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As I understand it a perfect electrical conductor (PEC) will expel all electric fields and time-varying magnetic fields. For this post I'm assuming low enough frequencies as to ignore displacement current and wave phenomenon. As a result the boundary conditions for the magnetic field become:

$$\hat n \times (H_1 - H_2) = \hat n \times H = K$$

$$\hat n \cdot (\mu_0 H_1 - \mu_0 H_2) = \hat n \cdot \mu_0 H = 0$$

I understand these to mean that there is no normal component to the magnetic field at the surface and that the surface current density $K$ is equal $H$. Does this also imply what the phase of $K$ should be?

Based on this reference I am going to say that $H_i$ is a field which already exists in the the environment before the PEC was introduced, and that no current sources are present locally. And, that $H_e$ is the field that is produced once a PEC is introduced. The total field is $H=H_i+H_e$. In this case $H_e$ is entirely responsible for enforcing the boundary condition of expelling $H_i$ from the material and that $K$ is the cause of $H_e$. Naively, without doing any math, I would have assumed that the phase of $H_i$, $H_e$ and $K$ should be the same.

But I also would have assumed, like this wiki artical on Eddy currents, that Faraday's law of induction ($\nabla\times E_e = -\partial B/ \partial t$) would describe the process by which $K$ was generated. Unless the spatial derivative is having an affect that I don't understand, then it seems that there'd be a phase difference between $B$ and $K$ of 90°.

In the reference listed earlier, which was specifically written for finite conductivity, you can see that the phase of the current near the surface is 45°. I reproduced those plots with a significantly higher conductivity (see figure below) and the phase remains 45°.

Current density in slab

What I am confused about is that if $K$ and $B$ are not in phase, then wouldn't there be times when $B$ would dip into the conductor violating the boundary condition? And since clearly I am wrong, why is it 45° and how does that not violate the boundary conditions?

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  • $\begingroup$ This might be drivel, I think there is some confusion between impressed and induced currents. I think an induced current is one which was generated by an electric field within the material, which was itself generated by an alternating magnetic field. In a PEC that electric field cannot exist. That means that the surface current that was impressed (?) by the field was responsible for repelling the magnetic field and is in phase with the impressed magnetic field. I'm confused how this works in either the MQS case, or when the conductivity is very high, but not infinite. $\endgroup$ – Jay Mar 20 at 12:05

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