Speed of light in lossy dielectric medium

 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. $$\nabla^2 E = \mu \epsilon \frac{\partial^2 E}{\partial t^2}$$ 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)}##. $$- k^2 E = - \mu_0 \epsilon_r \epsilon_0 \omega^2 E$$ 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. $$Re(k)\cdot x = Re(\omega)t$$ Differentiating both sides, we get the equation for phase velocity. $$v = \frac{dx}{dt} = Re\left(\frac{\omega}{k}\right)$$ 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. $$v = Re\left(\frac{c}{\sqrt{\epsilon_r}}\right)$$ 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##. $$a^2 - b^2 + 2iab = \epsilon' + i\epsilon''$$ 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. $$a = \frac{\sqrt{\epsilon' + \sqrt{\epsilon'^2 + \epsilon''^2}}}{\sqrt{2}}$$ So the correct formula for phase velocity is as follows. $$v = \frac{\sqrt{2}c}{\sqrt{\epsilon' + \sqrt{\epsilon'^2 + \epsilon''^2}}}$$ Which again simplifies to formula you have whenever ##\small \epsilon'' = 0##.