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Plane wave passing from vacuum to conductor

  1. Jul 31, 2015 #1


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    1. The problem statement, all variables and given/known data
    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.
    2. Relevant 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.

    3. 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.
  2. jcsd
  3. Jul 31, 2015 #2


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    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?
  4. Jul 31, 2015 #3


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    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.
  5. Aug 2, 2015 #4


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    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.
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