What is the order of magnitude of drift velocity of electrons in a conductor? Deduce the relation between the current flowing through a conductor and drift velocity of electrons in it.

Q: What is the order of magnitude of drift velocity of electrons in a conductor? Deduce the relation between the current flowing through a conductor and drift velocity of electrons in it.

**Order of magnitude** The drift velocity of electrons is extremely small — of the order of **$10^{-4} \text{ m/s}$** (about 0.1 mm/s) for ordinary currents. **Deriving $I = neAv_d$** Consider a conductor of cross-sectional area A with n free electrons per unit volume, each of charge e drifting with speed v_d. - In time Δt, an electron travels a distance $v_d \cdot \Delta t$. - Electrons within volume $A \cdot (v_d \cdot \Delta t)$ cross a section; their number $= n \cdot A \cdot v_d \cdot \Delta t$. - Charge crossing: $\Delta Q = (n \cdot A \cdot v_d \cdot \Delta t) \cdot e$. Hence current **$I = \Delta Q/\Delta t = n e A v_d$**.

Current Electricity

Drift velocity and current

Official Answer

Order of magnitude

The drift velocity of electrons is extremely small — of the order of $10^{-4} \text{ m/s}$ (about 0.1 mm/s) for ordinary currents.

Deriving $I = neAv_d$

Consider a conductor of cross-sectional area A with n free electrons per unit volume, each of charge e drifting with speed v_d.

In time Δt, an electron travels a distance $v_d \cdot \Delta t$ .
Electrons within volume $A \cdot (v_d \cdot \Delta t)$ cross a section; their number $= n \cdot A \cdot v_d \cdot \Delta t$ .
Charge crossing: $\Delta Q = (n \cdot A \cdot v_d \cdot \Delta t) \cdot e$ .

Hence current $I = \Delta Q/\Delta t = n e A v_d$ .

drift velocity10^-4 m/snumber density ncross-sectional areaI = neAv_dfree electronscharge carrierscurrent relation

Marking Scheme

10.5 mark: order of drift velocity $\approx 10^{-4} \text{ m/s}$ (accept $10^{-3}-10^{-5} \text{ m/s}$ ).
21 mark: correct derivation counting charge $\Delta Q = n e A v_d \Delta t$ through a cross-section.
30.5 mark: final relation $I = n e A v_d$ with symbols defined.

Hint

Count the charge in the volume $A \cdot v_d \cdot \Delta t$ that crosses a cross-section in time Δt, then divide by Δt.

Quick Oral Answer

Drift velocity is only about $10^{-4}$ metres per second; by counting the charge $n \cdot e \cdot A \cdot v_d$ that crosses a cross-section each second, the current comes out to be $I = neAv_d$ .

Analysis & Explanation

Concept

Even though electrons drift at only ~ $10^{-4} \text{ m/s}$ , currents flow the instant a circuit closes because the electric field (and hence the drift) is set up along the whole wire almost at the speed of light — every electron everywhere starts moving nearly together.

The counting argument

The relation $I = neAv_d$ is obtained by counting the charge that crosses a cross-section per second. The number density n (~ $10^{28}-10^{29} \text{ m}^{-3}$ for metals) is enormous, which is exactly why a tiny drift speed still delivers amperes of current.

Exam trap

Do not confuse drift velocity (~ $10^{-4} \text{ m/s}$ ) with the random thermal speed of electrons (~ $10^5-10^6 \text{ m/s}$ ) or with the signal/field propagation speed (~ $10^8 \text{ m/s}$ ).
Marks are lost when the derivation states the formula without the volume-and-charge counting steps.

Real-world link

The smallness of v_d explains why a torch lights instantly yet an individual electron takes hours to travel a few metres of wire — the message travels fast, the messengers slowly.

Common Mistakes

1Quoting the thermal (random) speed ~ $10^5 \text{ m/s}$ as the drift velocity — the drift speed is far smaller, ~ $10^{-4} \text{ m/s}$ .
2Writing $I = neAv_d$ directly without the derivation (counting charge in volume $A \cdot v_d \cdot \Delta t$ ) as required for full marks.
3Omitting or wrongly defining n as 'number of electrons' instead of 'number of free electrons per unit volume'.

Interesting Facts

For a copper wire carrying 1 A with area ~1 mm², the drift speed works out to roughly 0.07 mm/s — slower than a snail, yet the lamp lights instantly.

The free-electron number density in copper is about $8.5 \times 10^{28}$ per m³, so even a whisper of drift speed moves an Avogadro-scale flood of charge every second.

Spotted a mistake or something unclear?

Tell us — we fix reported answers fast.

Frequently Asked Questions

If drift velocity is so small, why does a bulb glow instantly?

Because the electric field that drives the electrons is established throughout the conductor almost at the speed of light. Every free electron along the wire starts drifting nearly simultaneously, so current — and light — appears instantly even though individual electrons crawl at ~ $10^{-4} \text{ m/s}$ .

What do the symbols in $I = neAv_d$ stand for?

n = number of free electrons per unit volume, e = magnitude of electronic charge, A = cross-sectional area of the conductor, and v_d = drift velocity of the electrons. Their product gives the current I.