In a conducting medium, such that the conductivityMaxwell's equations become
By neglecting resonant or other effects we may use the linear approximationsandwhereandare independent of time.
Maxwell's equations becomeand
Taking the curl of the last of these gives
We have now the wave equation in a conducting medium:
The last two equations are called the telegraph equations and incorporate damping termsandso electromagnetic waves travelling in a conducting medium experience attenuation proportianal too the conductance.
By assumingandare of complex exponential form the last two of Maxwells equations above becomeand
The first telegraph equation then becomeswhich has the form of the Helmhotz equationwith
We may use the identityto demonstrate the that the equations for conducting and non conducting media are the same if the dielectric constantis replaced by a complex dielectric constant
Since we have replacedby a complex equivalent, we must obtain a complex equivalent for the refractive index. This is done by writing
where k is a constant called the extinction coefficient.
We replace the propagation constant k by
Assuming thatis parallel to the– axis, then
This wave is attenuated by the factor