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°.
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?