Current, drift velocity and mobility
Electric current is the rate of flow of charge: I = Q/t. Unit: ampere (A) = C/s. By convention current flows in the direction positive charges would move, which is opposite to the motion of electrons.
In a metal, free electrons move randomly at high speeds, but with no field their average velocity is zero, so there is no current. When a field E is applied, the electrons gain a small average velocity opposite to E called the drift velocity v_d. If τ is the average time between collisions (relaxation time), v_d = eEτ/m.
Current in terms of drift velocity: I = n e A v_d, where n is the number of free electrons per m³ and A is the cross-sectional area. The current density is j = I/A = n e v_d.
Mobility μ is the drift speed per unit field: μ = v_d / E = eτ/m. Unit: m² V⁻¹ s⁻¹.
Drift velocity is very small (about 10⁻⁴ m/s), yet a bulb glows almost the instant you switch it on. This is because the electric field travels through the wire at nearly the speed of light, so electrons all along the wire start drifting at once.
Worked example: I = 1.5 A in a copper wire of area 1 mm² (10⁻⁶ m²) with n = 8.5 × 10²⁸ m⁻³. v_d = I/(neA) = 1.5 / (8.5 × 10²⁸ × 1.6 × 10⁻¹⁹ × 10⁻⁶) ≈ 1.1 × 10⁻⁴ m/s.
Key points
- I = Q/t; I = neAv_d.
- v_d = eEτ/m; μ = v_d/E.
- Drift velocity is of the order of 10⁻⁴ m/s; random thermal speeds are much larger.
- Current is a scalar, though it has a direction along the wire.
Exam tips
- For a wire of non-uniform thickness carrying a steady current, I is the same everywhere but v_d is larger where the wire is thinner.
- Convert mm² to m² by multiplying by 10⁻⁶.
Common mistakes
- Treating drift velocity as the speed at which the signal travels.
- Thinking current density is the same in thick and thin parts of a wire.
- Mixing up mobility and drift velocity.