# E&M Question: Total Transmission

1. May 31, 2007

### WolfOfTheSteps

Hello Physics Forum people! :)

1. The problem statement, all variables and given/known data

The three regions in the figure below contain perfect dialectrics. For a wave in medium 1, incident normally upon the boundary at z = -d, what combination of $\epsilon_{r2}$ and $d$ produces no reflection? Express your answers in terms of $\epsilon_{r1}$, $\epsilon_{r3}$, and the oscillation frequency of the wave, $f$.

http://img45.imageshack.us/img45/1571/problem89gi6.th.jpg [Broken]

2. Relevant equations

1/2 Wavelength matching equation:

$$Z_{in} = Z_L, \ \ \ \ \ \ \ \ \mbox{for } l = n\lambda/2$$

Wavelength/permitivity relation:

$$\lambda = \frac{c}{f\sqrt{\epsilon_{r}}}$$

3. The attempt at a solution

I used the above equations to get

$$\lambda_2 = \frac{c}{f\sqrt{\epsilon_{r2}}}$$

$$d = \frac{\lambda_2}{2} = \frac{c}{2f\sqrt{\epsilon_{r2}}}$$

I have no idea how to get $\epsilon_{r2}$ in terms of $\epsilon_{r1}$ and $\epsilon_{r3}$.

$$\epsilon_{r2} = \sqrt{\epsilon_{r1}\epsilon_{r3}}$$

$$d = \frac{c}{4f(\epsilon_{r1}\epsilon_{r3})^{1/4}}$$

So at least I'm close with the expression for d. (Except, I don't know why they have a 4 in the denominator instead of a 2).

But how do I find $\epsilon_{r2}$ in terms of $\epsilon_{r1}$ and $\epsilon_{r3}$??? I can't see a way!

Thanks for any help!!

Last edited by a moderator: May 2, 2017