# Charge carriers and drift velocity

> A-Level Physics · CIE 9702
> Source: https://www.owlsprep.com/study/cie-9702-u9-charge-carriers-and-drift-velocity/

This sub-topic explains how moving charge carriers produce electric current, introduces drift velocity, and derives the key relationship $I = nAve$ used to solve problems about current in different materials.

**Prerequisites:** [Definition of electric current](https://www.owlsprep.com/study/cie-9702-u9-electric-current/)

## Learning objectives

- Define charge carriers, number density and drift velocity
- Derive the $I = nAve$ relationship for current
- Compare drift velocity and number density across materials
- Solve calculation problems using the drift velocity formula

## Charge Carriers and Number Density

All electric current is produced by moving charged particles called charge carriers. In metallic conductors, charge carriers are free electrons that move through the stationary metal lattice. In electrolytes, charge carriers are positive and negative ions, while in semiconductors they are electrons and positively charged holes.

**Number density** — Number density is the number of charge carriers per unit volume of a material, measured in $\text{m}^{-3}$

*Notation:* $n$

*Example:* Metals have very high number density ($\approx 10^{29} \text{ m}^{-3}$), while insulators have effectively zero free charge carriers.

**Check your understanding**

Test your understanding of number density:

1. Which of the following has the highest number density of free electrons?

   - Intrinsic silicon
   - Copper wire
   - Pure germanium
   - Glass

   *Answer:* Copper wire

   *Why:* Copper is a metallic conductor with a very high concentration of free electrons, far higher than semiconductors or insulators.

**Worked example:** A $1.5 \times 10^{-6} \text{ m}^3$ block of copper contains $1.2 \times 10^{23}$ free electrons. Calculate the number density of charge carriers.

1. Number density equals total number of charge carriers divided by total volume:
2. $$n = \frac{N}{V}$$
3. Substitute the given values:
4. $$n = \frac{1.2 \times 10^{23}}{1.5 \times 10^{-6}} = 8.0 \times 10^{28} \text{ m}^{-3}$$

## What is Drift Velocity?

When no potential difference is applied across a conductor, free electrons move randomly at high speed due to thermal energy, with zero net displacement along the conductor. When a potential difference is applied, an electric field exerts a force on electrons, accelerating them in one direction. Electrons repeatedly collide with the stationary metal lattice, losing energy and changing direction. This results in a small net average motion along the conductor, called drift.

**Drift velocity** — Drift velocity is the average net velocity of charge carriers along the conductor, in the direction of the electric field.

*Notation:* $v$

**Worked example:** Explain why typical drift velocity in a metal is much smaller than the instantaneous speed of free electrons.

1. Free electrons have a high instantaneous random thermal speed (~$10^6 \text{ m s}^{-1}$) even when no current flows.
2. When an electric field is applied, it only adds a small net directional component to the random motion, because frequent collisions with the lattice repeatedly stop and reverse the acceleration of electrons.
3. The average of this small net directional motion is drift velocity, which is typically ~$10^{-4} \text{ m s}^{-1}$, much smaller than the instantaneous thermal speed.

> **Exam tip:** Examiners often test the explanation for why drift velocity is small. Always mention random thermal motion and collisions with the metal lattice.

## Derivation and Use of $I = nAve$

**Derivation:** Derive the relationship between current and drift velocity

*Starting from:* Definition of current: $I = \frac{\Delta Q}{\Delta t}$

1. Consider a conductor of cross-sectional area $A$, with number density $n$, charge per carrier $e$, drift velocity $v$.
2. In time $\Delta t$, charge carriers travel a distance $v\Delta t$. The volume of charge that passes a cross-section in this time is:
3. $$V = A v \Delta t$$
4. Number of charge carriers in this volume is $nV = n A v \Delta t$, so total charge passing the cross-section is:
5. $$\Delta Q = e \times n A v \Delta t$$

*Conclusion:* Substituting into $I = \frac{\Delta Q}{\Delta t}$ cancels $\Delta t$, giving the core relation: $I = n A v e$

**Worked example:** A copper wire of diameter 2.0 mm carries a current of 3.0 A. Number density of free electrons is $8.0 \times 10^{28} \text{ m}^{-3}$ and $e = 1.6 \times 10^{-19} \text{ C}$. Calculate drift velocity.

1. Convert diameter to radius: $r = \frac{2.0 \text{ mm}}{2} = 1.0 \text{ mm} = 1.0 \times 10^{-3} \text{ m}$
2. Calculate cross-sectional area:
3. $$A = \pi r^2 = \pi (1.0 \times 10^{-3})^2 \approx 3.14 \times 10^{-6} \text{ m}^2$$
4. Rearrange $I = nAve$ for $v$:
5. $$v = \frac{I}{n A e}$$
6. Substitute values:
7. $$v = \frac{3.0}{(8.0 \times 10^{28})(3.14 \times 10^{-6})(1.6 \times 10^{-19})} \approx 7.5 \times 10^{-5} \text{ m s}^{-1}$$

> **info**
>
> For the same current and cross-sectional area, $v \propto \frac{1}{n}$. Semiconductors have much lower $n$ than metals, so drift velocity is much higher in semiconductors.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using diameter directly in the area formula instead of converting to radius
  - Why it fails: This gives an area 4 times too large, leading to a drift velocity 4 times smaller than the correct value
  - Correct: Always halve the diameter to get radius, and convert all length units to metres (SI units)
- **Wrong:** Confusing total number of charge carriers with number density
  - Why it fails: Number density is per unit volume, not the total count in the whole material, so mixing these up gives wrong orders of magnitude
  - Correct: Check units: number density has units of $\text{m}^{-3}$, so this will confirm you have the right quantity
- **Wrong:** Assuming all charge carriers have charge equal to $e$
  - Why it fails: In electrolytes, ions can have charge of $2e$, $3e$ etc, so using $e$ gives the wrong current
  - Correct: Check the type of charge carrier given, and substitute the correct charge per carrier into the formula
- **Wrong:** Stating drift velocity is the speed of electrons in a wire
  - Why it fails: This ignores the high random thermal motion of electrons, which is what examiners test for
  - Correct: Always clarify that drift velocity is the average net velocity in the direction of the electric field

## Cheatsheet

| Quantity | Symbol | Unit | Description |
| --- | --- | --- | --- |
| Number density | $n$ | $\text{m}^{-3}$ | Charge carriers per unit volume |
| Cross-sectional area | $A$ | $\text{m}^2$ | Area of conductor cross-section |
| Drift velocity | $v$ | $\text{m s}^{-1}$ | Average net velocity of charge carriers |
| Charge per carrier | $e$ | $\text{C}$ | Charge of one charge carrier |
| Current | $I$ | $\text{A}$ | Electric current |
| Core formula | $I = nAve$ | - | Relationship between current and drift velocity |

## What's next

Charge carriers and drift velocity is the foundational concept for explaining why different materials have different resistivities, and forms the basis for understanding how semiconductors work. This topic is frequently tested in both multiple choice and structured questions in CIE 9702 exams, often combined with resistivity calculations. Next, you will build on this concept to learn about resistivity and Ohm's law, before moving on to semiconductor devices and other applications of current electricity.

- [Potential difference and e.m.f.](https://www.owlsprep.com/study/cie-9702-u9-potential-difference-and-e-m/)
- [Resistance and resistivity](https://www.owlsprep.com/study/cie-9702-u9-resistance-and-resistivity/)
- [I-V Characteristics](https://www.owlsprep.com/study/cie-9702-u9-i-v-characteristics/)

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