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 \gt1, 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.