1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Construction of an equivalent dielectric tensor

  1. Feb 10, 2010 #1
    1. The problem statement, all variables and given/known data

    A medium is described by the response functions [tex]\varepsilon (\omega )[/tex] and [tex]{\mu ^{ - 1}}(\omega )[/tex] in

    [tex]\textbf{D} = \varepsilon (\omega )\textbf{E}, \textbf{H} = {\mu ^{ - 1}}(\omega )\textbf{B}.[/tex]

    Construct the equivalent dielectric tensor [tex]{K_{ij}}(\omega ,\textbf{k})[/tex] in terms of [tex]\varepsilon (\omega )[/tex] and [tex]{\mu ^{ - 1}}(\omega )[/tex]

    2. Relevant equations

    [tex]\textbf{D} = \varepsilon (\omega )\textbf{E}[/tex]

    [tex]\textbf{H} = {\mu ^{ - 1}}(\omega )\textbf{B}[/tex]

    [tex]\textbf{P} = {\varepsilon _0}{\chi ^e}\textbf{E}[/tex]

    [tex]\textbf{M} = {\chi ^m}\textbf{B}/{\mu _0}[/tex]

    [tex]{P_i} = {\varepsilon _0}\chi _{ij}^e{E_j}[/tex]

    [tex]{M_i} = \chi _{ij}^m{B_j}/{\mu _0}[/tex]

    3. The attempt at a solution

    I seriously have no idea. I know what the answer should be but I only need a push in the right direction. Dont know where to start or how to attack the problem. All help is appreciated.
  2. jcsd
  3. Feb 13, 2010 #2
    Hi! I actually did this one just a moment ago. I guess you are also solving exercise 6.1 in Melrose, McPhedran's book "Electromagnetic processes in dispersive media". :) You should use the following equations:

    K_{i,j}(\omega, \textbf{k}) = \delta_{ij} + \frac{i}{\omega \epsilon_0}\sigma_{ij}(\omega, \textbf{k})[/tex]

    \left(\textbf{J}_{ind}\right)_i(\omega, \textbf{k}) = \sigma_{ij}(\omega, \textbf{k})E_j(\omega, \textbf{k})[/tex]

    \left(\textbf{J}_{ind}\right)_i(\omega, \textbf{k}) = -i\omega P_i(\omega, \textbf{k}) + i \epsilon_{ijk}k_j M_k(\omega, \textbf{k})[/tex]

    P_i(\omega, \textbf{k}) = D_i(\omega, \textbf{k}) - \epsilon_0 E_i(\omega, \textbf{k})[/tex]

    D_i(\omega, \textbf{k}) = \epsilon(\omega)E_i(\omega, \textbf{k})[/tex]

    M_i(\omega, \textbf{k}) = \frac{1}{\mu_0}B_i(\omega, \textbf{k}) - H_i(\omega, \textbf{k})[/tex]

    H_i(\omega, \textbf{k}) = \mu^{-1}(\omega)B_i(\omega, \textbf{k})[/tex]

    B_i(\omega, \textbf{k}) = \frac{1}{\omega}\epsilon_{ijk}k_j E_k(\omega, \textbf{k})[/tex]

    Then I guess the rest is straightforward. Good luck!
  4. Feb 13, 2010 #3
    Btw, is it possible that you also take T. Hellsten's course at KTH and have this exercise as a deadline until next thursday? Just wondering. :)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Construction of an equivalent dielectric tensor
  1. Construct a launcher (Replies: 4)

  2. ROtate construction (Replies: 0)

  3. Dielectric Constants (Replies: 1)

  4. Construction drawing (Replies: 2)