7. Waves in Lossy Media

In a lossy medium characterized by conductivity , the Helmholtz equation becomes

(1)

The equivalent permittivity is a complex quantity

(2)

The factor
is called the loss tangent, because it is equivalent to the tangent of the phase angle of the complex permittivity.

Our simplest solution can keep its original form

(3)

with a complex wavenumber

(4)

where and are the real and imaginary parts of the wavenumber , respectively.

The skin depth (for a good conductor) or the penetration depth (for a poor conductor)

(5)

is a measure of how far a plane wave can penetrate into a lossy medium. Because the envelope of the electric field decays to 1/e of its original amplitude at one penetration depth (or, skin depth) into a lossy medium.

Approximations

Good Conductor	Poor Conductor

Electron Plasma

In an isotropic lossless electron plasma, the Helmholtz equation becomes

(6)

where

(7)

is the permittivity of plasma, and

(8)

is the plasma frequency expressed in terms of electron charge, , electron mass, , and the number density of electrons, .

The corresponding wavenumber is

(9)

Note that when the wave frequency is lower than the plasma frequency, the wavenumber becomes purely imaginary, and the electromagnetic wave becomes evanescent.

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This page was last updated on February 1, 1998.