# EM wave with circular polarization interferes with conductor

Tags:
1. Feb 21, 2016

1. The problem statement, all variables and given/known data
EM wave with circular polarization travels in directon z interferes with perfect conducting surface xy.

1. find reflected wave
2. calculate induced charge density and surface current induced on conducting surface

Can you verify if I started point 1. correctly, and give some idea how to calculate point 2. Because in Griffiths I can only find info that there are volume currents which in limit of perfect coonductor become true surface current.

2. The attempt at a solution

I started the first point

$$E_I(z,t) = E_{0I}(\hat{x}+i\hat{y})e^{i(k_1z-\omega t)}$$
$$B_I(z,t) = \frac{1}{v_1}E_{0I}(-i\hat{x}+\hat{y})e^{i(k_1z-\omega t)}$$

$$E_R(z,t) = E_{0R}(\hat{x}-i\hat{y})e^{i(-k_1z-\omega t)}$$
$$B_R(z,t) = -\frac{1}{v_1}E_{0R}(i\hat{x}+\hat{y})e^{i(-k_1z-\omega t)}$$

$$E_T(z,t) = E_{0T}(\hat{x}+i\hat{y})e^{i(k_2z-\omega t)}$$
$$B_T(z,t) = \frac{k_2}{\omega}E_{0T}(-i\hat{x}+\hat{y})e^{i(k_2z-\omega t)}$$

Then I apply boundary conditions in $z=0$.
$B^\perp = 0$ and $E^\perp = 0$ so I only parallel conditions left
$$E^\parallel_1 - E^\parallel_2 = 0$$
and
$$\frac{1}{\mu_1}B^\parallel_1 - \frac{1}{\mu_2}B^\parallel_2 = K_{free} \times \hat{n}$$

Griffiths in 9.4.2 says that $K_{free} = 0$, but I'm not sure if its still true with perfect conductor. For $E$ I have
$$(\hat{x}+i\hat{y})E_{0T} + (\hat{x}-i\hat{y})E_{0R} = (\hat{x}+i\hat{y})E_{0T}$$
and for $B$
$$(-i\hat{x}+\hat{y})\frac{1}{\mu_1v_1}E_{0T} + (-i\hat{x}-\hat{y})\frac{1}{\mu_1v_1}E_{0R} = (-i\hat{x}+\hat{y})\frac{k_2}{\mu_2\omega}E_{0T}$$

which need to be solved, but I'm not sure if I treated polarization right …

2. Feb 26, 2016