where denotes average velocity, m the effective mass and q the charge of the charge carriers.
The steady state solution () of this differential equation is
is the mean free time of a charge carrier, and is the mobility. Now, introducing charge carrier density n (particles per unit volume), we can relate average velocity to current density:
The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency , in which case
Here it is assumed that
In other conventions, is replaced by in all equations. The imaginary part indicates that the current lags behind the electrical field, which happens because the electrons need roughly a time to accelerate in response to a change in the electrical field. Here the Drude model is applied to electrons; it can be applied both to electrons and holes; i.e., positive charge carriers in semiconductors.
This simple classical Drude model provides a very good explanation of DC and AC conductivity in metals, the Hall effect, and thermal conductivity (due to electrons) in metals. The model also explains the Wiedemann-Franz law of 1853. However, it greatly overestimates the electronic heat capacities of metals. In reality, metals and insulators have roughly the same heat capacity at room temperature. Although the model can be applied to positive (hole) charge carriers, as demonstrated by the Hall effect, it does not predict their existence.