What is drift velocity of charge carriers and how is it related to current?

In one line

Drift velocity is the small average velocity that free electrons (or holes) acquire in the direction set by an applied electric field, superimposed on their fast random thermal motion.

Explained simply

Picture a crowded railway platform where people move randomly in all directions. If a slight slope tilts the platform, everyone still moves randomly but the whole crowd slowly shifts downhill. That slow overall shift is the drift velocity. Electrons in a wire move randomly at very high speeds, but their net drift along the wire is only a fraction of a millimetre per second.

How it works

1. Without a field, electrons move randomly with high thermal speeds and collide with lattice ions; the average velocity is zero, so there is no current.
2. With a field \(E\), each electron accelerates by \(a = eE/m\) between collisions.
3. If the average time between collisions (relaxation time) is \(\tau\), the average extra velocity gained is \(v_d = eE\tau/m\).
4. Electrons drift opposite to \(E\) (negative charge); holes drift along \(E\).
5. Current relation: in time \(t\), charges in a length \(v_dt\) of a wire of area \(A\) pass a point: \(Q = neAv_dt\), so \(I = neAv_d\) and \(J = nev_d\).
6. Even though \(v_d\) is tiny, a bulb lights instantly because the electric field travels through the circuit at nearly the speed of light, moving all electrons at once.

Key formula

\(v_d = \dfrac{eE\tau}{m} = \mu E\), \(I = neAv_d\). \(v_d\): drift velocity (m/s), \(E\): field (V/m), \(\tau\): relaxation time (s), \(m\): electron mass (kg), \(\mu\): mobility (m²/V s), \(n\): carrier concentration (m⁻³), \(A\): area (m²), \(I\): current (A).

Quick example

A copper wire of area 1 mm² carries 1 A; \(n = 8.5\times10^{28}\) m⁻³. \(v_d = I/(neA) = 1/(8.5\times10^{28}\times1.6\times10^{-19}\times10^{-6}) \approx 7.4\times10^{-5}\) m/s, about 0.07 mm/s. At that speed an electron needs over 3 hours to drift one metre.

Common mistakes

• Confusing drift velocity with the random thermal speed, which is much larger.
• Thinking a bulb lights only when electrons from the switch reach it.
• Forgetting that electrons drift opposite to the field direction.

How MU asks it

Asked as "Define drift velocity and derive \(I = neAv_d\)" (3 to 5 marks) and as numericals on drift velocity. Define it, explain random motion plus drift, derive \(I = neAv_d\), and state \(v_d = \mu E\).