Electromagnetic Radiation of Single frequency Incident on a Slab

[MattIverson]

Homework Statement

The problem is state in the attachment.

Homework Equations

None to add.

3. The Attempt at a Solution
Struggling with this problem because of the phasor part of it. Perhaps this is justified because the angle is very small and so that relates to a low frequency? Is it that you add the x and y vectors together? I'm not really sure. I have some little ideas like that, but honestly I'm pretty lost on this problem.

I would so much appreciate some help with this problem! Thank you so much!

 

[Charles Link]

What this refers to is basically a Huygens principle for determining the speed of light through the dielectric which normally gets slowed by the presence of the polarization that occurs in the material with, with $n=\sqrt{\epsilon_r}$ where $P=\epsilon_o \chi E$, and $\epsilon_r=1+\chi$. $\\$ I don't have the answer at my fingertips either, but I seem to recall something about the index $n$ being less than 1 for some plasmas above the cut-off frequency, so that the phase velocity would be faster than $c$. This would (presumably) make the phase angle positive in the phasor diagram. (I'm just giving you a place to start googling, or researching E&M textbooks). See section 5.2 of this "link" and equation (5.53) https://ocw.mit.edu/courses/nuclear...a-physics-i-fall-2003/lecture-notes/chap5.pdf For frequencies below $\omega_p$, the dielectric constant is negative, making $n$ imaginary, so the wave does not propagate, but rather gets quickly attenuated.