Current density and drift speed

Take a conductor holding n mobile carriers per cubic metre, each of charge q, all drifting at average velocity v(d) along the wire. Mark a cross-section of area A and ask what crosses it in time Δt. Every carrier inside a slab of length v(d)Δt behind the face gets there, and no carrier outside it does, so the slab volume A v(d)Δt contains n A v(d)Δt carriers and delivers ΔQ = n q v(d) A Δt. Divide by Δt: I = n q v(d) A. Each factor is separately knowable. For copper, n comes from density and molar mass on the assumption of one conduction electron per atom: (8960 kg m⁻³ ÷ 0.06355 kg mol⁻¹) × 6.022 × 10²³ mol⁻¹ = 8.5 × 10²⁸ m⁻³. That means n|q| = 1.36 × 10¹⁰ C of mobile charge in every cubic metre of copper. The figure is enormous, which already tells you that the remaining factor, v(d), will have to be very small.

I = n q v(d) A mixes a material property with a geometric one. Divide the area out and what remains belongs to the point: J = n q v(d), the current density, in A m⁻². Unlike current, it is a vector, and it is defined everywhere inside the conductor rather than only on a chosen surface. Its direction is that of the conventional current, and that survives the sign of the carrier: in a metal q = −e and v(d) points against E, so the two minus signs cancel and J lies along E. In an electrolyte, cations drift with E and anions against it, and both terms of J = n₊q₊v₊ + n₋q₋v₋ add rather than cancel. Working in J rather than I is what lets you describe conductors that are not neat wires — current spreading through a grounding plate, through seawater, or through tissue, where 'the' cross-sectional area is not a well-posed quantity.

Given J everywhere, the current through a surface is I = ∫ J ⋅ dA, with dA the area element along the chosen normal. The dot product earns its place. Only the component through the face counts, so tilting a plane cross-section of a wire by θ multiplies its area by 1/cos θ and its normal component of J by cos θ, leaving I unchanged — as it must be, since the same charge per second passes either way. If J is uniform and normal, the integral collapses to I = JA. If it is not, do the integral: a wire of radius R with J(r) = J₀ r/R gives I = ∫₀ᴿ (J₀ r/R)(2πr dr) = (2/3)πR²J₀, so the mean current density is two-thirds of the peak; with R = 1.0 mm and J₀ = 3.0 × 10⁶ A m⁻², I = 6.3 A. Steady flow also forces J₁A₁ = J₂A₂. Charge cannot accumulate, so a wire that narrows carries the same I with proportionally larger J and drift speed.

Put numbers in. A 1.0 mm² copper wire — ordinary lamp cord — carrying 1.0 A has J = 1.0 × 10⁶ A m⁻², so v(d) = J/(n|q|) = 1.0 × 10⁶ / 1.36 × 10¹⁰ = 7.3 × 10⁻⁵ m s⁻¹. That is 0.073 mm s⁻¹, about 26 cm in an hour; an electron entering a 2.0 m cord reaches the lamp 2.7 × 10⁴ s later, more than seven hours. The electrons themselves are not slow at all. A conduction electron in copper moves at roughly the Fermi speed, 1.6 × 10⁶ m s⁻¹, in a direction re-randomised every few tens of femtoseconds; the drift is a bias on that motion of order one part in 10¹⁰. The lamp still lights immediately because the field that sets up the drift propagates along the wire at a good fraction of c — about 10 ns for that 2.0 m cord — and carriers were already present everywhere, filament included. Nothing has to arrive from the switch.

Why is v(d) proportional to E instead of growing without limit? Between collisions a carrier accelerates at qE/m; each collision randomises its velocity, so on average it restarts from scratch and accumulates v(d) = (|q|E/m)τ over a mean free time τ. Define the mobility μ = v(d)/E = |q|τ/m, in m² V⁻¹ s⁻¹, and the rest follows: J = n|q|μE = σE with σ = n q² τ/m. That is Ohm's law written at a point; V = IR is what it integrates to for a uniform conductor. Copper's measured σ = 5.96 × 10⁷ S m⁻¹ divided by n|q| = 1.36 × 10¹⁰ C m⁻³ gives μ = 4.4 × 10⁻³ m² V⁻¹ s⁻¹, hence τ = μm/|q| = 2.5 × 10⁻¹⁴ s and a mean free path v(F)τ ≈ 40 nm, roughly 150 atomic spacings. The field inside our 1.0 A wire is only J/σ = 0.017 V m⁻¹, and the net forward creep per free flight is v(d)τ ≈ 2 × 10⁻¹⁸ m.

The Drude argument behind μ = |q|τ/m is classical and wrong in detail: metal electrons are degenerate, so the speed setting the mean free path is the Fermi speed, not the thermal √(3kT/m) = 1.2 × 10⁵ m s⁻¹ at 300 K. It survives because the derivation needs only 'accelerate, then randomise'. Three limits matter. In a metal n is essentially fixed and temperature acts through τ, so resistivity rises with T; in intrinsic silicon n itself climbs steeply with T and resistivity falls — and with n ≈ 10¹⁶ m⁻³ against copper's 8.5 × 10²⁸, a mobility some 30 times larger still leaves σ smaller by some 3 × 10¹¹. Next, v(d) = μE is linear only at low field: carriers in silicon saturate near 10⁵ m s⁻¹ once they can shed energy to optical phonons. Finally, alternating current crowds into a skin depth δ = √(ρ/(π f μ₀)) — 8.4 mm in copper at 60 Hz, so mains wiring stays near-uniform, but 65 μm at 1.0 MHz, where I = JA fails.