The drift velocity formula is vd = I/(nqA), where I is the current, n is the number of free charge carriers per cubic meter, q is the charge on each carrier and A is the conductor's cross-sectional area. Drift velocity is the average velocity that free charges pick up along a conductor when an electric field pushes on them. In ordinary copper wires it is usually well under a millimeter per second.
This guide defines drift velocity, derives the formula from current density, and connects it to the alternative form vd = eEτ/m and to electron mobility. It includes a variables and units table, a fully worked copper-wire example using a sourced free-electron density, an explanation of why signals travel fast even though electrons crawl, the factors that change drift speed, and practice problems with answers.
Free electrons in a metal are never at rest. They move randomly at very high speeds and constantly scatter off the vibrating lattice and impurities. With no field applied, those random motions cancel out and there is no net flow of charge.
When a voltage creates an electric field in the wire, each electron gains a small push opposite to the field between collisions. The average of all those small pushes is a slow, steady net motion along the wire. OpenStax University Physics, Volume 2 describes this as charges losing kinetic energy to the material and reaching a constant velocity known as the drift velocity.
To define drift velocity in one line, it is the average velocity of the free charge carriers in a conductor due to an applied electric field. For electrons, the drift velocity points opposite to the electric field and opposite to conventional current, because electrons carry negative charge.
To see how small it is in comparison, consider the electrons' random motion. Copper's free-electron Fermi energy is 7.00 eV, according to the free electron model table in University Physics Volume 3 (the same table used for copper's electron density in the worked example below). Solving E = ½mv2 for that energy gives a speed of about 1.6 × 106 m/s. Typical drift velocities are smaller by a factor of a billion or more.
The drift velocity formula can be written in four equivalent ways. Which one you use depends on what the problem gives you.
| Form | Equation | Use it when you know |
|---|---|---|
| Current form | vd = I/(nqA) | Current, carrier density and wire area |
| Current density form | vd = J/(nq) | Current density (current per unit area) |
| Relaxation-time form | vd = eEτ/m | Electric field and mean time between collisions |
| Mobility form | vd = μE | Electric field and mobility |
For electrons, q is the elementary charge, e = 1.602 176 634 × 10−19 C, which NIST lists as an exact value. Most textbooks round it to 1.60 × 10−19 C.
| Symbol | Quantity | SI unit | Typical value for copper |
|---|---|---|---|
| vd | Drift velocity | m/s | About 10−5 to 10−3 m/s |
| I | Current | A (C/s) | Problem-specific |
| n | Free charge carrier density | m−3 | 8.47 × 1028 m−3 |
| q or e | Charge per carrier | C | 1.602 × 10−19 C |
| A | Cross-sectional area | m2 | Problem-specific |
| J | Current density | A/m2 | Problem-specific |
| E | Electric field in the conductor | V/m (N/C) | Problem-specific |
| τ | Relaxation time (mean time between collisions) | s | About 2.5 × 10−14 s |
| m | Electron mass | kg | 9.109 × 10−31 kg |
| μ | Mobility | m2/(V·s) | About 4.4 × 10−3 m2/(V·s) |
The electron mass is from NIST's CODATA 2022 value, 9.109 383 7139 × 10−31 kg. The copper τ and μ values are calculated later in this guide.
The derivation follows OpenStax University Physics, section 9.2. Picture a straight wire with cross-sectional area A, carrying n free carriers per cubic meter, each with charge q, all drifting at speed vd.
Distance moved. In a short time dt, each carrier moves a distance vd dt along the wire.
The derivation uses the derivative I = dQ/dt. If the dt notation is unfamiliar, our guide to dx and other calculus terms explains what a differential means.
The current form tells you how fast carriers drift for a given current, but not why they reach that speed. The Drude model answers that. An electric field E exerts a force of magnitude eE on each electron. Between collisions the electron accelerates. Each collision randomizes its direction and erases the extra velocity it gained.
The average time between collisions is the relaxation time, τ. In steady state, the momentum the field adds per unit time balances the momentum lost to collisions. The LibreTexts Drude model page writes this balance as mvd/τ = qE. Solving for the drift speed:
vd = eEτ/m
Written with direction, electrons drift opposite to the field, so some textbooks show vd = −eEτ/m. The same model gives the conductivity σ = ne2τ/m, which links the microscopic picture to the resistivity values you look up in tables.
Mobility measures how readily a carrier drifts in a given field. It is the drift velocity per unit electric field:
μ = vd/E, so vd = μE
Comparing with vd = eEτ/m shows that μ = eτ/m. The LibreTexts mobility page gives the same result, μ = qτ/m. The SI unit is m2/(V·s). Semiconductor work often uses cm2/(V·s), where 1 m2/(V·s) = 104 cm2/(V·s).
Because σ = nqμ and resistivity ρ = 1/σ, you can estimate copper's mobility from its resistivity. OpenStax Table 9.1 lists copper's resistivity at 20 °C as 1.68 × 10−8 Ω·m.
Problem: A copper wire with a diameter of 1.00 mm carries a steady current of 3.00 A. Find the electron drift velocity, the electric field inside the wire, and the time an electron takes to drift 1.00 m.
Copper contributes about one free electron per atom. The free electron model table in University Physics Volume 3 lists copper's conduction electron density as 8.47 × 1028 m−3. OpenStax Volume 2 computes 8.34 × 1028 m−3 in its own example from an assumed density and atomic mass. Both round to about 8.5 × 1028 m−3, so use whatever value your course gives. This example uses n = 8.47 × 1028 m−3.
Both routes give the same answer, which is a good sign the numbers are consistent.
For comparison, OpenStax's own example finds a drift velocity of 4.54 × 10−4 m/s for a 12-gauge copper wire (2.053 mm diameter) carrying 20.0 A, with the minus sign showing that electrons move opposite to conventional current.
If electrons need an hour to move a meter, why does a lamp come on the moment you flip the switch? Two facts explain it.
A garden hose that is already full of water is a useful picture. When you open the tap, water comes out the far end at once, even though the water that entered at the tap has barely moved.
The tiny drift speed is also why large currents need so many carriers. Because n is enormous, even a slow drift moves a lot of charge per second.
| Factor | Effect on vd | Why |
|---|---|---|
| Current (same wire) | Increases in proportion | vd = I/(nqA), so doubling I doubles vd |
| Cross-sectional area (same current) | Decreases as 1/A | The same charge flow is spread over more carriers |
| Carrier density n (material) | Decreases as 1/n | More carriers share the same current |
| Electric field or voltage | Increases in proportion | vd = μE, and E = V/L in a uniform wire |
| Wire length (same voltage) | Decreases as 1/L | The same voltage across a longer wire gives a weaker field |
| Temperature | Decreases (same field) | Stronger lattice vibrations shorten τ, which lowers μ |
One result surprises many students. If the voltage across a wire is fixed, then vd = μV/L, which does not depend on the wire's area at all. A thicker wire carries more current at the same voltage, but each electron drifts at the same speed.
The table below shows how material alone changes the drift speed. Each row uses the same current density, J = 1.00 × 106 A/m2, with vd = J/(ne) and densities from the University Physics Volume 3 table.
| Metal | Conduction electron density (m−3) | vd at J = 1.00 × 106 A/m2 |
|---|---|---|
| Aluminum | 18.1 × 1028 | 3.45 × 10−5 m/s |
| Iron | 17.0 × 1028 | 3.67 × 10−5 m/s |
| Copper | 8.47 × 1028 | 7.37 × 10−5 m/s |
| Gold | 5.90 × 1028 | 1.06 × 10−4 m/s |
| Silver | 5.86 × 1028 | 1.07 × 10−4 m/s |
Use e = 1.602 × 10−19 C and the carrier densities above. Try each problem before reading the answer.
Answer: A = π(1.00 × 10−3 m)2 = 3.14 × 10−6 m2. nqA = (18.1 × 1028)(1.602 × 10−19)(3.14 × 10−6) = 9.11 × 104 C/m. vd = 5.00 ÷ 9.11 × 104 = 5.49 × 10−5 m/s.
Find the current. Electrons in a copper wire with A = 1.00 mm2 drift at 1.00 × 10−4 m/s. What is the current?
Answer: A = 1.00 × 10−6 m2. I = nqAvd = (8.47 × 1028)(1.602 × 10−19)(1.00 × 10−6)(1.00 × 10−4) = 1.36 A.
Double the diameter. The copper wire in the worked example is replaced by one with twice the diameter, carrying the same 3.00 A. What is the new drift velocity?
Answer: Doubling the diameter multiplies the area by 4, so vd drops by a factor of 4: 2.81 × 10−4 ÷ 4 = 7.04 × 10−5 m/s.
Mobility. Carriers in a sample drift at 4.00 × 10−4 m/s in a field of 0.100 V/m. Find the mobility.
Answer: μ = vd/E = 4.00 × 10−4 ÷ 0.100 = 4.00 × 10−3 m2/(V·s).
Gold wire. A gold wire with a diameter of 2.053 mm carries 10.0 A. Find the drift velocity.
Answer: A = π(1.0265 × 10−3 m)2 = 3.31 × 10−6 m2. vd = 10.0 ÷ [(5.90 × 1028)(1.602 × 10−19)(3.31 × 10−6)] = 3.20 × 10−4 m/s.
Travel time. How long does an electron in the worked example take to drift along a 2.50 m lamp cord?
Answer: t = 2.50 m ÷ 2.81 × 10−4 m/s = 8.9 × 103 s, or about 2.5 hours.
If you are working these in an online platform, our notes on Mastering Physics explain how its hints and feedback work. For lab courses that measure current and resistance, our lab report format guide shows how to present calculations like these. Before a test, the exam preparation strategies guide offers a plan for reviewing formulas.
Drift velocity problems are mostly unit bookkeeping, but one wrong exponent can throw an answer off by a factor of a million. If you keep getting stuck, a tutor can go through your steps and show where the powers of ten slip. Our physics homework help page explains how step-by-step support works.
Drift speed is the magnitude of the drift velocity, the average speed at which free charges move along a conductor because of an applied electric field. It is not the speed of any single electron, which moves randomly much faster. In copper wires, drift speed is typically a fraction of a millimeter per second. The drift speed formula is the same as the velocity version, vd = I/(nqA), taken as a magnitude.
The main drift velocity formula is vd = I/(nqA). Equivalent forms are vd = J/(nq), vd = eEτ/m and vd = μE. Pick the one that matches the quantities given in the problem.
In terms of the field, vd = eEτ/m, where τ is the relaxation time and m is the electron mass. Grouping eτ/m as the mobility μ gives the shorter form vd = μE.
Metals contain an enormous number of free electrons, about 8.5 × 1028 per cubic meter in copper. Because so many carriers share the current, each one only needs to move slowly. Frequent collisions with the lattice also keep resetting their extra velocity.
For a fixed current, yes: a thicker wire has a larger area, so vd = I/(nqA) is smaller. For a fixed voltage across a wire of fixed length, no: vd = μV/L, and the thicker wire simply carries more current.
Drift velocity is a speed in m/s that depends on how strong the field is. Mobility is the drift velocity per unit field, in m2/(V·s), and is a property of the material and temperature. They are linked by vd = μE.
No. Electrons drift at roughly 10−4 m/s in typical wires, while the electrical signal travels through the circuit at roughly 108 m/s. The signal is a change in the electric field, not the electrons themselves racing along the wire.
Formulas and values come from OpenStax University Physics Volumes 2 and 3, LibreTexts and NIST CODATA 2022 constants, as of Nov. 2026.
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