility](https://en.wikipedia.org/wiki/Electron_mobility)
- [AVS 61: Mean free path and the size of metal wires](https://www2.avs.org/symposium2014/Papers/Paper_EM-TuM12.html)
- [Lời Giải Hay: Theory of resistance, Physics 11 (in Vietnamese)](https://loigiaihay.com/ly-thuyet-dien-tro-vat-li-11-canh-dieu-a150864.html)