Power of Wavefront Penetrating Dielectric Material

[DemoniWaari]

Homework Statement

How large proportion of wavefront's power penetrates dielectric material's surface in a perpendicular collision from air. The only parameter that I have is \varepsilon_r = 16 where \varepsilon_r is the relative permittivity.

Homework Equations

\varepsilon = \varepsilon_r \varepsilon_0
\eta = \sqrt{\frac{\mu}{\varepsilon}} Where \mu = \mu_0 Because of the dielectric material.
\vec E(\vec r) = \vec E_0 e^{-j \vec k \cdot \vec r}
\vec H(\vec r) = \frac{1}{\eta} \vec E_{0p} e^{-j \vec k \cdot \vec r} Where \vec E_{0p} is perpendicular to \vec E_0 and has the same magnitude.
\vec S(\vec r) = \frac{1}{2} \vec E(\vec r) \times \vec H(\vec r)^{*} The complex poyinting vector.

The Attempt at a Solution

\vec E_+(\vec r) = \vec E_0 e^{-j \vec k_1 \cdot \vec r} wavefront in the air.
And
\vec E_-(\vec r) = \vec E_0 e^{-j \vec k_2 \cdot \vec r} wavefront in the material
And \eta_1 = \sqrt{\varepsilon_0 \mu_0} is the wave impedance.
Thus we get
\vec H_+(\vec r) = \frac{1}{\eta_1} \vec E_{0p} e^{-j \vec k_1 \cdot \vec r}
And
\vec H_-(\vec r) = \frac{1}{\eta_2} \vec E_{0p} e^{-j \vec k_2 \cdot \vec r}
Now the complex poyinting vectors are
\vec S_+(\vec r) = \frac{1}{2} \vec E_+(\vec r) \times \vec H_+(\vec r)^{*}
We're not interested in the directions so we can just check the magnitudes thus we get.
\vec S_+(\vec r) = \frac{1}{2\eta_1} |E_0|^2
And same thing for the other wavefront...
\vec S_-(\vec r) = \frac{1}{2\eta_2} |E_0|^2

So the passing fraction is
\frac{ \vec S_-(\vec r)}{\vec S_+(\vec r)} =<br /> \frac{\frac{1}{\eta_2}}{\frac{1}{\eta_1}} = \frac{\eta_1}{\eta_2} =<br /> \frac{\sqrt{\frac{\mu_0}{\varepsilon_0}}}<br /> {\sqrt{ \frac{\mu_0}{\varepsilon_r \varepsilon_0} }} = \sqrt{\epsilon_r}=4

So interestingly output is four times higher than input... So I have a problem here =(

 

[fluidistic]

I suggest to use [itex.] rather than [tex.] or the latex will be displayed on a new line for each expression.
When you close the [itex.], use [/itex.] rather than [\itex.]. This is the reason why it doesn't work.
Without the . inside the []'s.

 

[DemoniWaari]

OH YES!
Thank you kind sir.

 

[rude man]

Penetrated power = incident power minus reflected power.