Dielectric interface in a waveguide

[radonballoon]

Homework Statement

Consider a wave guide with a square cross section of dimensions a x a. Let the z axis be the axis of the wave guide. Suppose the region z < 0 is vacuum, and the region z >0 is a dielectric with permittivity [tex] \epsilon [/tex]. Write a solution of the wave equations and boundary conditions such that there is an incident and reflected wave for z < 0 and a transmitted wave fro z > 0. All three waves are TE(1,0) waves. Determine the transmitted power as a fraction of the incident power. [Answer: [tex] S_{trans}/S_{inc} = 4kk^{'}/(k+k^{'})^2[/tex]]

Homework Equations

The Attempt at a Solution

So I wrote the electric field as a superposition of the TE electric field for z < 0:
[tex]\vec{E} = \left[-\hat{y}\frac{\pi}{a}sin\left(\frac{\pi x}{a}\right)\right][\Psi_0e^{i(kz-\omega t)}+\Psi_0^{''}e^{-i(kz+\omega t)}][/tex]
Applying boundary condition that the parallel components of E must be continuous on the boundary at z=0, I got that [tex]\Psi_0^{'} = \Psi_0 + \Psi_0^{''} [/tex]
Where the double primed one is the reflected wave, and the single prime is transmitted. This, however, is the only equation I can get by applying boundary conditions. Any thoughts? Also I get a really messy equation if I take the real part of E and B and cross them, nothing like the simple answer they give me.

 

[Jhero]

Thank you for your interesting question. I am always excited to see people exploring new concepts and pushing the boundaries of our knowledge. I have taken a look at your attempt at a solution and I believe I can offer some guidance to help you reach the answer provided.

Firstly, your expression for the electric field is correct for the incident and reflected waves, but it is missing the transmitted wave. The transmitted wave should have a similar form as the incident wave, but with a different amplitude, which we can call \Psi_0^{''}. This amplitude is related to \Psi_0 by a transmission coefficient, which we can call T, such that \Psi_0^{''} = T\Psi_0.

Now, to apply the boundary condition at z=0, we need to consider both the electric field and the magnetic field. The electric field boundary condition you have already correctly applied, but we also need to consider the magnetic field. Applying the same logic, we can write the magnetic field as a superposition of the TE magnetic field for z < 0:
\vec{B} = \left[\hat{x}\frac{\pi}{a}cos\left(\frac{\pi x}{a}\right)\right][\Psi_0e^{i(kz-\omega t)}+\Psi_0^{''}e^{-i(kz+\omega t)}]
And applying the boundary condition that the parallel components of B must be continuous on the boundary at z=0, we get \Psi_0^{'} = \Psi_0 - \Psi_0^{''}.

Now we have two equations for \Psi_0^{'} and \Psi_0^{''}, which we can solve simultaneously to get the transmission coefficient T. Once we have T, we can easily calculate the transmitted power as a fraction of the incident power using the formula provided in the question.

I hope this helps you in finding the correct solution. Keep up the good work and don't be afraid to ask for help when needed.