Dispersion Relation for EM Waves: Skin Depths in Good and Poor Conductors

Thread starter CAF123
Start date Nov 20, 2013
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Em Em waves Waves

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The discussion focuses on deriving the dispersion relation for electromagnetic (EM) waves and understanding skin depths in conductors. The participants explore the expansion of the wave vector k for both good and poor conductors, noting that the imaginary part of k is essential for calculating skin depth. There is a consensus that for good conductors, the term involving permittivity can often be neglected, while for poor conductors, it must be included. Additionally, the phase relationship between the electric and magnetic fields is examined, revealing a π/4 lag in good conductors. The conversation emphasizes the importance of correctly handling complex numbers in these calculations to derive accurate results.

Nov 22, 2013

vanhees71

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rude man said:

This equation indicates B leading E by 45 deg. which is incorrect unless you're making an assumption that I can't fathom. B should lag by 45 deg. I, vanhees & the problem's statement all agree on this. So your error must have taken place ahead of this relation unless, again, somewhere there's an inconsistency regarding convention.

That's all correct. If you use my convention of the signs, i.e.,
\vec{B} \propto \exp(-\mathrm{i} \omega t).
Your case is for a good conductor, i.e., \sigma \mu \omega \gg \epsilon \mu \omega^2 or
\frac{\sigma}{\epsilon} \gg 1.
Then you have
k \simeq \sqrt{\frac{\sigma \mu}{\omega}} \exp(+\mathrm{i} \pi/4).
That means
B_0 \exp(-\mathrm{i} \omega t)=\sqrt{\frac{\sigma \mu}{\omega}} E_0 \exp[-\mathrm{i}(\omega t-\pi/4)].
This means the phase of the B field is behind that of the E field by an amount of \pi/4 (in the limit of good conductivity).