Plane wave passing from vacuum to conductor

[fluidistic]

Homework Statement

A monochromatic plane wave with frequency $\omega$ and real amplitude $E_0$ passes from medium 1 to medium 2 orthogonally with the surface between the 2 media. Both media are semi-infinite; the indices of refraction are $n_1=\sqrt{\mu_0 \varepsilon _0}$ and $n_2=\sqrt{\mu_0 \varepsilon \left ( 1+ \frac{i4\pi \sigma}{\omega \varepsilon} \right ) }$ respectively.
1)Find the system of equations that allows to get the value of all the electric fields in both media.
2)Find the transmitted $\vec E$ field in terms of the incident one.
3)Calculate their phase difference.

Homework Equations

$\vec E_I = \vec E_R + \vec E_T$. In words, the incident electric field is equal to the transmitted plus reflected electric fields.

The Attempt at a Solution

I notice that the problem is basically a plane wave passing from vacuum to a metal, with normal incidence.
I must apply the matching conditions for E and H in order to establish the system of equations asked in part 1).
So: $\hat n \cdot (\vec E_2 - \vec E_1 )=0$ and $\hat n \times (\vec H_2 - \vec H_1)=\vec 0$.
Now, $\vec E_2 = \vec E_T$ and $\vec E_1 = \vec E_I + \vec E_R$. I could now go on by writing down the H_i's in terms of the E_i's and answer to the question I suppose.
But I have a doubt: are my matching conditions correct? Because if it's a non perfect conductor, there should be some surface charge density and also a surface current or so... And the matching conditions would not be worth 0, but I am not sure.
I'd appreciate any comment.

 

[DEvens]

Well... I don't want to do too much of your homework for you. Ok, consider he case where $\sigma$ is much smaller than $\epsilon$. That is, the magnitude of the imaginary part is very small. What should you observe about the amplitude of the wave as you move farther into material 2? Correspondingly, let $\sigma$ have a somewhat larger value, but still a lot smaller than $\epsilon$. How would the wave amplitude change in this case? And how would it be different compared to the first case?

In other words, what does that $\sigma$ represent about a wave traveling through a material where it is non-zero?

What is the equation of the wave in a perfectly transmitting material? That is, how does the wave behave as it moves through a material with $\sigma$ zero? What is corresponding equation in the case where $\sigma$ is not zero?

 

[fluidistic]

Well... I don't want to do too much of your homework for you. Ok, consider he case where $\sigma$ is much smaller than $\epsilon$. That is, the magnitude of the imaginary part is very small. What should you observe about the amplitude of the wave as you move farther into material 2? Correspondingly, let $\sigma$ have a somewhat larger value, but still a lot smaller than $\epsilon$. How would the wave amplitude change in this case? And how would it be different compared to the first case?

In other words, what does that $\sigma$ represent about a wave traveling through a material where it is non-zero?

What is the equation of the wave in a perfectly transmitting material? That is, how does the wave behave as it moves through a material with $\sigma$ zero? What is corresponding equation in the case where $\sigma$ is not zero?

Here are my thoughts although not backed up by any math:
When the imaginary part is small, the metal behaves each time less as a perfect conductor, it behaves more like a dielectric. So I guess the amplitude in region 2 should increase.
If sigma starts to increase, the metal starts to behave more like a good conductor and the amplitude of the wave should be lesser.
Basically sigma is related to the skin depth, i.e. how far the wave will travel before falling below a threshold of amplitude compared to its maximum amplitude at the surface of the metal. The smaller sigma, the further the wave can penetrate into the metal without falling below a threshold. The bigger sigma, the less the wave can penetrate without having its amplitude decreased until a threshold.

But I do not see how this helps me and especially how this answers my doubt.

 

[fluidistic]

Any idea about the matching conditions? Shall I assume no induced current nor charge density? (If so, why? Since it's a metal they should be non zero, right?).
If I must assume that there are induced charges and current, then I would obtain 2 equations with 4 unknowns... no idea how to solve this problem.
Any idea is greatly appreciated.