GATE 2018 EC – Question 59
A uniform plane wave traveling in free space and having the electric field
$$\vec E=(\sqrt2\hat a_x-\hat a_z)\cos\left[6\sqrt3\pi\times10^8t-2\pi(x+\sqrt2z)\right]\ \text{V/m}$$
is incident on a dielectric medium (relative permittivity $>1$, relative permeability $=1$) as shown in the figure below and there is no reflected wave.
The relative permittivity (correct to two decimal places) of the dielectric medium is ________.
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Correct answer: 2
Explanation
The phase has $k_x=2\pi$ and $k_z=2\pi\sqrt2$, so the incidence angle from the interface normal ($x$ axis) satisfies $\tan\theta_i=\frac{k_z}{k_x}=\sqrt2$. (Check: $|k|=2\pi\sqrt3=\frac{\omega}{c}$.) The field is in the plane of incidence, so there is no reflection only at the Brewster angle, where $\tan\theta_B=\sqrt{\varepsilon_r}$. So $\sqrt{\varepsilon_r}=\sqrt2$ and $\varepsilon_r=2$.