deletions | additions       diff --git a/resistance.tex b/resistance.tex  index 6b12a7a..05c2219 100644  --- a/resistance.tex  +++ b/resistance.tex 
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The two key elements for conducting a current are mobile charged particles and an energy dissipation mechanism.\footnote{We're discussing here about non-superconducing metals. Superconductivity is a totally different beast.} Since a current is just the motion of charged particles, you need the former by definition. In order to prevent these particles accelerating indefinitely, you also need an energy dissipation mechanism -- without it, you have no steady state current for nonzero electric field.  We'll now derive Ohm's law starting only from these assumptions. Assume that we have a uniform density $n$ of particles, carrying a charge $q$. These carriers are accelerated by an electric field $\mathbf{E}$ and scatter via a certain mechanism with a characteristic lifetime $\tau$. Their Assuming they have a well defined momentum $\mathbf{p}$, their  equation of motion is therefore: is:\footnote{As long as we only consider small changes in momentum, \mathbf{p} can also be crystal momentum}  \begin{equation}  \frac{d\mathbf{p}}{dt} = - \frac{1}{\tau} \mathbf{p} + q \mathbf{E}  \end{equation}  Let's assume we have You can view this equation as  a certain medium that has Langevin equation, with  a uniform density $n$ of mobile carriers of charge $q$. When an electric field $\mathbf{E}$ is applied, these carriers accelerate dissipation constant $1 / {\tau}$  and therefore conduct electricity. a constant (rather than random) force.  The current they conduct is proportional to their steady state  velocity $\mathbf{v}$, as follows: $\mathbf{v}$ is:  \begin{equation}  \mathbf{j}_{i} \frac{d\mathbf{p}}{dt}  = 0 \Leftrightarrow \mathbf{v}  \end{equation}