Mean Drift Velocity
Specification: OCR A H556 | Section: 4.1.2 | Focus: mean drift velocity of charge carriers, the equation I = nAve, and the distinction between conductors, semiconductors and insulators in terms of the number density of charge carriers.
- define mean drift velocity and explain why electrons in a wire move slowly even when the current is large
- derive and use the equation I = nAve to calculate current or drift velocity
- explain the physical meaning of each term in the equation
- distinguish between conductors, semiconductors and insulators based on the number density n of charge carriers
- apply the equation to real exam-style problems involving wires, semiconductors and different materials
Big idea: current is not about how fast individual electrons zoom along. It is about how many charge carriers pass a point each second. A huge number of very slow electrons can give a large current.
What is mean drift velocity?
When a potential difference is applied across a conductor, the free electrons inside experience a force and start to drift in one direction. However, they do not move in straight lines. They constantly collide with the vibrating ions in the lattice.
The mean drift velocity v is the average velocity of the charge carriers along the conductor, taking all these collisions into account.
Electrons in a wire do not travel at the speed of light. Their drift velocity is typically only a few millimetres per second. The electrical signal (the electric field) travels very quickly, but the charge carriers themselves move slowly.
Why is drift velocity so small?
- electrons have very large random thermal velocities (around 106 m s−1)
- they constantly collide with the positive ions in the metal lattice
- after each collision they are scattered in random directions
- the applied electric field only adds a tiny extra drift component on top of the random motion
Thermal motion
Random, very fast, in all directions. Averages to zero net movement.
Drift motion
Slow, directed, superimposed on the thermal motion. Responsible for the net current.
Deriving the equation I = nAve
Consider a wire of cross-sectional area A containing free charge carriers each with charge e. The number of free charge carriers per unit volume is n (the number density). The charge carriers move with a mean drift velocity v along the wire.
Electron path through a metal lattice
Derivation of I = nAve
- In a time t, each charge carrier travels a distance L = v × t.
- The volume of wire swept out in time t is volume = A × L = A × v × t.
- The number of charge carriers in this volume is number = n × volume = n × A × v × t.
- The total charge passing a point in time t is Q = number × e = n × A × v × t × e.
- Current is charge per unit time: I = Q / t = n × A × v × e.
| Symbol | Quantity | SI unit | Typical value (copper) |
|---|---|---|---|
| I | current | A (ampere) | depends on circuit |
| n | number density of charge carriers | m−3 | ~8.5 × 1028 m−3 |
| A | cross-sectional area | m2 | ~1 mm² = 1 × 10−6 m² |
| v | mean drift velocity | m s−1 | ~10−4 m s−1 |
| e | elementary charge | C (coulomb) | 1.60 × 10−19 C |
Conductors, semiconductors and insulators
The value of n varies enormously between different materials. This is the main reason conductors, semiconductors and insulators behave so differently.
| Material type | n (m−3) | Examples | Charge carriers |
|---|---|---|---|
| Conductor | ~1028 – 1029 | copper, aluminium, silver | free electrons from the outer shells of atoms |
| Semiconductor | ~1016 – 1019 | silicon, germanium | electrons and holes; increases sharply with temperature |
| Insulator | ~0 (effectively) | rubber, glass, plastic | almost no free charge carriers |
For the same current I and cross-sectional area A, a material with a smaller n must have a larger drift velocity v. This is why drift velocities in semiconductors are often much larger than in metals.
Why does n matter?
- Conductors: huge n means even a tiny v gives a large current. Electrons are abundant.
- Semiconductors: n is much smaller, so v must be larger to carry the same current. Heating or doping increases n dramatically.
- Insulators: n ≈ 0, so effectively no current can flow even with a large potential difference.
Worked examples
Drift velocity explorer
Try selecting the semiconductor option. Notice how the same current requires a much larger drift velocity when n is small.
Common misconceptions
Thermal velocity vs drift velocity
"Electrons travel at the speed of light"
No. The electric field propagates quickly, but drift velocity is only ~10−4 m s−1.
"Larger current means faster electrons"
Not necessarily. Current depends on n, A and e as well as v. A larger area or larger n can give more current at the same v.
"Insulators have no electrons"
They have electrons, but they are not free to move. n ≈ 0 for conduction.
"I = nAve only works for metals"
It works for any material with mobile charge carriers, including semiconductors and electrolytes. For ions, replace e with the ion charge q.
Knowledge Check
- The average velocity of charge carriers along a conductor
- Taking into account their random thermal motion and collisions
- I = nAve
- In semiconductors, only a small fraction of electrons are free to move
- Most electrons are bound in covalent bonds; only those excited across the band gap are free
- n = I / (A v e)
- A = 4.0 × 10−6 m²
- n = 3.0 / (4.0 × 10−6 × 1.5 × 10−4 × 1.60 × 10−19) = 3.1 × 1028 m−3
Exam-Style Questions
- A = πr² = π × (0.40 × 10−3)² = 5.03 × 10−7 m²
- v = I / (n A e) = 2.5 / (8.5 × 1028 × 5.03 × 10−7 × 1.60 × 10−19)
- v = 3.7 × 10−4 m s−1
- A = I / (n v e) = 1.9 × 10−7 / (1.5 × 1016 × 80 × 1.60 × 10−19)
- A = 9.9 × 10−7 m² (about 1 mm²)
- Metals have n ~ 1028 m−3 whereas semiconductors have n ~ 1016 m−3
- Metals have many more free charge carriers per unit volume, so can carry much larger currents for the same v and A
Topic Summary
Definition
Average velocity of charge carriers along a conductor, after accounting for collisions and random thermal motion.
Key equation
I = n A v e
Current depends on number density, area, drift velocity and charge.
Materials
Conductors: n ~ 1028 m−3
Semiconductors: n ~ 1016 m−3
Insulators: n ≈ 0
Key insight
Drift velocity is tiny (~mm/s) but current can be large because n is enormous in metals.