GATE EC 2018 Electromagnetics Solution

GATE EC 2018 Electromagnetics Solution

A uniform plane wave travelling in free space and having electric field,

\overrightarrow E=(\sqrt2\;{\overrightarrow a}_x-{\overrightarrow a}_z)\;\cos\;\lbrack6\sqrt3\pi\times10^8t-2\pi(x+\sqrt2z)\rbrack\frac Vm

is incident on a dielectric medium (relative permittivity > 1, relative permittivity = 1) as shown in the figure and there is no reflected wave.

GATE EC 2018 Electromagnetics Solution img1 - Grad Plus

The relative permittivity (correct to two decimal places) of the dielectric medium is _.

Ans:- 2

Explanation

Propagation vector is,

\begin{array}{l}\overrightarrow K=2\pi(\widehat x+\sqrt2\widehat z)\\\;\;\;=2\pi\sqrt3\left(\frac1{\sqrt3}\widehat x+\sqrt{\frac23}\widehat z\right)\end{array} \begin{array}{l}\therefore\cos\theta_{ix}=\frac1{\sqrt3}\\So,\;\tan\theta_{ix}=\sqrt2\end{array}

In the question it is stated that, to have no reflected wave, means the incidence angle is the Brewester angle as wave is partially polarized.

For polarization, \;\tan\theta_B=\mu

\therefore\tan\theta_B=\mu=\sqrt{\frac{\varepsilon_r\varepsilon_0}{\varepsilon_0}}=\sqrt{\varepsilon_r} \therefore\varepsilon_r=2

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