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Drift Velocity

Average velocity acquired by electrons in a conductor under field E. τ is the mean relaxation time, m is electron mass, e is electron charge. Typically ~10⁻⁴ m/s — far slower than random thermal speeds.
Class 11Class 12
Derivation

Free electron model

In a conductor, conduction electrons move randomly with high thermal speeds (~10610^6 m/s). In the absence of a field, their average velocity is zero — no net current.

When field E⃗\vec{E} is applied, each electron experiences force F⃗=−eE⃗\vec{F} = -e\vec{E} and gains acceleration:

a⃗=−eE⃗m\vec{a} = \frac{-e\vec{E}}{m}

Relaxation time

Electrons collide with lattice ions, losing the acquired velocity. Between collisions, an electron accelerates for an average time τ\tau (mean relaxation time, typically ∼10−14\sim 10^{-14} s).

Velocity gained between collisions: a⃗τ=−eE⃗τ/m\vec{a}\tau = -e\vec{E}\tau/m

Drift velocity

The average extra velocity acquired by the electron in the field direction:

v⃗d=−eE⃗τm\vec{v}_d = \frac{-e\vec{E}\tau}{m}

Magnitude:

vd=eEτm\boxed{v_d = \frac{eE\tau}{m}}

Direction is opposite to E⃗\vec{E} (electrons drift against the field).

Order of magnitude

For copper: τ≈2.5×10−14\tau \approx 2.5 \times 10^{-14} s, E∼1E \sim 1 V/m:

vd≈1.6×10−19×1×2.5×10−149.1×10−31≈4.4×10−3 m/sv_d \approx \frac{1.6\times10^{-19} \times 1 \times 2.5\times10^{-14}}{9.1\times10^{-31}} \approx 4.4 \times 10^{-3} \text{ m/s}
Remember
Drift speed ($\sim$mm/s) is far less than thermal speed ($\sim 10^6$ m/s). Yet the electric signal (field propagation) travels at close to the speed of light — the field is established almost instantly throughout the conductor, setting all electrons drifting simultaneously.