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In the solution of Maxwell equations inside conductors,

$J=\sigma E$ is used.

In the derivation of e.g phase difference, and skin depth, $ \sigma$ is assumed to be real, e.g

$e^{-b\cdot r}$ is the term responsible for the wave attenuation,

and Tan-1(|b||a|) is used to determine the phase difference between the magnetic and electric components

Now, Ohms law breaks down for high frequencies, where $ \sigma$ is real, thus the need to replace it with the complex conductivity if E has a $e^{-i\omega t}$ dependance

So, because B is dependant on $\sigma$ doesn't this mean that the normal skin depth formula (1/|b|) is actually wrong for high frequencies. B is dependant on conductivity and conductivity is complex, then b is complex therefore you need to need to break it down into real and imaginary components b= c +d i Where c,d are vectors

The term responsible for attenuation is $e^{-b\cdot r}$

however since now b is complex, to get the values for the real wave instead of in complex notation we must take the real part of it

re{ $e^{-b\cdot r}$} =re{ $e^{-(c+di)\cdot r}$}

re{$e^{-c\cdot r}$ $e^{i(-d\cdot r)}$}

= $e^{-c\cdot r}$ $Cos(-d\cdot r)$

In this form we obviously get a slightly more complex function where the skin depth is actually 1/|a| where a is the real part of B and a new $cos(-k\cdot r)$ term that makes the function even more complex

Is this thinking correct? that the standard treatment of conductivity being real is wrong in the context of wave attenuation in the context of extremely high frequencies.

the term $e^{i(-d\cdot r)}$

also has an impact of the wave speed formula

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