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Classical electrodynamics -Good conductor

  1. Oct 26, 2008 #1
    1. The problem statement, all variables and given/known data
    The question given is an electromagnetic wave incident on a vacuum metal interface. The wave is incident normally. We're given that the metal is a good conductor i.e. [tex]\omega \tau <<1 [/tex] where [tex]\tau[/tex] is the collision time of the metal and omega is the angular frequency. The metal is also non-magnetic and the conductivity is of the order of [tex] 10^8 [/tex] Siemens per metre

    The first part is easy enough it's just to show that T the transmissivity is equal to 2/n where n is the real part of N the refractive index.

    The part that has me stumped is to find the value of [tex]\omega[/tex] so that the fraction of incident power deposited beyond a depth d is maximised


    2. Relevant equations

    I think the relevant equation is the poynting vector.

    I'm taking the time averaged poynting vector for a wave in vacuum as [tex] S_{avg} = \frac{1}{2} \sqrt{\frac{\epsilon}{\mu}} E_{i}^{2}[/tex] and inside the metal I assume the form[tex] S_{avg} = \frac{1}{4} \sigma_{0} \delta E_{t}^{2} e^{\frac{-2 z}{\delta}[/tex]

    Where delta is given as[tex]\sqrt{\frac{2}{\mu \omega \sigma_{0}}}[/tex]

    I'm also taking[tex]n=\frac{c}{\omega \delta}[/tex]


    3. The attempt at a solution
    I started by saying that since the time averaged poynting vectors is independent of x and y in both case we can say
    [tex]\int S_{avg} dA=S_{avg} A[/tex]

    Using this I divided the power incident on the surface by the power incident on hte same area but at a distance d below the surface to obtain


    [tex]\frac{\delta \sigma_{0}}{2}\sqrt{\frac{\mu}{\epsilon}}\frac{E_{t}}{E_{i}}\frac{E_{t}}{E_{i}}e^{\frac{-2 d}{\delta}}[/tex]


    but with this I differentiate w.r.t. omega and I can't obtain an answer? It's been bugging me for a while so I hope someone can help me out. Hope the format of the question is ok It's my first time posting here. =D

    edit: I forgot to say that I replaced {E_{t}/E_{i}}^2 with 2/n I tried changing the latex code but it won't edit for some reason.
     
    Last edited: Oct 26, 2008
  2. jcsd
  3. Oct 26, 2008 #2

    marcusl

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    You have already assumed that the field decays exponentially on the scale of the skin depth

    [tex]\delta=\sqrt{\frac{2}{\mu \omega \sigma_{0}}} [/tex]

    But this already gives you the answer: to maximize power deposition beyond d, choose a frequency to make the skin depth greater than d.
     
  4. Oct 26, 2008 #3
    That makes sense but the question asks for a specific value of the frequency. Initially I thought I'd obtain an equation of the form [tex]\omega e^{-\omega}[/tex] which has a definite maximum value.
     
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