Editing Drude model (section)

=== Constant electric field ===
At time {{math|''t'' {{=}} ''t''<sub>0</sub> + ''dt''}} the average electron's momentum will be
<math display="block">\langle\mathbf{p}(t_0+dt)\rangle=\left( 1 - \frac{dt}{\tau} \right) \left(\langle\mathbf{p}(t_0)\rangle + q\mathbf{E} \, dt\right),</math>
and then
<math display="block">\frac{d}{dt}\langle\mathbf{p}(t)\rangle = q\mathbf{E} - \frac{\langle\mathbf{p}(t)\rangle}{\tau},</math>
where {{math|⟨'''p'''⟩}} denotes average momentum and {{mvar|q}} the charge of the electrons. This, which is an inhomogeneous differential equation, may be solved to obtain the general solution of
<math display="block">\langle\mathbf{p}(t)\rangle = q \tau \mathbf{E}(1-e^{-t/\tau}) + \langle\mathbf{p}(0)\rangle e^{-t/\tau}</math>
for {{math|''p''(''t'')}}. The [[steady state]] solution, {{math|{{sfrac|''d''|''dt''}}⟨'''p'''⟩ {{=}} 0}}, is then
<math display="block">\langle\mathbf{p}\rangle = q \tau \mathbf{E}.</math>

As above, average momentum may be related to average velocity and this in turn may be related to current density,
<math display="block">\begin{align}
\langle\mathbf{p}\rangle &= m \langle\mathbf{v}\rangle, \\
\mathbf{J} &= n q \langle\mathbf{v}\rangle,
\end{align}</math>
and the material can be shown to satisfy Ohm's law <math>\mathbf{J} = \sigma_0 \mathbf{E}</math> with a [[Direct current|DC]]-conductivity {{math|''σ''<sub>0</sub>}}:
<math display="block">\sigma_0 = \frac{n q^2 \tau}{m}</math>