---
title: "Drift velocity, mobility and conductivity in metals"
subject: "Electrical Engineering Material"
chapter: 2
key: "semester_3/electrical_engineering_material:drift-mobility-conductivity-metals"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/drift-mobility-conductivity-metals
site: BCT Engineering
---

# Drift velocity, mobility and conductivity in metals

Electrical Engineering Material · Chapter 2: Free Electron Theory of Conduction in Metal

- Asked in: 2081 Baisakh, 6 marks
- Video: [Electrical conductivity of metals](https://www.youtube.com/watch?v=O2QHL4ZR8E8) — Taylor Sparks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/drift-mobility-conductivity-metals
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-2

## Question

What are drift velocity, mobility and conductivity for the electrons in metals?

## Short answer

Drift velocity v_d is the average velocity electrons gain under a field: v_d = e tau E / m_e. Drift mobility mu_d = v_d/E = e tau / m_e. Electrical conductivity sigma = n e mu_d = n e^2 tau / m_e, where n is the free electron density.

## Answer

#### Drift velocity

Without a field, conduction electrons in a metal move randomly at high speed (about $10^6$ m/s) and their average velocity is zero, so there is no net current. When a field $\mathcal{E}$ is applied, each electron is accelerated opposite to the field between collisions with vibrating ions and defects. The small **average velocity** gained in the direction of the force is the **drift velocity** $v_d$.

```text
  E field  -------->
  e- path:  \/\/\/\_/\/\/\_/\  (random + slow drift)
            <--- net drift v_d
```

Between collisions the acceleration is $a = e\mathcal{E}/m_e$. If $\tau$ is the mean free time (relaxation time) between collisions, and each collision randomises the velocity, the average gained velocity is

$$
v_d = \frac{e\tau}{m_e}\mathcal{E}
$$

Typical $v_d$ is only mm/s to cm/s, far less than the thermal speed.

#### Drift mobility

The **drift mobility** $\mu_d$ is the drift velocity per unit electric field:

$$
\mu_d = \frac{v_d}{\mathcal{E}} = \frac{e\tau}{m_e} \qquad (\text{m}^2\text{V}^{-1}\text{s}^{-1})
$$

It measures how easily electrons move through the lattice. It is large when $\tau$ is long (few collisions). For copper, $\mu_d \approx 43\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$.

#### Conductivity

With $n$ free electrons per unit volume moving at $v_d$, the current density is

$$
J = e n v_d = e n \mu_d \mathcal{E}
$$

Comparing with Ohm's law $J = \sigma\mathcal{E}$:

$$
\sigma = e n \mu_d = \frac{n e^2 \tau}{m_e}, \qquad \rho = \frac{1}{\sigma}
$$

**Example (copper):** $n = 8.5 \times 10^{28}\ \text{m}^{-3}$, $\mu_d = 43 \times 10^{-4}\ \text{m}^2/\text{V s}$ gives $\sigma = 8.5 \times 10^{28} \times 1.6 \times 10^{-19} \times 43 \times 10^{-4} \approx 5.9 \times 10^{7}\ \text{S/m}$.

#### Factors

| Quantity | Depends on | Effect of rising temperature |
|---|---|---|
| $v_d$ | $\mathcal{E}$, $\tau$ | Falls (more collisions) |
| $\mu_d$ | $\tau$, $m_e$ | Falls ($\mu \propto 1/T$ for lattice scattering) |
| $\sigma$ | $n$, $\mu_d$ | Falls; $n$ fixed in metals |

In metals $n$ does not change with temperature, so the fall in $\sigma$ (rise in resistivity, $\rho \propto T$) comes entirely from the fall in mobility due to increased lattice scattering.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is drift velocity in a metal?
   A. The random thermal speed of electrons (~10^6 m/s)
   B. The small average velocity gained along the force direction, on top of random motion
   C. The speed of the electric field in the wire
   D. The velocity of the ions
   Answer: B. Random motion averages to zero; the net average velocity due to the field is v_d. (Key point: Drift velocity)

2. If τ is the mean free time, v_d equals
   A. eτE/m_e
   B. eE/(m_e τ)
   C. m_e E/(eτ)
   D. eτ/(m_e E)
   Answer: A. Acceleration eE/m_e acting for time τ between collisions gives v_d = (eτ/m_e)E. (Key point: Drift velocity formula)

3. Drift mobility is defined as
   A. μ = v_d E
   B. μ = e n τ/m_e
   C. μ = E/v_d
   D. μ = v_d/E = eτ/m_e
   Answer: D. Mobility is drift velocity per unit field, and is larger for longer τ. (Key point: Mobility)

4. Conductivity of a metal is
   A. σ = e μ_d/n
   B. σ = n μ_d/e
   C. σ = e n μ_d = n e^2 τ/m_e
   D. σ = e n E
   Answer: C. J = e n v_d = e n μ_d E, compared with J = σE. (Key point: Conductivity)

## Related in Electrical Engineering Material

- [Drift mobility of electrons in potassium](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Apotassium-drift-mobility-39-5) — Electrical Engineering Material, chapter 2, asked 2×
- [Copper drift mobility from scattering cross-section](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-mobility-from-cross-section) — Electrical Engineering Material, chapter 2
- [Copper drift mobility and scattering time](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-mobility-scattering-time) — Electrical Engineering Material, chapter 2
- [Mean free path from drift mobility](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Amean-free-path-43-mobility) — Electrical Engineering Material, chapter 2, asked 4×
- [Drift mobility in potassium (molar mass 39.95)](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Apotassium-drift-mobility-39-95) — Electrical Engineering Material, chapter 2, asked 2×
- [Einstein relation; mobility, conductivity, resistivity](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aeinstein-mobility-conductivity) — Electrical Engineering Material, chapter 6

## Related in other subjects

- [Drift velocity of electrons in copper](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adrift-velocity-copper-two-hundred-amp) — Engineering Physics, chapter 8
- [Drift speed of electrons, 20 A](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adrift-speed-copper-twenty-amp) — Engineering Physics, chapter 8, asked 2×
- [Drift speed of electrons, 10 A](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adrift-speed-copper-ten-amp) — Engineering Physics, chapter 8, asked 2×
- [Current from drift velocity](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Adrift-velocity-free-electron-current) — Basic Electrical Engineering, chapter 1