E&M Question: Total Transmission

[WolfOfTheSteps]

Hello Physics Forum people! :)

Homework Statement

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

Homework Equations

1/2 Wavelength matching equation:

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

Wavelength/permitivity relation:

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

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 $\epsilon_{r3}$.

Here are the book's answers:

$$\epsilon_{r2} = \sqrt{\epsilon_{r1}\epsilon_{r3}}$$$$d = \frac{c}{4f(\epsilon_{r1}\epsilon_{r3})^{1/4}}$$

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!

 

[KataKoniK]

Thank you for your question. It is great to see so many dedicated scientists on this forum!

In order to find the combination of \epsilon_{r2} and d that produces no reflection, we must first understand the conditions for no reflection to occur. In this case, we are dealing with a wave incident normally on the boundary at z = -d, which means that the angle of incidence is 0 degrees.

To have no reflection, the impedance of the two media must match, which means that their characteristic impedances must be equal. Using the 1/2 wavelength matching equation, we can write:

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

where Z_{in} is the input impedance of medium 1 and Z_L is the characteristic impedance of medium 2.

Now, let's express Z_{in} and Z_L in terms of the respective characteristic impedances of the three media (Z_{in1}, Z_{in2}, and Z_L3). We can write:

Z_{in1} = \sqrt{\frac{\mu_0}{\epsilon_{r1}}} = \frac{\sqrt{\mu_0}}{\sqrt{\epsilon_{r1}}}

Z_{in2} = \sqrt{\frac{\mu_0}{\epsilon_{r2}}} = \frac{\sqrt{\mu_0}}{\sqrt{\epsilon_{r2}}}

Z_L3 = \sqrt{\frac{\mu_0}{\epsilon_{r3}}} = \frac{\sqrt{\mu_0}}{\sqrt{\epsilon_{r3}}}

Since the impedance must match, we can set Z_{in1} equal to Z_L3:

\frac{\sqrt{\mu_0}}{\sqrt{\epsilon_{r1}}} = \frac{\sqrt{\mu_0}}{\sqrt{\epsilon_{r3}}}

Solving for \epsilon_{r3}, we get:

\epsilon_{r3} = \frac{\epsilon_{r1}}{\epsilon_{r2}}

Substituting this into the expression for Z_L3, we get:

Z_L3 = \frac{\sqrt{\mu_0}}{\sqrt{\epsilon_{r1}\epsilon_{r2}}}

Now, using the 1/2 wavelength matching equation, we can write:

Z_{in1} = Z

 

[piercas]

Hello Physics Forum people! It looks like you're working on a problem related to total transmission in a wave with perfect dielectrics. This is a common topic in electromagnetism and can be solved using the equations for wavelength and permitivity relations.

You have correctly calculated the wavelength in medium 2 and the distance d in terms of the oscillation frequency and \epsilon_{r2}. However, to find \epsilon_{r2} in terms of \epsilon_{r1} and \epsilon_{r3}, you can use the boundary conditions for total transmission. These conditions state that the impedance of medium 2 must equal the impedance of medium 1 in order to have no reflection at the boundary. Using this condition, you can set the impedance of medium 2 (calculated using \epsilon_{r2}) equal to the impedance of medium 1 (calculated using \epsilon_{r1}).

Solving for \epsilon_{r2} in terms of \epsilon_{r1} and \epsilon_{r3} will give you the desired expression. Remember to also consider the boundary conditions for total transmission at the other boundary (z = 0) to get the correct value for d.

I hope this helps and good luck with your problem!