Physical Origin of Ohm's Law

Ohm's law can be derived from the microscopic picture of electron drift introduced in section 11.4. Electrical conduction in a metal is due to the mobile conduction electrons, assumed free to move throughout the volume of the conductor. During this motion the electrons repeatedly collide with the fixed ion cores of the lattice (electron-electron collisions are assumed negligible), and this succession of random collisions, on average, produces zero net motion by itself.

When an electric field $\vec{E}$ is applied, an electron's actual motion becomes a combination of this random collisional motion PLUS a systematic drift caused by the field, in the direction opposite to $\vec{E}$. Consider a single electron of mass $m$ in the field $\vec{E}$: the force on it is $\vec{F}=e\vec{E}$ (magnitude), giving it an acceleration

between collisions. Crucially, the TYPE of collision an electron undergoes with an ion core is such that whatever drift velocity it had built up just before the collision has no bearing on its velocity immediately after -- after each collision, the electron essentially starts building up its drift velocity afresh in a new random direction, though it still drifts, on average, opposite to $\vec{E}$.

Let $\tau$ (tau) be the average time between two successive collisions (the mean free time). On average, an electron therefore acquires a drift speed

Combining this with $V_d=J/(ne)$ from Eq. (11.6) and rearranging gives

which is written as $E=\rho J$, where

$\rho=\frac{m}{ne^2\tau}\qquad\text{--- (11.15)}$ …