TEM mode propagation between 2 infinite conductor plates

[fluidistic]

Homework Statement

I am trying to solve a problem from Jackson's book (in chapter 8). I must describe the propagation of a TEM mode through a transmission line that consists of two infinite conductor plates that are parallel to each other and separated by a distance a. There's a dielectric between the 2 plates.

At first I did not know how to start the problem. Maybe solve the wave equation with boundary conditions that $\vec E$ must be perpendicular to the conductor surfaces and $\vec B$ must be parallel to it and both fields should have no component in the z direction, the direction of propagation. On the top of this, since it's a TEM, Jackson's demonstrate that in a hollow cylinder the fact that $E_z$ and $B_z$ vanish implies that these fields satisfy electrostatics Maxwell equations and I believe it is also the case for the exercise I'm dealing with. This is also confirmed by page 14 of http://www.ece.msstate.edu/~donohoe/ece4333notes3.pdf for the particular set up I'm facing in this problem.

Homework Equations

Not sure...

The Attempt at a Solution

Following the document linked above I reach that $\vec E=-\frac{V}{d}e^{-ikz}\hat y$ and $\vec H=\sqrt{\frac{\varepsilon}{\mu}}\frac{V}{a}e^{-ikz}\hat x$.
So I see no propagation, no time dependence... However in the document I linked above one read

The current directions in the two plates correspond to the direction of propagation assumed for the waves.

which confuses me even more. How come there is a current in the plates? They seemed to act like a capacitor to me and as far as I know there is no current in the plates... What I am missing?

I've also found another solution to the problem but it did not convince me since the guy assumes time dependence of the fields and for some unknown reason to me, he assumes from the start that the current in the plates if of the form $\vec K=(z,t)=K_0\mu e^{i(kz-\omega t)}\hat x$. See http://www-personal.umich.edu/~pran/jackson/P506/P506W02HW02.pdf.

So which answer is correct? Thank you.

 

[fluidistic]

Still stuck at the very same point in this problem. I don't know which solution is correct and whether there is some time dependence on the fields.