I have to thinking about how the skin effect come into play in transmission lines with two conductors and dielectric in the middle. For normal skin effect, we consider a potential across a conductor as shown: Where the tangential E right above and right below the surface at z=0, are equal. The E will cause J0 to flow as show in orange arrows. The J decrease as it goes deeper into the +z direction as E attenuates with depth. But for a parallel plate transmission line as shown: As TEM wave travel from left to right, E is in y direction and H in x direction. Boundary condition produce Js as shown in purple color at the surface of the conductors by H, and ρ in red from E. We know we need to consider skin effect in the two plates. How do I calculate the skin effect? The E is normal to the surface of the conductor, but current flow is parallel to the conductor plates. I have no question the E attenuate as it penetrates normal to the conductor plates, BUT the current flow is parallel to the plates. Please help me understand this. Thanks
Also I don't understand all the Maxwell's eq talked about surface current and surface charge. But we know that there is skin effect that current do penetrates into the metal, not just on the surface.
The diagram of the parallel plate transmission line is a more detail drawing from Field and Wave Electromagnetics by David Cheng. The charge is from boundary condition [itex] \nabla \cdot \vec D = \rho_v [/itex] and surface current from [itex]\nabla X\vec H=\vec J[/itex]. But when comes to current derivation, the book only use the J component, there is no mention the contribution of the ρ from the E!!! How is the ρ plays in the transmission line that transform from EM wave to V and I?