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

Thread starter CAF123
Start date Nov 20, 2013
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SUMMARY

The discussion centers on the dispersion relation for electromagnetic (EM) waves in conductors, specifically the skin depth in good and poor conductors. The derived expression for the wave vector is given as $$\hat{k} = (\mu \epsilon w^2 + i\mu \sigma w)^{1/2}$$. Participants clarify that skin depth, defined as δ = 1/(real part of k), varies with conductivity, frequency, and permeability. The phase difference between electric and magnetic fields in good conductors is confirmed to be π/4, indicating that the magnetic field lags the electric field.

PREREQUISITES

Understanding of electromagnetic wave propagation
Familiarity with complex numbers and their applications in physics
Knowledge of skin depth concepts in conductive materials
Proficiency in binomial expansion and Taylor series

NEXT STEPS

Study the derivation of the complex intrinsic impedance in conducting media
Learn about the implications of skin depth in various materials
Explore the relationship between conductivity, frequency, and skin depth
Investigate the phase relationships in electromagnetic fields in conductors

USEFUL FOR

Physicists, electrical engineers, and students studying electromagnetic theory, particularly those focusing on wave propagation in conductive materials.

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).