Drude current relaxation ($\mathbf{J}(t)={\sigma }_0{\mathbf{E}}_0+[\mathbf{J}(0)-{\sigma }_0{\mathbf{E}}_0]e^{-t/\tau }$)

Switch on a constant electric field ${\mathbf{E}}_0$ at $t=0$. For prescribed initial current density $\mathbf{J}(0)$, the exact Drude model solution is the displayed formula. For Electrons, the corresponding drift velocity isStarting from zero drift velocity, the current density reaches $1-e^{-1}\approx 0.632$ of its limiting value after one momentum relaxation time. Switching the electric field off at $T$ instead gives $\mathbf{J}(t)=\mathbf{J}(T)e^{-(t-T)/\tau }$ for $t\ge T$. A finite step of electric field changes the acceleration immediately but leaves drift velocity and current density continuous.

More generally, multiplying the current density equation by an integrating factor givesThis is the causal response of a first-order relaxation equation: only earlier electric fields enter the integral. It also shows why instantaneous Ohm's law is an approximation when the electric field changes appreciably during a momentum relaxation time.