I'm having difficulties solving boundary conditions for an infinitely thin conducting layer in a presence of an alternating field.
I use the Maxwell equations:
$\nabla \cdot \mathbf B = 0$
$\nabla \cdot \mathbf D = \rho$
$\nabla \times \mathbf E = i \omega \mathbf B$
$\nabla \times \mathbf H = \mathbf j - i \omega \mathbf D$
The problem arises with the last two ones. For stationary fields the terms $i \omega \mathbf B$ and $i \omega \mu \mathbf D$ vanish, and I can use the "textbook" boundary conditions like: $\hat n \times (\mathbf H_1 - \mathbf H_2)=\mathbf j$
Do those equations get an additional term $-i \omega \mathbf D$ in the case of time-varying fields?
If they do, those terms are different above and below the sheet, and I am not sure what value should be used.
The current on the sheet $\mathbf j = \sigma \mathbf E$ turns out to be discontinued as well.
What field (current) should be used as the field on the very boundary?
I would assume it's the average of the fields directly above and below the boundary, but I'm mostly guessing and don't know an exact proof. I am deeply grateful for any help.