Model

Classical model

(E5.3.1) E5.3 · A Model for Electrical Conduction →

v_{d} = \frac{q E}{m _{e}} τ, ρ = \frac{m _{e}}{n q ^{2} τ}

Statement

τ is the average time between collisions. It sets both the drift speed and the resistivity.

Symbols and units

$v_{d}$	drift speed	m/s
$τ$	mean time between collisions	s (≈10⁻¹⁴ in Cu)
$m_{e}$	electron mass	9.11×10⁻³¹ kg
$n$	free electrons per unit volume	m⁻³
$ρ$	resistivity	Ω·m
$q$	charge per carrier	C
$E$	applied field	V/m

When it applies

Classical free-electron picture. Right order of magnitude, wrong temperature trend.

Don't use when

Gets the order of magnitude and the existence of Ohm's law right, but predicts the wrong temperature trend for resistivity in detail — a quantum treatment is needed for that.

Derivation

Between collisions the field accelerates an electron:

a = \frac{q E}{m _{e}}

Each collision wipes out the accumulated drift, so on average the electron has been accelerating for a time τ:

v_{d} = a τ = \frac{q E τ}{m _{e}}

Feed that into the current density:

J = n q v_{d} = \frac{n q ^{2} τ}{m _{e}} E

Comparing with J = σE identifies the conductivity, and its inverse the resistivity:

σ = \frac{n q ^{2} τ}{m _{e}}, ρ = \frac{m _{e}}{n q ^{2} τ}

Worked examples

Mean free path in copper

τ ≈ 2.5 × 10⁻¹⁴ s at a thermal speed of 1.6 × 10⁶ m/s:

ℓ = v τ \approx 4 \times 1 0^{- 8} m

Roughly 150 atomic spacings — further than a classical collision picture can account for.

Watch out for

The model explains why Ohm's law holds at all — σ contains no E, so J stays proportional to E. It gets the temperature dependence wrong in detail; that needs quantum mechanics.

Connects to

Ohm's law Microscopic model Temperature coefficient