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Homework Help: Anisotropic dielectric medium

  1. Apr 6, 2006 #1
    we have a non magnetic but anisotropic dielectric medium which has the following relationships between D and E

    Dx = k1*Ex, Dy = k2*Ey, Dz = k3*Ez

    we have to show that waves propogate in the z-dir'n at one speed only.

    I can't get the wave eq'n to fall out. Usually you just use Maxwell's equations and the vector identity
    curl(curl(E)) = grad(divE)- grad squared E
    and set divE = 0. You can usually say this because in a dielectric there is no free charge therefore divD = 0 and in an LIH medium E is proportional to D therefore divE = 0 . Instead now I have to include the grad(divE) expression and the algebra doesn't even resemble the wave eq'n. I'd appreciate any hints on where I've gone wrong.
     
  2. jcsd
  3. Apr 7, 2006 #2

    dextercioby

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    Well, you'll have to see what impact does

    [tex] \left(\begin{array}{c} D_{x}\\D_{y}\\D_{z}\end{array}\right)=\left(\begin{array}{ccc} k_{1} & 0 & 0 \\ 0 & k_{2} & 0\\ 0 & 0 & k_{3} \end{array}\right) \left(\begin{array}{c} E_{x}\\E_{y}\\E_{z}\end{array}\right) [/tex]

    have upon the wave equations. I assume the Permittivity matrix has constant elements, if not, you'd have to be more careful with the calculus.

    Daniel.
     
    Last edited: Apr 7, 2006
  4. Apr 8, 2006 #3

    Meir Achuz

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    If k1=k2=k, then there is a relatively simple solution.
    curl(curl B)=-del^2 B=d_t curl{[k]E}. (d_t is the partial wrt t, etc.)
    For the z component in Cartesian coords:
    curl{[k]E}_z=d_x(k2 E_y)-d_y(k1 E_x)=k (curl E)_x, if k1=k2=k.
    Then, -del^2 B_z=-k(d_t)^2 B_z, and B_z has a wave solution with
    velocity=1/\sqrt{k} (All with c=1)
     
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