What is the Relationship Between Impedance of Dielectric and Material?

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Homework Statement

Part(a): Show relation between impedance of dielectric and impedance of material:

Part (b): Show the ratio between reflected and incident amplitude is:

Part (c): Three layers now from left: vacuum, dielectric 2 sandwiched inbetween, and dielectric 1.

Part (d): Give reasons why dielectric coatings are painted on glass transmission devices.

Homework Equations

The Attempt at a Solution

Part(c)
$$Z_{in} = \frac {E_{(-l)}}{I_{(-l)}} = \frac {E_i e^{jkl} + E_r e^{-jkl} } {E_i e^{jkl} - E_r e^{-jkl} - E_r e^{-jkl} }Z = \frac { (Z_1 + Z_2)e^{jkl} + (Z_1 - Z_2)e^{-jkl} } {(Z_L + Z) e^{jkl} - (Z_L - Z)e^{-jkl} } Z_2$$
$$Z_{in} = \frac {Z_2^2}{Z_1}$$
$$n_2^2 = n_1$$

To calculate ratio:
Consider boundary at z=0 between Z₂ and Z₁:
$$\frac{E_T}{E_I} = \frac {2Z_1}{Z_1 + Z_2} = \frac {2\sqrt{n_1}}{\sqrt{n_1} + n}$$

Consider Boundary at z = -λ/4 between Z₀ and Z₂:
$$\frac{E_T}{E_I} = \frac {2Z_2}{Z_0 + Z_2} = \frac {2}{\sqrt{n} + 1}$$

Then they are the same! Which is strange because there should be a standing wave in the sandwiched layer, which destructively interferes to poduce a smaller transmitted wave onto boundary z>0..

Part(d)
For impedance matching, so that maximum power is transmitted; no reflection.

 

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