rework the drude derivation

maybe that works better

src/3_drude_model.md

@@ -135,20 +135,22 @@ Our goal is then to compute the *average* velocity.
 happens with each individual element.**
 
 Let us compute how the average velocity changes with time.
-The equation with the Lorentz force we just average right away:
+Consider the effect that scattering has over a small time $dt$.
+A fraction $dt/τ$ of the electrons scatters, and that their average velocity becomes zero.
+The rest of the electrons $(1 - dt/τ)$ accelerates by the Lorentz force, and after $dt$ their velocity becomes
 $$
-m\frac{d⟨\mathbf{v}⟩}{dt} = -e\left(\mathbf{E}+⟨\mathbf{v}⟩×\mathbf{B}\right).
+m\mathbf{v}(t + dt) - m\mathbf{v}(t) = - dt e (\mathbf{E} + \mathbf{v} × \mathbf{B}).
 $$
-Almost there, but we still need to do something with the change of the average velocity due to scattering.
-Consider the effect that scattering has over a small time $dt$.
-Most electrons continue with the same velocity, however a fraction $dt/τ$ will scatter, and that their average velocity becomes zero.
-Therefore we get
+Combining the two groups of particles, we get
 $$
-⟨\mathbf{v}(t+dt)⟩ = ⟨\mathbf{v}(t)⟩(1 - dt/τ) + 0⋅(dt/τ) ⇒ \frac{d⟨\mathbf{v}⟩}{dt} = -\frac{⟨\mathbf{v}⟩}{τ}.
+\begin{align}
+m⟨\mathbf{v}(t+dt)⟩ &= (m⟨\mathbf{v}(t)⟩ - dt e (\mathbf{E} + \mathbf{v} × \mathbf{B}))(1 - dt/τ) + 0⋅(dt/τ)\\
+                    &= m⟨\mathbf{v}(t)⟩ - dt [e (\mathbf{E} + \mathbf{v} × \mathbf{B}) - m⟨\mathbf{v}(t)⟩/τ] \\
+                    &+ dt² [e (\mathbf{E} + \mathbf{v} × \mathbf{B}) m⟨\mathbf{v}(t)⟩/τ
+\end{align}
 $$
-
-That's it!
-We now combine both contributions into a single equation and get
+We now neglect the term proportional to $dt²$ (it vanishes faster when $dt → ∞$).
+Finally, we recognize that $(⟨\mathbf{v}(t+dt)⟩ - (⟨\mathbf{v}(t)⟩)/dt = d⟨\mathbf{v}(t)⟩)/dt$, and arrive to
 $$
 m\frac{d⟨\mathbf{v}⟩}{dt} = -m\frac{⟨\mathbf{v}⟩}{τ} -e\left(\mathbf{E}+⟨\mathbf{v}⟩×\mathbf{B}\right).
 $$