Electric current, drift velocity and mobility
Free electrons in a metal drift slowly against the field at v_d = eEτ/m. This drift carries a current I = neAv_d.
Free electrons in a metal
Electric current through an area is the net charge crossing it per unit time: I = q/t. If the current changes, I = dq/dt. In a metal, the charge carriers are free electrons. They move randomly at high speeds and keep colliding with the fixed positive ions. With no field, their velocities point every way and average to zero, so there is no net current. When a cell is joined, an electric field E is set up inside the wire. Between collisions, each electron is now pushed opposite to E. After every collision its direction is random again. The result is a slow, steady drift opposite to E, at the drift velocity v_d. The average time between two successive collisions is the relaxation time τ, about 10⁻¹⁴ s in metals. The mobility μ of a carrier is its drift speed per unit field: μ = v_d/E.
Derivation: drift velocity v_d = −eEτ/m
Take a conductor with N free electrons, each of charge −e and mass m, in a uniform electric field E.
- 1Force on each electron: F = −eE. So its acceleration is a = −eE/m.
- 2Take electron i. Just after its last collision its velocity is vᵢ, in a random direction. After a time tᵢ its velocity is Vᵢ = vᵢ − eEtᵢ/m.
- 3Average over all N electrons: v_d = (Vᵢ)average = (vᵢ)average − (eE/m)(tᵢ)average.
- 4The random velocities cancel, so (vᵢ)average = 0. The average time since the last collision is the relaxation time, so (tᵢ)average = τ.
- 5★So, v_d = −eEτ/m. The minus sign shows that electrons drift opposite to the field.
★ key step
Derivation: current I = neAv_d
Take a wire of area of cross-section A with n free electrons per unit volume, all drifting with speed v_d.
- 1In a time Δt, each electron moves a distance v_d Δt along the wire.
- 2So every electron within a length v_d Δt of the area A crosses it in that time. The volume of this slice is A v_d Δt.
- 3Number of electrons crossing = n A v_d Δt. Charge crossing = e n A v_d Δt.
- 4Current = charge ÷ time: I = e n A v_d Δt ÷ Δt.
- 5★So, I = neAv_d, and the current density is j = I/A = nev_d.
★ key step
How to draw the drift diagram
- Draw a cylinder for the wire. Shade a slice of length v_d Δt next to one cross-section, and label its area A.
- Draw a long arrow along the wire and label it E. Draw the current arrow I in the same direction.
- Draw a short arrow the other way and label it v_d (electrons).
- To show random motion, draw the zigzag path of one electron that slowly shifts opposite to E.
Worked example: drift speed in a copper wire
A copper wire of area of cross-section 2.0 mm² carries a current of 3.4 A. Copper has 8.5 × 10²⁸ free electrons per m³. Find the drift speed of the electrons. How long does an electron take to drift 1 m? (e = 1.6 × 10⁻¹⁹ C)
- Given: I = 3.4 A, A = 2.0 mm² = 2.0 × 10⁻⁶ m², n = 8.5 × 10²⁸ m⁻³.
- Formula: I = neAv_d, so v_d = I/(neA).
- neA = 8.5 × 10²⁸ × 1.6 × 10⁻¹⁹ × 2.0 × 10⁻⁶ = 27.2 × 10³ = 2.72 × 10⁴ C m⁻¹.
- v_d = 3.4 A ÷ (2.72 × 10⁴ C m⁻¹) = 1.25 × 10⁻⁴ m s⁻¹.
- Time to drift 1 m: t = 1 m ÷ (1.25 × 10⁻⁴ m s⁻¹) = 8.0 × 10³ s, a little over 2 hours.
v_d = 1.25 × 10⁻⁴ m s⁻¹, about 1.3 × 10⁻⁴ m s⁻¹ (0.13 mm per second); about 8.0 × 10³ s to drift 1 m
Worked example: mobility and relaxation time
In a metal wire the electric field is 0.20 V m⁻¹ and the electrons drift at 8.0 × 10⁻⁴ m s⁻¹. Find (a) the mobility of the electrons and (b) their relaxation time. (e = 1.6 × 10⁻¹⁹ C, m = 9.1 × 10⁻³¹ kg)
- (a) Formula: μ = v_d/E.
- μ = (8.0 × 10⁻⁴ m s⁻¹) ÷ (0.20 V m⁻¹) = 4.0 × 10⁻³ m² V⁻¹ s⁻¹.
- (b) Formula: μ = eτ/m, so τ = μm/e.
- τ = (4.0 × 10⁻³ × 9.1 × 10⁻³¹) ÷ (1.6 × 10⁻¹⁹) = (36.4 × 10⁻³⁴) ÷ (1.6 × 10⁻¹⁹) s.
- τ = 22.75 × 10⁻¹⁵ s, which rounds to 2.3 × 10⁻¹⁴ s.
(a) μ = 4.0 × 10⁻³ m² V⁻¹ s⁻¹; (b) τ ≈ 2.3 × 10⁻¹⁴ s
Electrons in your phone charger cable creep along at less than a millimetre per second, yet charging starts at once. The electric field travels through the wire almost at the speed of light, so electrons all along the cable start drifting together.
1 marks: “Define relaxation time. What happens to it when a metal wire is heated?” The average time between two successive collisions of a free electron. It decreases on heating, as the ions vibrate more and collisions become more frequent.
2 marks: “Define mobility of a charge carrier. Write its SI unit and its relation with relaxation time.” Mobility is the drift speed per unit electric field, μ = v_d/E. SI unit m² V⁻¹ s⁻¹. For electrons μ = eτ/m, so a longer relaxation time means a higher mobility.
3 marks: “Derive the relation I = neAv_d for a current-carrying conductor.” Take area A and n electrons per m³ drifting at v_d. In time Δt, the electrons in volume A v_d Δt cross A. Charge = neAv_d Δt. Divide by Δt: I = neAv_d. One mark each for the volume, the charge and the result.
Drift is slow, the signal is fast. For current, say 'I = n e A v': number, charge, area, velocity.