Is it possible to derive Ohm's law (perhaps in some appropriate limit) from Maxwell's Equations?
[Physics] Derivation of Ohm’s Law
electric-circuitselectric-currentelectrical-resistanceelectromagnetismsolid-state-physics
electric-circuitselectric-currentelectrical-resistanceelectromagnetismsolid-state-physics
Is it possible to derive Ohm's law (perhaps in some appropriate limit) from Maxwell's Equations?
Best Answer
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.