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\section{Physical basis for the resistance of metals}  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.}  Let's assume we have a certain medium that has a uniform density $n$ of mobile carriers of charge $q$. When an electric field $\mathbf{E}$ is applied, these carriers accelerate and therefore conduct electricity. The current they conduct is proportional to their velocity $\mathbf{v}$, as follows:  \begin{equation}  \mathbf{j}_{i} =