Mobility and its Relation with Electric Current

Mobility $\mu$ of a charge carrier is defined as the magnitude of its drift velocity per unit applied electric field:

Its SI unit is $\text{m}^2\,\text{V}^{-1}\,\text{s}^{-1}$ (metre-squared per volt-second). From the expression for drift velocity found in Section 3.3, $v_d = eE\tau/m$, mobility can be written directly in terms of the microscopic quantities of the material:

so that mobility depends only on the charge carrier's charge-to-mass ratio and the relaxation time $\tau$ of the material -- it does NOT depend on the applied field itself, since $v_d$ is directly proportional to $E$ and the two dependences on $E$ cancel.

Relation between drift velocity and current. Consider a conductor of uniform cross-sectional area $A$, with $n$ free electrons per unit volume (the free-electron number density), carrying a current $I$ due to an applied field. In a small time interval $\Delta t$, every free electron drifts, on average, a distance $v_d\,\Delta t$ along the wire. So every free electron that lies within a distance $v_d\,\Delta t$ of a chosen cross-section will cross that section within the interval $\Delta t$. These electrons occupy a cylindrical volume of length $v_d\,\Delta t$ and cross-sectional area $A$, i.e. a volume $A\,v_d\,\Delta t$, and so the number of free electrons in this volume is $n\,A\,v_d\,\Delta t$.

Each electron carries a charge of magnitude $e$, so the total charge crossing the section in time $\Delta t$ is

and hence the current is

This is the central relation connecting the microscopic drift velocity to the macroscopic, measurable current $I$. It also defines the current density, $J = I/A = n e v_d$, the current per unit cross-sectional area, a vector quantity pointing along the direction of conventional current flow, which is a more fundamental quantity than $I$ itself since it does not depend on the particular size of wire chosen. …