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Is it possible to derive Ohm's law (perhaps in some appropriate limit) from Maxwell's Equations?

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For a simple theory behind Ohm's law, see e.g. the Drude model at Wikipedia or this Phys.SE post. –  Qmechanic Oct 16 '12 at 0:09
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Ohm's law $\vec\jmath=\sigma\vec{E}$ can be derived rigorously in the limit of small electric fields using linear response theory. This leads to Kubo's formula for the electric conductivity, which relates $\sigma$ to the zero frequency limit of the retarded current-current correlation function.

$$ \sigma^{\alpha\beta}(q)=\lim_{\omega\to0}\frac{1}{-i\omega}\left\{\frac{ne^2}{m}\delta^{\alpha\beta} - i\langle[j^\alpha(\omega,q),j^\beta(-\omega,-q)]\rangle \right\} $$

(This derivation, of course, involves more than just Maxwell's equation. This is properly derived in the context of non-equilibrium field theory.) The Drude model is a model for the spectral function of the current-current correlation function in terms of a single ``collision time''. This model can be derived within kinetic theory, which is applicable when interactions are weak and the correlation function can be computed in terms of quasi-particles.

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No, it is an approximation and not derived from first principles. It is based on empirical observations.

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Are you talking about Ohm's law of the Maxwell equations? :D –  NiftyKitty95 Oct 16 '12 at 7:36
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