Find Parallel Polarized E Field Reflection Direction

[yungman]

For a parallel polarization EM hitting the conductor boundary in an oblique angle. z axis is perpendicular to the boundary and point into the conductor. y-axis it out of the page which give x pointing up. Let the boundary surface by xy plane. With this:

The direction of the incident is:

$$\hat n_i \;=\; \hat x sin \theta_i + \hat z cos \theta_i \;\hbox { and direction of }\; \hat {E_i} \;= \hat x cos \theta_i - \hat z sin \theta_i$$

I know

$$\hat {E_r} \;=\; \hat x cos \theta_i + \hat z sin \theta_i$$

My question is how can I derive the direction of $\vec {E_r}$ by using formulas? I got this by looking at the reflection as I move the incident E towards the boundary...by drawing. I want to find this mathametically. Please help.

Thanks

Alan

 

[ednobj]

You know from reflections that
$$\hat{n}_r=\hat{x}\sin(\theta_i)-\hat{z}\cos(\theta_i)$$
If your E-field is in free space, it must be normal to this outgoing wave vector. Since you had a parallel incident E-field, there are only two possibilities and one is just the negative of the other. Which one is correct depends on the relative index of reflection; see Fresnel equations for parallel plane incidence to show which.