Drift Velocity — Physica

Equivalent forms

I = n A q v_d

\vec{J} = n q \vec{v}_d

v_d = \mu E

One equation reconciles two facts that feel contradictory: electricity is near-instant, yet the electrons creep — because current is carried by the whole sea, not a single swift particle.

Symbol	Name	SI unit	Dimension	Range
$v_d$	Drift velocityoutput Average net electron speed along the wire	m/s	L·T⁻¹	0 – 0.01
$I$	Current Total current through the wire	A	I	0.1 – 10
$n$	Carrier density Free electrons per unit volume	m⁻³	L⁻³	1e+28 – 1e+29
$A$	Cross-section Wire cross-sectional area	m²	L²	1e-7 – 0.00001
$q$	Carrier charge Elementary charge (electron)	C	I·T	1.602176634e-19 – 1.602176634e-19

Unit systems

Symbol	SI	CGS	Imperial
$v_d$	m/s	cm/s	m/s
$I$	A	statA	A
$n$	m⁻³	cm⁻³	m⁻³
$A$	m²	cm²	in²

Where it holds

Valid for steady (DC) current in an ohmic conductor with uniform carrier density. Breaks down for semiconductors with two carrier types, very high frequencies (skin effect), and ballistic nanoscale conductors where carriers travel without scattering.

Dimensional analysis

[I]/([n][A $][q]) = A / (m^{-3} \cdot m^{2} \cdot A\cdot s) = A / (m^{-1}\cdot A\cdot s) = m/s \checkmark$

Discovery

Paul Drude · 1900

Drude modeled a metal as a gas of free electrons bouncing off fixed ions. Treating conduction as drift between collisions, he derived Ohm's law microscopically and gave the first estimate of how slowly electrons actually drift.

Try this

Flip a switch and the bulb lights instantly — yet the electrons themselves crawl slower than a snail. How fast do they actually move?

A copper wire of cross-section 1 mm² carries 1 A. With n = 8.5×10²⁸ free electrons per m³, find the average drift speed of the electrons.

Research status: stable

Real-world applications

Sizing conductors for current capacity
Hall-effect sensors inferring carrier density
Understanding electromigration failure in chips
Designing busbars and power distribution

Common misconceptions

Electrons race through the wire — in fact they drift $at \sim 0.1\,\mathrm{mm/s}$
The lamp lights because an electron travels from switch to bulb — it lights because the field propagates near light speed and pushes the local sea everywhere at once
Higher voltage means faster individual electrons by a huge factor — drift stays tiny; it's the count of carriers that makes current large

Experimental verification

The Hall effect measures n directly; plugging measured n into nAqv_d reproduces the wire's current. Tonomura-style and classic Tolman-Stewart inertia experiments confirm electrons are the mobile carriers in metals.

Derivation

In time dt, all electrons within

v_d\cdot dt

of a cross-section pass through it.

That volume is

A\cdot v_d\cdot dt

, containing

n\cdot A\cdot v_d\cdot dt

electrons, each of charge q.

Charge through

= q\cdot n\cdot A\cdot v_d\cdot dt

, so

I = dQ/dt

= nAqv_d, giving

v_d = I/(nAq)

.

Limiting cases

A \to

large⟶

v_d \to 0

A thick wire spreads the same current over many carriers, so each barely moves.

n \to

large⟶

v_d \to 0

More carriers share the load; metals (huge n) have tiny drift speeds.

I \to 0

⟶

v_d \to 0

No net field, no net drift — only random thermal motion remains.

What if…

What if you double the current?

Drift velocity doubles — it's linear in I for a fixed wire.

What if the wire were a semiconductor with far fewer carriers?

With n a billion times smaller, the same current needs drift speeds a billion times larger — which is why semiconductors heat and saturate.

What if you used a thicker wire for the same current?

Larger A means slower drift and lower current density, reducing heating and electromigration.

1

Copper wire carrying 1 A

Given ·

I:: 1
n:: 8.5e+28
A:: 0.000001
q:: 1.602176634e-19

Find · v_d

Steps

Multiply denominator: $nAq = 8.5\times 10^{28} \times 10^{-6} \times 1.602\times 10^{-19} \approx 1.36\times 10^{4}$
$v_d = I/(nAq) = 1/1.36\times 10^{4} \approx 7.3\times 10^{-5} m/s$
That' $s \sim 0.07\,\mathrm{mm/s}$ — about 4 cm per minute

Result ·

v_d = 1 / (8.5\times 10^{28} \times 10^{-6} \times 1.602\times 10^{-19}) \approx 7.3\times 10^{-5} m/s

Ohm's Law→Electric Field of a Point Charge→Magnetic Force on a Current-Carrying Wire→