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Speed of light in lossy dielectric medium 
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#1
Jan3113, 09:21 PM

P: 5

It is usually written that the speed of light in a dielectric medium is ##v=\frac{c}{\sqrt{\epsilon_r}}##, where ##c## is the speed of light in vacuum and ##\epsilon_r## is the relative permittivity. But, how can it be calculated for lossy and not necessarily lowloss dielectrics, i.e. those with a complex permittivity ##\epsilon_r=\epsilon'j\epsilon"##?
a) ##v=\frac{c}{\sqrt{\epsilon'}}##? b) ##v=\frac{c}{\sqrt{\epsilon_r}}##? c) None of above? Related to previous question: If the natural resonance frequency of a resonant mode in an empty (vacuum) microwave cavity is ##f_0##, which is the natural resonance frequency for the same mode, but with the cavity completelyfilled with the previous lossy dielectric material? a) ##f=\frac{f_o}{\sqrt{\epsilon'}}##? b) ##f=\frac{f_o}{\Re{(\sqrt{\epsilon'j\epsilon"})}}##? b) ##f=\frac{f_o}{\sqrt{\epsilon_r}}##? c) None of above? All of the textbooks and webs (that I've found) about this topic, consider only lossless dielectrics or approximations for lowloss dielectrics, but no general expressions. 


#2
Feb113, 01:59 AM

Sci Advisor
P: 2,470

It helps to understand the equation, not just know it. Ignoring, for the moment, frequency dependence of permittivity and permeability, electromagnetic wave is described by the following equation.
[tex]\nabla^2 E = \mu \epsilon \frac{\partial^2 E}{\partial t^2}[/tex] In your case, ##\small \mu = \mu_0## and ##\small \epsilon = \epsilon_r \epsilon_0##. And to make things easier, lets take solution of the form ##E = E_0 e^{i(k\cdot x  \omega t)}##. [tex] k^2 E =  \mu_0 \epsilon_r \epsilon_0 \omega^2 E[/tex] Since ##\small c^2 = 1/\mu_0\epsilon_0##, the above gives you ##k^2 = \frac{\epsilon_r \omega^2}{c^2}##. Now we can look at velocity. Since you seem to be interested in the phase velocity, we are looking at points in space and time where ##\small E## has same phase as ##\small E_0##. In other words, ##\small k \cdot x  \omega t## is purely imaginary. Realizing that ##\small x## and ##\small t## are always real, that gives you a simple enough formula. [tex]Re(k)\cdot x = Re(\omega)t[/tex] Differentiating both sides, we get the equation for phase velocity. [tex]v = \frac{dx}{dt} = Re\left(\frac{\omega}{k}\right)[/tex] From earlier, we have ##\frac{\omega^2}{k^2} = \frac{c^2}{\epsilon_r}##. So we have to take the square root here, keeping in mind that we are working with complex numbers. [tex]v = Re\left(\frac{c}{\sqrt{\epsilon_r}}\right)[/tex] If ##\small \epsilon_r## is real, that trivially simplifies to your formula. It is, however, not. We need to take square root of a complex number. In other words, we are looking for some ##\small z = a + ib## such that ##\small z^2 = \epsilon_r##. [tex]a^2  b^2 + 2iab = \epsilon' + i\epsilon''[/tex] This is a system of equations with four possible solutions, but only one is physical. It's a bit messy, so I only give the result for a, which is all we need for the real part above. [tex]a = \frac{\sqrt{\epsilon' + \sqrt{\epsilon'^2 + \epsilon''^2}}}{\sqrt{2}}[/tex] So the correct formula for phase velocity is as follows. [tex]v = \frac{\sqrt{2}c}{\sqrt{\epsilon' + \sqrt{\epsilon'^2 + \epsilon''^2}}}[/tex] Which again simplifies to formula you have whenever ##\small \epsilon'' = 0##. 


#3
Feb413, 05:37 PM

P: 5

Thank you very much.



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