Surface Current and Electric Field

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BnayaMeir
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Hi everyone!

I'm pretty new in this forum, I found the topics here very relevant to my physics course. And here is my question:

Given the following drawing, two infinite sheets (in y and z axis) of ideal conductive material. their thickness is infinitesimal (dx->0).

Diagram of two parallel vertical lines in the x-y plane, labelled x = -a/2 and x = a/2, with dash-dot horizontal x-axis and vertical y-axis through the origin.


The electric field is defined:

Piecewise electric field equation: E-vector equals E0 cos(omega t minus beta z) cos(pi x / a) y-hat for |x| <= a/2, and zero for |x| > a/2.

I have askes to find the surface charge and current density.

well.. I tried to apply the integral gauss law
Handwritten-style equation: the closed surface integral of D dot dA equals the volume integral of rho dV, Gauss's law in integral form.
but in 2 dimensions (didn't work).
I have also tries the derivative version of the law
Equation div D = rho, written as V · D⃗ = ρ, showing the divergence of the electric displacement field equals the charge density.
which gave me zero. it looks right for me, since the electric field should be perpendicular to the surface. but the only electric field is parallel to the surfaces. Then where this field is come from?? I'm very confused..

I thought to find the current density after finding the charge density by the following equation in 2-D:
Handwritten-style equation for charge conservation: the closed surface integral over A of J dot dA plus the time derivative of the volume integral over V of rho dV equals zero.


I hope you will be able to help me, thanks a lot!
 
Have you considered that this could be the electric field of a propagating electromagnetic wave?