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The Drude model describes classical charge carriers accelerated by an applied electric field and randomized by collisions of constant mean waiting time $\tau$. Its mean drift obeys $\dot{v}+v/\tau =eE/m$, and harmonic conductivity is $n{e}^2\tau /[m(1-i\omega \tau )]$ in the negative-frequency exponential convention.

The Drude model relative permittivity is $1-{\omega }_p^2/[\omega (\omega +i/\tau )]$. At frequencies large compared with the collision rate but below ${\omega }_p$, its leading real part is negative and the physical spatial wave-number branch has positive imaginary part. The field therefore decays into the conductor; close to the threshold the full complex response must be retained.

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.