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Flaviu Cipcigan edited resistance.tex
over 10 years ago
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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 \cite{Drude_1900}. assumptions\cite{Drude_1900}. 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 $\tau$. Assuming they have a well defined momentum $\mathbf{p}$, their equation of motion 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}