Drift of electrons and resistivity

The slow drift

Narrated video. Transcript:

Zoom into a copper wire. Free electrons fly about between the fixed ions. At room temperature their random speeds are about 10⁵ metres per second.

Every so often an electron hits an ion and flies off in a new, random direction. Just after a collision, its velocity can point anywhere, so averaged over all the electrons, it is zero.

Now switch on a field E. Each electron feels a force minus e E, so between collisions it accelerates at a equals minus e E over m.

Follow electron i. It left its last collision with velocity v zero i, a time t i ago. Its velocity now is v i equals v zero i minus e E t i over m.

Average over all the electrons. The v zero i average to zero. The times t i average to τ, the average time between collisions, called the relaxation time.

So on average the electrons have a small velocity, the drift velocity: v d equals minus e E τ over m. The minus sign means they drift opposite to the field.

Watch one electron. It zig-zags wildly, but every short flight is bent a little against the field. Over many collisions it creeps steadily backwards along the wire.

This creep is very slow, about a millimetre per second, while the random speed is about 10⁵ metres per second. The drift is a tiny bias on a furious random motion.

From drift to resistivity

Narrated video. Transcript:

Now find the current. Take a wire of cross-section A, with n free electrons per cubic metre, all drifting at speed v d.

In a time Δt, every electron within a distance v d Δt of a cross-section crosses it. They fill a cylinder of volume A v d Δt, which holds n A v d Δt electrons.

Each carries charge e, so the charge crossing is e n A v d Δt, and the current is I equals n e A v d. The current density is j equals n e v d.

Put in v d equals e E τ over m. We get j equals n e squared τ over m, times E. Compare with j equals σ E: the conductivity is σ equals n e squared τ over m.

So we have derived Ohm’s law! And the resistivity is ρ equals m over n e squared τ. More free electrons, or longer times between collisions, mean a lower resistivity.

The drift speed per unit field is called the mobility: μ equals v d over E, which is e τ over m. Its SI unit is metre squared per volt second.

Worked example. A copper wire of cross-section 1.0 × 10⁻⁷ square metre carries 1.5 ampere. Copper has about 8.5 × 10²⁸ free electrons per cubic metre.

v d equals I over n e A: 1.5 divided by 8.5 × 10²⁸, times 1.6 × 10⁻¹⁹, times 1.0 × 10⁻⁷. The bottom is 1.36 × 10³, so v d is about 1.1 × 10⁻³ metre per second, 1.1 millimetre per second.

At that speed an electron takes about 15 minutes to cross a metre of wire. Yet the bulb lights at once. Why? The field travels along the wire almost at the speed of light, and starts every electron drifting at the same moment.

Try it: drift in a wire

Interactive simulation. Things to try:

• Follow the highlighted electron: wild zig-zags, but slowly creeping against E.
• Compare the measured average velocity with −eEτ/m. Double the voltage and the drift doubles.
• Compare the drift with the random (thermal) speed. In the real copper panel the drift is millimetres per second; the thermal speed is about 10⁵ m/s.
• Raise the temperature: collisions come more often, τ gets shorter, the drift falls, and the resistivity ρ = m/(ne²τ) rises.

How fast do electrons drift?

A copper wire of cross-section 1.0 × 10⁻⁷ m² carries a current of 1.5 A. If copper has 8.5 × 10²⁸ free electrons per m³, what is the drift speed of the electrons? (e = 1.6 × 10⁻¹⁹ C)

1. 1.1 × 10⁻³ m/s
2. 1.1 × 10⁻⁶ m/s
3. 1.1 × 10⁵ m/s
4. 3 × 10⁸ m/s

Double the voltage

The potential difference across a metal wire is doubled, while its temperature is kept the same. What happens to the drift speed of the electrons and to the resistivity of the wire?

1. Drift speed doubles; resistivity stays the same
2. Drift speed doubles; resistivity halves
3. Drift speed stays the same; resistivity doubles
4. Drift speed is multiplied by four; resistivity stays the same

Summary

• Drift velocity: v_d = −eEτ/m
• Current: I = neA v_d, j = ne v_d
• Conductivity and resistivity: σ = ne²τ/m, ρ = m/(ne²τ)
• Mobility: μ = |v_d|/E = eτ/m

• Free electrons move randomly; a field adds a small average drift velocity opposite to E.
• τ, the relaxation time, is the average time between collisions.
• The drift model gives Ohm’s law: j is proportional to E, with σ = ne²τ/m.
• Drift speeds are only about a millimetre per second, but the field spreads almost at the speed of light, so all electrons start drifting at once.