Editing Drude model (section)

=== DC field ===
The simplest analysis of the Drude model assumes that electric field {{math|'''E'''}} is both uniform and constant, and that the thermal velocity of electrons is sufficiently high such that they accumulate only an infinitesimal amount of momentum {{math|''d'''''p'''}} between collisions, which occur on average every {{mvar|τ}} seconds.<ref name=":0" group="Ashcroft & Mermin" />

Then an electron isolated at time {{mvar|t}} will on average have been travelling for time {{mvar|τ}} since its last collision, and consequently will have accumulated momentum
<math display="block">\Delta\langle\mathbf{p}\rangle= q \mathbf{E} \tau.</math>

During its last collision, this electron will have been just as likely to have bounced forward as backward, so all prior contributions to the electron's momentum may be ignored, resulting in the expression
<math display="block">\langle\mathbf{p}\rangle = q \mathbf{E} \tau.</math>

Substituting the relations
<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>
results in the formulation of Ohm's law mentioned above:
<math display="block">\mathbf{J} = \left( \frac{n q^2 \tau}{m} \right) \mathbf{E}.</math>