Construction of an equivalent dielectric tensor

[billybomb87]

Homework Statement

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

$$\textbf{D} = \varepsilon (\omega )\textbf{E}, \textbf{H} = {\mu ^{ - 1}}(\omega )\textbf{B}.$$

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

Homework Equations

$$\textbf{D} = \varepsilon (\omega )\textbf{E}$$

$$\textbf{H} = {\mu ^{ - 1}}(\omega )\textbf{B}$$

$$\textbf{P} = {\varepsilon _0}{\chi ^e}\textbf{E}$$

$$\textbf{M} = {\chi ^m}\textbf{B}/{\mu _0}$$

$${P_i} = {\varepsilon _0}\chi _{ij}^e{E_j}$$

$${M_i} = \chi _{ij}^m{B_j}/{\mu _0}$$

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.

 

[sith]

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:

$$K_{i,j}(\omega, \textbf{k}) = \delta_{ij} + \frac{i}{\omega \epsilon_0}\sigma_{ij}(\omega, \textbf{k})$$

$$\left(\textbf{J}_{ind}\right)_i(\omega, \textbf{k}) = \sigma_{ij}(\omega, \textbf{k})E_j(\omega, \textbf{k})$$

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

$$P_i(\omega, \textbf{k}) = D_i(\omega, \textbf{k}) - \epsilon_0 E_i(\omega, \textbf{k})$$

$$D_i(\omega, \textbf{k}) = \epsilon(\omega)E_i(\omega, \textbf{k})$$

$$M_i(\omega, \textbf{k}) = \frac{1}{\mu_0}B_i(\omega, \textbf{k}) - H_i(\omega, \textbf{k})$$

$$H_i(\omega, \textbf{k}) = \mu^{-1}(\omega)B_i(\omega, \textbf{k})$$

$$B_i(\omega, \textbf{k}) = \frac{1}{\omega}\epsilon_{ijk}k_j E_k(\omega, \textbf{k})$$

Then I guess the rest is straightforward. Good luck!

 

[sith]

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. :)