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The Drude model treats carriers as classical independent particles, accelerated between collisions by the electric force. Collisions occur at constant rate ${\tau }^{-1}$ and randomize the directed velocity to zero mean; the ions are an effectively stationary background. Over a short time $dt$, directed velocity gains $(e/m)\mathbf{E}dt$ while the fraction $dt/\tau$ that collides loses its mean directed motion. ThereforeHere $e$ is the signed carrier charge. For the stated complex harmonic convention,The conductivity is positive in its dissipative real part regardless of the sign of $e$.

For zero charge density, Gauss's law makes the electric amplitude transverse. Faraday's and Ampère-Maxwell's laws, with displacement current retained, giveSubstitution of $e^{i\mathbf{k}\cdot \mathbf{x}-i\omega t}$ yieldsFor propagation in a fixed direction, $k$ is the possibly complex scalar wave number; equivalently the spatial equation uses $\mathbf{k}\cdot \mathbf{k}$, without complex conjugation. The printed real-vector notation $|\mathbf{k}|$ must be analytically continued in an absorbing or evanescent medium.

In the high-frequency Drude plasma cutoff regime, $\omega \tau \gg 1$ gives ${ϵ}_r\simeq 1-{\omega }_p^2/{\omega }^2$. Below the plasma frequency, the physical decaying branch isFinite collisions add a small phase component while maintaining positive attenuation. Very near the threshold one should retain the full complex permittivity rather than the purely imaginary approximation.