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E&M Question: Total Transmission

  1. May 31, 2007 #1
    Hello Physics Forum people! :)

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

    The three regions in the figure below contain perfect dialectrics. For a wave in medium 1, incident normally upon the boundary at z = -d, what combination of [itex]\epsilon_{r2}[/itex] and [itex]d[/itex] produces no reflection? Express your answers in terms of [itex]\epsilon_{r1}[/itex], [itex]\epsilon_{r3}[/itex], and the oscillation frequency of the wave, [itex]f[/itex].

    [​IMG]


    2. Relevant equations

    1/2 Wavelength matching equation:

    [tex]Z_{in} = Z_L, \ \ \ \ \ \ \ \ \mbox{for } l = n\lambda/2[/tex]

    Wavelength/permitivity relation:

    [tex]\lambda = \frac{c}{f\sqrt{\epsilon_{r}}}[/tex]

    3. The attempt at a solution

    I used the above equations to get

    [tex] \lambda_2 = \frac{c}{f\sqrt{\epsilon_{r2}}}[/tex]

    [tex] d = \frac{\lambda_2}{2} = \frac{c}{2f\sqrt{\epsilon_{r2}}}[/tex]

    I have no idea how to get [itex]\epsilon_{r2}[/itex] in terms of [itex]\epsilon_{r1}[/itex] and [itex]\epsilon_{r3}[/itex].

    Here are the book's answers:

    [tex]\epsilon_{r2} = \sqrt{\epsilon_{r1}\epsilon_{r3}}[/tex]


    [tex] d = \frac{c}{4f(\epsilon_{r1}\epsilon_{r3})^{1/4}}[/tex]

    So at least I'm close with the expression for d. (Except, I don't know why they have a 4 in the denominator instead of a 2).

    But how do I find [itex]\epsilon_{r2}[/itex] in terms of [itex]\epsilon_{r1}[/itex] and [itex]\epsilon_{r3}[/itex]??? I can't see a way!

    Thanks for any help!!
     
    Last edited: May 31, 2007
  2. jcsd
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