GATE EC 2005 Set 1 Q30 — Plane Waves and Properties

The electric field of a monochromatic plane wave travelling in a lossless isotropic and homogenous medium is given by $$\vec{E}(z, t)=E_{0}[\hat{x} \cos (\omega t-k z)+\hat{y} \sin (\omega t-k z)]$$ in a right-handed orthogonal co-ordinate system. Which of the following is the correct polarization of the electromagnetic wave?

A uniform plane wave with electric field $\vec{E}(x)=A_{y} \hat{a}_{y} e^{-j \frac{2 \pi x}{3}} \mathrm{~V} / \mathrm{m}$ is travelling in the air (relative permittivity, $\epsilon_{r}=1$ and relative permeability, $\mu_{r}=1$ ) in the $+x$ direction ( $A_{y}$ is a positive constant, $\hat{a}_{y}$ is the unit vector along the $y$ axis). It is …

Consider a narrow band signal, propagating in a lossless dielectric medium $\left(\varepsilon_{r}=4, \mu_{r}=1\right)$, with phase velocity $v_{p}$ and group velocity $v_{g}$. Which of the following statement is true? ($c$ is the velocity of light in vacuum.)

The electric field of a plane electromagnetic wave is $\boldsymbol{E}=\boldsymbol{a}_{x} C_{1 x} \cos (\omega t-\beta z)+\boldsymbol{a}_{y} C_{1 y} \cos (\omega t-\beta z+\theta) \quad \mathrm{V} / \mathrm{m}$. Which of the following combination(s) will give rise to a left handed elliptically polarized $\text{(LHEP)}$ wave?

A transparent dielectric coating is applied to glass $\left(\varepsilon_r=4, \mu_r=1\right)$ to eliminate the reflection of red light $\left(\lambda_0=0.75 \; \mu \mathrm{m}\right)$. The minimum thickness of the dielectric coating, in $\mu \mathrm{m}$, that can be used is_____________(rounded off to two decimal places).

Consider a rectangular coordinate system $(x,y,z)$ with unit vectors $a_{x}\:a_{y}$ and $a_{z}$. A plane wave traveling in the region $z\geq 0$ with electric field vector $E=10\cos\left ( 2\times 10^{8}t\:+\:\beta z\right )a_{y}$ is incident normally on the plane at $z=0$, where $\beta$ is the phase constant. The region $z\geq 0$ is in free …