EM wave with circular polarization interferes with conductor

[adremja]

Homework Statement

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 [itex]z=0[/itex].
[itex]B^\perp = 0[/itex] and [itex]E^\perp = 0[/itex] 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 [itex]K_{free} = 0[/itex], but I'm not sure if its still true with perfect conductor. For [itex]E[/itex] 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 [itex]B[/itex]
$$
(-i\hat{x}+\hat{y})\frac{1}{\mu_1v_1}E_{0T} + (-i\hat{x}-\hat{y})\frac{1}{\mu_1v_1}E_{0R} = (-i\hat{x}+\hat{y})\frac{k_2}{\mu_2\omega}E_{0T}
$$

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

 

[Asim Riaz]

For point 2 I can only think of that the charge density and surface current will appear on the conductor due to the wave, but I'm not sure how to calculate it.