# Mean-free-time between collisions and drift-velocity

In a physics text book I need help to make sense of the part highlighted in yellow: This is out of context of course, so just to make it clearer:

$\tau$ is the mean free time of the electrons in a conductor (the average time between collisions with ions in the material).

This text snippet is part of a derivation of a microscale expression of drift velocity. Before time $t<0$ there is no field $\vec E=0$, so electrons move randomly as always with no average drift.

At time $t=0$ a field $\vec E \neq 0$ is applied and drift starts. Electrons are accelerated $\vec Eq=\vec F=m \vec a$ and speed up.

The question:

What I find unclear in their method is the postulate marked in yellow.

Electrons are accelerated until time $t=\tau$ since they (on average) don't collide with anything that could absorb their kinetic energy. But why are we sure that when the first collisions happen, the electrons are decelerated exactly as much as the acceleration caused by the field, so it cancels out (it just balances, as the yellow marked text says)?

Because their net acceleration must be zero since we now assume to reach a steady drift velocity $\vec v_d$.

Apparently we expect constant acceleration before the first collision (during the time $\tau$) and suddenly zero net acceleration after that. How do we know that drift velocity is steady from now on and not only after $2 \tau$ or $5 \tau$ or more?

• I suspect what the book's saying is that, on average, a collision happens every $\tau$ which reduces the component of the electron's velocity in the drift direction by the same amount that the $\vec E$ field accelerates; and that is how $\tau$ is defined in the presence of the electric field. I bet a better book (or maybe on the next page :-) ) there's some statistical mechanics equations to derive this. Jan 4, 2015 at 13:06

The collision with an ion doesn't necessarily stop an electron, just kicks it in some random direction, s.t. the electron doesn't move anymore in the direction of the field. But after being kicked away, the electron is again accelerated in the field direction for a time which is on average $\tau$. And so on, an almost periodic movement. Averaging on this movement, and over all the electrons, one gets the drift velocity that you say, for a given strength of the field $\vec E$.