# Uniform Plane Waves

 Question 1
The magnetic field of a uniform plane wave in vacuum is given by

$\vec{H}(x,y,z,t)=(\hat{a}_x+2\hat{a}_y+b\hat{a}_z) \cos (\omega t+3x-y-z)$

The value of b is _____
 A 0 B 1 C -1 D -2
GATE EC 2020   Electromagnetics
Question 1 Explanation:
For uniform plane wave
$\hat{a}_{H}\cdot \hat{a}_{\rho }=0$
$\hat{a}_{H}$is unit vector in magnetic field direction $\hat{a}_{\rho }$ is unit vector in power flow direction
$\hat{a}_{H }=\frac{1\hat{a}_{x}+2\hat{a}_{y}+b\hat{a}_{z}}{\sqrt{1^{2}+2^{2}+b^{2}}}$
$\hat{a}_{\rho }=\frac{-3\hat{a}_{x}+\hat{a}_{y}+\hat{a}_{z}}{\sqrt{3^{2}+1^{2}+1^{2}}}$
$\hat{a}_{H}\cdot \hat{a}_{\rho }=0 (\hat{a}_{x}+2\hat{a}_{y}+b\hat{a}_{z})\cdot (-3\hat{a}_{x}+\hat{a}_{y}+\hat{a}_{z})=0$
$-3+2+b=0$
$b=1$
 Question 2
A uniform plane wave traveling in free space and having the electric field
$\vec{E}=(\sqrt{2}\hat{a}_{x}-\hat{a}_{z})cos[6\sqrt{3}\pi \times 10^{8}t-2\pi (x+\sqrt{2}z)]V/m$
is incident on a dielectric medium (relative permittivity $\gt$1, relative permeability = 1) as shown in the figure and there is no reflected wave.

The relative permittivity (correct to two decimal places) of the dielectric medium is___________.
 A 2 B 2.9 C 1 D 3
GATE EC 2018   Electromagnetics
Question 2 Explanation:
The wave has $E_{i}$ direction and propagation direction both in same plane (ZX).
The wave is plane of incidence (P) polarized.
\begin{aligned} \beta_{x} &=1, \quad \beta_{2}=\sqrt{2} \\ \tan \theta_{i} &=\frac{\beta_{z}}{\beta_{x}}=\sqrt{2} \end{aligned}
No reflection in P polarized wave under Brewster's angle of incidence.
\begin{aligned} \theta_{i} &=\theta_{B} \\ \tan \theta_{B} &=\sqrt{2}=\sqrt{\epsilon_{r}} \\ \epsilon_{r} &=2 \end{aligned}
 Question 3
The distance (in meters) a wave has to propagate in a medium having a skin depth of 0.1 m so that the amplitude of the wave attenuates by 20 dB, is
 A 0.12 B 0.23 C 0.46 D 2.3
GATE EC 2018   Electromagnetics
Question 3 Explanation:
Attenuation constant,
\begin{aligned} \alpha &=\frac{1}{\text { skin depth }}=10 \mathrm{Np} / \mathrm{m} \\ \text { 20log }_{10}\left(\frac{E_{0}}{E_{x}}\right) &=20 \mathrm{dB} \\ \frac{E_{o}}{E_{x}} &=10 \Rightarrow E_{x}=\frac{E_{0}}{10} \\ E_{x} &=E_{0} e^{-\alpha x}=E_{0} e^{-10 x}=\frac{E_{0}}{10} \\ e^{-10 x} &=\frac{1}{10} \\ x &=\frac{1}{10} \ln (10)=0.23 \mathrm{m} \end{aligned}
 Question 4
The permittivity of water at optical frequencies is 1.75 $\varepsilon_{0}$. It is found that an isotropic light source at a distance d under water forms an illuminated circular area of radius 5m, as shown in the figure. The critical angle is $\theta_{c}$.

The value of d (in meter) is _____________
 A 1.3 B 2.8 C 4.3 D 5.8
GATE EC 2017-SET-2   Electromagnetics
Question 4 Explanation:
Critical angle,
\begin{aligned} \theta_{c} &=\sin ^{-1} \sqrt{\frac{\epsilon_{2}}{\epsilon_{1}}} \\ &=\sin ^{-1} \sqrt{\frac{\epsilon_{0}}{1.75 \epsilon_{0}}}=49.1^{\circ} \end{aligned}

\begin{aligned} \tan 49.1 &=1.547=\frac{5}{d} \\ \Rightarrow \quad d &=\frac{5}{1.1547}=4.33 \mathrm{m} \end{aligned}
 Question 5
The expression for an electric field in free space is $E=E_{0}=(\hat{x}+\hat{y}+j2\hat{z})e^{-j(\omega t-kx+ky)}$, where x, y, z represent the spatial coordinates, t represents time, and $\omega$, k are constants. This electric field
 A does not represent a plane wave B represents a circular polarized plane wave propagating normal to the z-axis C represents an elliptically polarized plane wave propagating along x-y plane. D represents a linearly polarized plane wave
GATE EC 2017-SET-1   Electromagnetics
Question 5 Explanation:
$\vec{E}$ field direction $\Rightarrow\left(\hat{a}_{x}+\hat{a}_{y}+j 2 \hat{a}_{z}\right)$
Propagation direction $\Rightarrow k \hat{a}_{x}-k \hat{a}_{y}$
$\vec{E}$ is perpendicular to propagation
$\vec{E} \cdot \vec{P}=0$
Component in $\hat{a}_{z}$ has magnitude of 2.
Component in X-Y plane has magnitude of $\sqrt{2}.$
These two components are out of phase by $90^{\circ}$ and have unequal amplitudes. So, it is elliptically polarized wave.

 Question 6
Faraday's law of electromagnetic induction is mathematically described by which one of the following equations?
 A $\bigtriangledown \cdot \vec{B}=0$ B $\bigtriangledown \cdot \vec{D}=\rho _{v}$ C $\bigtriangledown \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$ D $\bigtriangledown \times \vec{H}=\sigma \vec{E}+\frac{\partial \vec{D}}{\partial t}$
GATE EC 2016-SET-3   Electromagnetics
Question 6 Explanation:
Rate of change of magnetic field results in induced voltage,
$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$
 Question 7
If a right-handed circularly polarized wave is incident normally on a plane perfect conductor, then the reflected wave will be
 A right-handed circularly polarized B left-handed circularly polarized C elliptically polarized with a tilt angle of $45^{\circ}$ D horizontally polarized
GATE EC 2016-SET-3   Electromagnetics
Question 7 Explanation:
Left circularly polarized, due to direction change after the reflection from conductor.
 Question 8
Let the electric field vector of a plane electromagnetic wave propagating in a homogenous medium be expressed as $E=\hat{x}E_{x}e^{-j(\omega t-\beta z)}$, where the propagation constant $\beta$ is a function of the angular frequency $\omega$. Assume that $\beta (\omega)$ and $E_{x}$ are known and are real. From the information available, which one of the following CANNOT be determined?
 A The type of polarization of the wave. B The group velocity of the wave C The phase velocity of the wave. D The power flux through the z = 0 plane.
GATE EC 2016-SET-2   Electromagnetics
Question 8 Explanation:
$\beta(\omega)$ is known so $V_{g}$ can be calculated.
$v_{p}=\omega / \beta$ can be calculated.
Polarization can be identified.
$\mu_{r}$ and $\epsilon_{r}$ cannot be found, due to which power flux cannot be calculated as power flux
$P=\frac{1}{2} \frac{|E|^{2}}{\eta}, \text{ where }\eta=120 \pi \times \sqrt{\frac{\mu_{r}}{\epsilon_{r}}}$
 Question 9
The electric field of a plane wave propagating in a lossless non-magnetic medium is given by the following expression
$E(z,t)=a_{x}5 cos(2\pi \times 10^{9}t+\beta z)+$ $a_{y} 3 cos(2\pi \times 10^{9}t+\beta z-\pi/2)$
The type of the polarization is
 A Right Hand Circular. B Left Hand Elliptical C Right Hand Elliptical D Linear.
GATE EC 2015-SET-2   Electromagnetics
Question 9 Explanation:
$\vec{E}(z, t)=\hat{a}_{x} 5 \cos \left(2 \pi \times 10^{9} t+\beta z\right)$
$+\hat{a}_{y} 3 \cos \left(2 \pi \times 10^{9} t+\beta z-\frac{\pi}{2}\right)$
$\Rightarrow$ Wave is travelling in $-\hat{a}_{z}$ direction.
$\Rightarrow$ Wave has orthogonal components with unequal amplitudes and checking the time trace.
$\therefore$ Wave is left hand elliptically polarized.
 Question 10
The electric field of a uniform plane electromagnetic wave is
$\vec{E}=(\vec{a}_{x}+j4\vec{a}_{y})exp[j(2\pi \times 10^{7}t-0.2z)]$
The polarization of the wave is
 A right handed circular B left handed circular C right handed elliptical D left handed elliptical
GATE EC 2015-SET-2   Electromagnetics
Question 10 Explanation:
\begin{aligned} \vec{E}(z, t)=& \hat{a}_{x} 5 \cos \left(2 \pi \times 10^{9} t+\beta z\right) \\ &+\hat{a}_{y} 3 \cos \left(2 \pi \times 10^{9} t+\beta z-\frac{\pi}{2}\right) \end{aligned}
$\Rightarrow$ Wave is travelling in $-\hat{a}_{z}$ direction.
$\Rightarrow$ Wave has orthogonal components with unequal amplitudes and checking the time trace.
$\therefore$ Wave is left hand elliptically polarized.
There are 10 questions to complete.