Microscopic Ohm's Law and Drift Velocity

Core Idea

Microscopic Ohm's law states J = σE. This emerges from charge carrier motion through a medium. Carriers accelerate under the electric force but collide with atoms, reaching average drift velocity v_d = μE where μ is mobility. The conductivity σ = nq²τ/m*, where τ is collision time and m* is effective mass.

How It's Best Learned

Estimate drift velocity in copper wire carrying household current and compare to thermal velocities. Derive σ = nq²τ/m* from classical mechanics.

Common Misconceptions

• Electrons drift at significant light speed fractions (drift velocity is typically mm/s).
• Collision time τ is constant (it depends on temperature and impurities).
• Drift velocity is the same as thermal velocity (drift is a slow superimposed bias).

Explainer

You already know that resistivity σ (or its inverse ρ) is a material property that relates current density to an applied electric field, and that current density J describes how charge flows per unit area. What the microscopic Ohm's law does is derive that relationship from first principles — starting with individual electrons and building up to the macroscopic formula J = σE. This derivation turns a phenomenological observation (Ohm's law works) into a mechanical explanation (here's why it works).

In a metal, conduction electrons are not stationary — they move at high thermal speeds (roughly 10⁶ m/s) in random directions, constantly colliding with the ion lattice. When there is no electric field, the average velocity is zero: as many electrons go left as right. Apply an electric field E and each electron accelerates: a = qE/m. But the electron doesn't accelerate freely for long — it collides with a lattice ion after an average time τ (the collision time, or relaxation time), which for copper at room temperature is about 25 femtoseconds. After each collision, the electron's velocity is randomized again. The result is that between collisions, a small net velocity builds up in the direction of the field; after a collision it resets. The average net velocity that survives, accumulated over the collision interval, is the drift velocity: v_d = (qEτ)/m.

Now count up the current. There are n charge carriers per unit volume, each carrying charge q and moving with average velocity v_d. The current density is J = nqv_d = (nq²τ/m)E. This is J = σE with conductivity σ = nq²τ/m — derived entirely from mechanics and the definition of current density. The Drude model packages the same physics with an effective mass m* to account for quantum corrections to electron behavior in real crystals. The key insight is that σ is large when n is large (many carriers), q is large (each carries more charge), τ is long (carriers travel far before colliding), and m is small (carriers accelerate quickly).

A crucial number to internalize: in a copper wire carrying a typical household current of 1 A with cross-section 1 mm², the drift velocity is roughly 0.07 mm/s — about the speed of a slow ant. The electrons themselves are racing around at thermal speeds a billion times faster, but those random velocities cancel and only this tiny directed bias survives. The electric field, by contrast, propagates at nearly the speed of light — which is why a light switch responds instantly even though the electrons barely crawl. Resistivity increases with temperature because higher temperatures shorten τ (more vigorous lattice vibrations cause more frequent collisions) and decreases with added impurities for the same reason. Superconductivity, by contrast, corresponds to τ → ∞: electrons propagate without scattering, and σ diverges.