Drift Velocity of Free Electrons

Consider a free electron in a metal at some instant just after it has suffered a collision with a lattice ion. Immediately after the collision, its velocity is entirely random in direction (this is what a collision does -- it randomises the electron's velocity). Between this collision and the next one, if an electric field $E$ is present, the electron experiences a constant force $F = -eE$ and hence a constant acceleration

where $m$ is the electron's mass. If the time elapsed since the last collision is $t$, the extra velocity gained due to the field alone is $\vec{a}t$, directed opposite to $\vec{E}$ (since the electron's charge is negative). Because collisions are random, different electrons have been travelling for different lengths of time since their last collision at any given instant; averaging $\vec{a}t$ over ALL the free electrons in the metal, at any instant, replaces $t$ by its average value, called the relaxation time, $\tau$ -- the average time that elapses between two successive collisions of a free electron.

The drift velocity, $\vec{v}_d$, is defined as this average extra velocity acquired by the free electrons due to the applied field:

Its magnitude is directed opposite to $\vec{E}$ (since the electron charge is negative), and it is extraordinarily small compared with the electrons' own random thermal speed -- typically of the order of $10^{-4}\ \text{m/s}$ (a fraction of a millimetre per second), against a thermal speed of order $10^5\ \text{m/s}$. It is remarkable that so slow a drift can nonetheless account for the familiar, effectively instantaneous "switching on" of a lamp the moment a switch is closed: what propagates almost instantly along the wire is not any individual electron itself, but the electric field (and the resulting disturbance in the electron sea) that is set up throughout the conductor as soon as the circuit is completed, very much like a wave. …