# GATE Questions & Answers of Electromagnetics Electronics and Communication Engg

#### Electromagnetics 101 Question(s) | Weightage 10 (Marks)

Concentric spherical shells of radii 2 m, 4 m, and 8 m carry uniform surface charge densities of 20 nC/m2, −4 nC/m2 and ${\mathrm{\rho }}_{\mathrm{s}}$, respectively. The value of ${\mathrm{\rho }}_{\mathrm{s}}$(nC/m2) required to ensure that the electric flux density $\stackrel{\to }{D}=\stackrel{\to }{0}$ radius 10 m is _________

The propagation constant of a lossy transmission line is (2+$j$5) m−1 and its characteristic impedance is (50+$j$0) Ω at $\omega ={10}^{6}$ rad s−1. The values of the line constants L,C,R,G are, respectively,

The current density in a medium is given by

$\overrightarrow J=\frac{400\;\sin\theta}{2\mathrm\pi\left(\mathrm r^2+4\right)}{\widehat a}_r\mathrm{Am}^{-2}$

The total current and the average current density flowing through the portion of a spherical surface r = 0.8 m, $\frac{\mathrm\pi}{12}\leq\theta\leq\frac{\mathrm\pi}4,0\leq\phi\leq2\mathrm\pi$ are given, respectively, by

Two lossless X-band horn antennas are separated by a distance of $200\lambda$. The amplitude reflection coefficients at the terminals of the transmitting and receiving antennas are 0.15 and 0.18, respectively. The maximum directivities of the transmitting and receiving antennas (over the isotropic antenna) are 18 dB and 22 dB, respectively. Assuming that the input power in the lossless transmission line connected to the antenna is 2 W, and that the antennas are perfectly aligned and polarization matched, the power ( in mW) delivered to the load at the receiver is ________

The electric field of a uniform plane wave travelling along the negative z direction is given by the following equation:

${\stackrel{\to }{E}}_{w}^{i}=\left({\stackrel{^}{a}}_{x}+j{\stackrel{^}{a}}_{y}\right){E}_{0}{e}^{jkz}$

This wave is incident upon a receiving antenna placed at the origin and whose radiated electric field towards the incident wave is given by the following equation:

${\stackrel{\to }{E}}_{a}=\left({\stackrel{^}{a}}_{x}+2{\stackrel{^}{a}}_{y}\right){E}_{I}\frac{1}{r}{e}^{-jkr}$

The polarization of the incident wave, the polarization of the antenna and losses due to the polarization mismatch are, respectively,

The far-zone power density radiated by a helical antenna is approximated as:

${\overrightarrow W}_{rad}={\overrightarrow W}_{average}\approx\widehat{a_r}C_0\frac1{r^2}\cos^4\theta$

The radiated power density is symmetrical with respect to $\phi$ and exists only in the upper hemisphere: $0\;\leq\;\theta\;\leq\frac\pi2;\;0\;\leq\;\phi\;\leq2\pi;\;C_0$is a constant. The power radiated by the antenna (in watts) and the maximum directivity of the antenna, respectively, are

A uniform and constant magnetic field $\mathrm{B}=\stackrel{^}{z}\mathrm{B}$ exists in the $\stackrel{^}{z}$ direction in vacuum. A particle of mass m with a small charge q is introduced into this region with an initial velocity $\mathrm{V}=\stackrel{^}{\mathrm{x}}{\mathrm{v}}_{x}+\stackrel{^}{\mathrm{z}}{\mathrm{v}}_{z}$. Given that B, m, q, vx and vz are all non-zero, which one of the following describes the eventual trajectory of the particle?

Let the electric field vector of a plane electromagnetic wave propagating in a homogenous medium be expressed as $\mathrm{E}=\stackrel{^}{x}{E}_{x}{e}^{-j\left(\omega t-\beta z\right)}$, where the propagation constant $\beta$ is a function of the angular frequency $\omega$. Assume that $\beta \left(\omega \right)$ and Ex are known and are real. From the information available, which one of the following CANNOT be determined?

Light from free space is incident at an angle ${\theta }_{i}$ to the normal of the facet of a step-index large core optical fibre. The core and cladding refractive indices are n1=1.5 and n2=1.4, respectively.
The maximum value of ${\theta }_{i}$ (in degrees) for which the incident light will be guided in the core of the fibre is ________

The parallel-plate capacitor shown in the figure has movable plates. The capacitor is charged so that the energy stored in it is E when the plate separation is d. The capacitor is then isolated electrically and the plates are moved such that the plate separation becomes 2d.

At this new plate separation, what is the energy stored in the capacitor, neglecting fringing effects?

A lossless microstrip transmission line consists of a trace of width w. It is drawn over a practically infinite ground plane and is separated by a dielectric slab of thickness t and relative permittivity ${\epsilon }_{r}>1$. The inductance per unit length and the characteristic impedance of this line are L and Z0, respectively.

Which one of the following inequalities is always satisfied?

A microwave circuit consisting of lossless transmission lines T1 and T2 is shown in the figure. The plot shows the magnitude of the input reflection coefficient $\mathrm{\Gamma }$ as a function of frequency f. The phase velocity of the signal in the transmission lines is 2×108 m/s.

The length L (in meters) of T2 is ________

A positive charge q is placed at x=0 between two infinite metal plates placed at x=-d and at x=+d respectively. The metal plates lie in the yz plane

The charge is at rest at t=0, when a voltage +V is applied to the plate at -d and voltage -V is applied to the plate at x=+d. Assume that the quantity of the charge q is small enough that it does not perturb the field set up by the metal plates. The time that the charge q takes to reach the right plate is proportional to

If a right-handed circularly polarized wave is incident normally on a plane perfect conductor, then the reflected wave will be

Faraday’s law of electromagnetic induction is mathematically described by which one of the following equations?

Consider an air-filled rectangular waveguide with dimensions a = 2.286 cm and b = 1.016 cm. At 10 GHz operating frequency, the value of the propagation constant (per meter) of the corresponding propagating mode is __________

Consider an air-filled rectangular waveguide with dimensions a = 2.286 cm and b = 1.016 cm. The increasing order of the cut-off frequencies for different modes is

A radar operating at 5 GHz uses a common antenna for transmission and reception. The antenna has a gain of 150 and is aligned for maximum directional radiation and reception to a target 1 km away having radar cross-section of 3 m2. If it transmits 100 kW, then the received power (in μW) is __________

Consider the charge profile shown in the figure. The resultant potential distribution is best described by

The electric field component of a plane wave traveling in a lossless dielectric medium is given by $\stackrel{\to }{E}\left(z,t\right)={\stackrel{^}{a}}_{y}2\mathrm{cos}\left({10}^{8}t-\frac{z}{\sqrt{2}}\right)\mathrm{V}/\mathrm{m}$. Thewavelength (in m) for the wave is_______.

A vector $\stackrel{\to }{P}$ is given by $\stackrel{\to }{P}={x}^{3}y{\stackrel{\to }{a}}_{x}-{x}^{2}{y}^{2}{\stackrel{\to }{a}}_{y}-{x}^{2}yz{\stackrel{\to }{a}}_{z}$. Which one of the following statements is TRUE?

The longitudinal component of the magnetic field inside an air-filled rectangular waveguide made of a perfect electric conductor is given by the following expression

The cross-sectional dimensions of the waveguide are given as a = 0.08 m and b = 0.033 m. The mode of propagation inside the waveguide is

The electric field intensity of a plane wave traveling in free space is given by the following expression

$\boldsymbol E(x,t)={\boldsymbol a}_y\;24\mathrm\pi\;\cos(\mathrm{ωt}-{\mathrm k}_0\mathrm x)(\mathrm V/\mathrm m).$

In this field, consider a square area 10 cm X 10 cm on a plane $\mathrm x+\mathrm y=1.$ The total time-averaged power (in mW) passing through the square area is _______.

Consider a uniform plane wave with amplitude (E0) of 10 V/m and 1.1 GHz frequency travelling in air, and incident normally on a dielectric medium with complex relative permittivity (${\mathrm{\epsilon }}_{\mathrm{r}}$) and permeability (${\mathrm{\mu }}_{\mathrm{r}}$) as shown in the figure.

The magnitude of the transmitted electric field component (in V/m) after it has travelled a distance of 10 cm inside the dielectric region is_______.

In a source free region in vacuum, if the electrostatic potential $\varphi=2x^2+y^2+cz^2$, the value of constant $c$ must be _______.

The electric field of a uniform plane electromagnetic wave is

$\stackrel{\to }{E}=\left({\stackrel{\to }{a}}_{x}+j4{\stackrel{\to }{a}}_{y}\right)\mathrm{exp}\left[j\left(2\mathrm{\pi }×{10}^{7}t-0.2z\right)\right]$

The polarization of the wave is

An air-filled rectangular waveguide of internal dimension a cm X b cm (a > b) has a cutoff frequency of 6 GHz for the dominant TE10 mode. For the same waveguide, if the cutoff frequency of the TM11 mode is 15 GHz, the cutoff frequency of the TE01 mode in GHz is __________.

Two half –wave dipole antennas placed as shown in the figure are excited with sinusoidally varying currents of frequency 3 MHz and phase shift of $\mathrm{\pi }$/2 between them (the element at the origin leads in phase) . If the maximum radiated E –field at the point P in the x-y plane occurs at an azimuthal angle of 60o, the distance d (in meters) between the antennas is________.

The electric field of a plane wave propagating in a lossless non-magnetic medium is given by the following expression

The type of the polarization is

The directivity of an antenna array can be increased by adding more antenna, as a larger number of elements

A coaxial cable is made of two brass conductors. The spacing between the conductors is filled with Teflon (${\epsilon }_{r}$ = 2.1, tan $\delta$ = 0). Which one of the following circuits can represent the lumped element model of a small piece of this cable having length $\bigtriangleup$z ?

A vector field exists inside a cylindrical region enclosed by the surfaces ρ = 1, z = 0 and z=5. Let S be the surface bounding this cylindrical region. The surface integral of this field on is______.

Consider the 3 m long lossless air-filled transmission line shown in the figure. It has a characteristic impedance of 120 $\mathrm{\pi \Omega }$, is terminated by a short circuit, and is excited with a frequency of 37.5 MHz. What is the nature of the input impedance (Zin)?

A 200 m long transmission line having parameters shown in the figure is terminated into a load RL. The line is connected to a 400 V source having source resistance RS Through a switch, which is closed at t=0. The transient response of the circuit at the input of the line (z=0) is also drawn in the figure. The value of RL (in Ω) is_______.

A coaxial capacitor of inner radius 1 mm and outer radius 5 mm has a capacitance per unit length of 172 pF/m. If the ratio of outer radius to inner radius is doubled, the capacitance per unit length (in pF/m) is ________.

A two-port network has scattering parameters given by $\left[S\right]=\left[\begin{array}{cc}{s}_{11}& {s}_{12}\\ {s}_{21}& {s}_{22}\end{array}\right]$. If the port-2 of the two- port is short circuited , the ${s}_{11}$ parameter for the resultant one-port network is

The force on a point charge +q kept at a distance d from the surface of an infinite grounded metal plate in a medium of permittivity $\in$ is

In spherical coordinates, let ${\widehat a}_\theta,{\widehat a}_\phi$; denote unit vectors along the $\theta,\phi$ directions

$E=\frac{100}r\sin\theta\;\cos\;\left(\omega t-\beta r\right){\widehat a}_\theta V/m$

and

$H=\frac{0.265}r\sin\theta\;\cos\;\left(\omega t-\beta r\right){\widehat a}_\phi A/m$

represent the electric and magnetic field components of the EM wave at large distances r from a dipole antenna, in free space. The average power (W) crossing the hemispherical shell located at is _______.

For a parallel plate transmission line, let v be the speed of propagation and Z be the characteristic impedance. Neglecting fringe effects, a reduction of the spacing between the plates by a factor of two results in

The input impedance of a $\frac{\lambda }{8}$ section of a lossless transmission line of characteristic impedance 50Ω is found to be real when the other end is terminated by a load ${Z}_{L}\left(=R+jx\right)\Omega$ If X is 30Ω, the value of R (in Ω) is_________

To maximize power transfer, a lossless transmission line is to be matched to a resistive load impedance via a $\lambda /4$ transformer as shown.

The characteristic impedance (in Ω) of the $\lambda /4$ transformer is _________.

Which one of the following field patterns represents a TEM wave travelling in the positive x direction?

if $\overrightarrow r=x{\widehat a}_x+y{\widehat a}_y+z{\widehat a}_z\;and\;\left|\overrightarrow r\right|=r$ then div $\left(r^2\nabla\left(\ln\;r\right)\right)$ = ______ .

If the electric field of a plane wave is

$\overrightarrow{\mathrm E}\left(z,t\right)=\widehat x3\cos\left(\omega t-kz+30^\circ\right)-\widehat y4\sin\left(\omega t-kz+45^\circ\right)\left(mV/m\right)$

the polarization state of the plane wave is

In the transmission line shown, the impedance Zin (in ohms) between node A and the ground is _________.

For a rectangular waveguide of internal dimensions a × b (a>b), the cut-off frequency for the TE11 mode is the arithmetic mean of the cut-off frequencies for TE10 mode and TE20 mode. If $a=\sqrt{5}$cm, the value of b (in cm) is _____.

Consider an air filled rectangular waveguide with a cross-section of 5 cm × 3 cm. For this waveguide, the cut-off frequency (in MHz) of TE21 mode is _________.

In the following figure, the transmitter Tx sends a wideband modulated RF signal via a coaxial cable to the receiver Rx. The output impedance ZT of Tx, the characteristic impedance Z0 of the cable and the input impedance ZR of Rx are all real.

Which one of the following statements is TRUE about the distortion of the received signal due to impedance mismatch?

Given the vector A = $\left(\cos\;x\right)\left(\sin\;y\right){\widehat a}_x+\left(\sin\;x\right)\left(\cos\;y\right){\widehat a}_y$ where ${\widehat a}_x{\widehat a}_y$ denote unit vectors along x,y directions, respectively. The magnitude of curl of A is ________

A region shown below contains a perfect conducting half-space and air. The surface current $\overrightarrow{K_s}$ on the surface of the perfect conductor is $\overrightarrow{K_s}=\widehat X2$; amperes per meter. The tangential $\overrightarrow H$ field in the air just above the perfect conductor is

Assume that a plane wave in air with an electric field $\overrightarrow E=10\cos\left(\omega t-3x-\sqrt3z\right){\overset\frown a}_y\;V/m$ is incident on a non-magnetic dielectric slab of relative permittivity 3 which covers the region z > 0. The angle of transmission in the dielectric slab is ________ degrees.

The magnitude of the gradient for the function $f\left(x,y,z\right)={x}^{2}+3{y}^{2}+{z}^{3}$ at the point (1,1,1) is _________.

The directional derivative of $f\left(x,y\right)=\frac{xy}{\sqrt{2}}\left(x+y\right)$at (1, 1) in the direction of the unit vector at an angle of $\frac{\mathrm{\pi }}{4}$ with y-axis, is given by ______ .

For an antenna radiating in free space, the electric field at a distance of 1 km is found to be 12 mV/m. Given that intrinsic impedance of the free space is 120πΩ, the magnitude of average power density due to this antenna at a distance of 2 km from the antenna (in nW/m2 ) is ____.

Match column A with column B.

 Column A Column B 1. Point electromagnetic source P. Highly directional 2. Dish antenna Q. End fire 3. Yagi-Uda antenna R. Isotropic

The electric field (assumed to be one-dimensional) between two points A and B is shown. Let ${\psi }_{A}$ and ${\psi }_{B}$ be the electrostatic potentials at A and B, respectively. The value of ${\psi }_{B-}{\psi }_{A}$ in Volts is ________.

Given$\overrightarrow F=z{\overset\frown a}_x+x{\overset\frown a}_y+y{\overset\frown a}_z\;$. If Srepresents the portion of the sphere for $z\ge 0$, then $\int_s\nabla\times\overrightarrow F.\overrightarrow{ds}$ is______________.

If $\overrightarrow E=-\left(2y^3-3yz^2\right)\widehat x-\left(6xy^2-3xz^2\right)\widehat y+\left(6xyz\right)\widehat z$ is the electric field in a source free region, a valid expression for the electrostatic potential is

A monochromatic plane wave of wavelength $\lambda$ = 600 $\mu$m is propagating in the direction as shown in the figure below. and $\stackrel{\to }{{E}_{t}}$ denote incident, reflected, and transmitted electric field vectors associated with the wave.

The angle of incidence ${\theta }_{i}$ and the expression for $\stackrel{\to }{{E}_{i}}$ are

A monochromatic plane wave of wavelength $\lambda$ = 600 $\mu$m is propagating in the direction as shown in the figure below. and $\stackrel{\to }{{E}_{t}}$ denote incident, reflected, and transmitted electric field vectors associated with the wave.

The expression for $\stackrel{\to }{{E}_{r}}$ is

A plane wave propagating in air with is incident on a perfectly conducting slab positioned at $x\le 0$.The $\stackrel{\to }{E}$ field of the reflected wave is

The electric field of a uniform plane electromagnetic wave in free space, along the positive $x$ direction, is given by $\stackrel{\to }{E}=10\left({\stackrel{^}{a}}_{y}+j{\stackrel{^}{a}}_{z}\right){e}^{-j25x}$.The frequency and polarization of the wave, respectively, are

A coaxial cable with an inner diameter of 1 mm and outer diameter of 2.4 mm is filled with a dielectric of relative permittivity 10.89. Given , the characteristic impedance of the cable is

The radiation pattern of an antenna in spherical co-ordinates is given by

$F\left(\theta \right)={\mathrm{cos}}^{4}\theta ;0\le \theta \le \mathrm{\pi }/2$

The directivity of the antenna is

A transmission line with a characteristic impedance of 100 $\Omega$ is used to match a 50 $\Omega$ section to a 200 $\Omega$ section. If the matching is to be done both at 429 MHz and 1 GHz, the length of the transmission line can be approximately

The direction of vector A is radially outward from the origin, with $\left|\mathrm{A}\right|=k{r}^{n}$ where r2 = x2 +y2 +z2 and k is a constant. The value of n for which $\nabla ·\mathrm{A}=0$ is
${\mathrm{H}}_{\mathrm{z}}=3\mathrm{cos}\left(2.094×{10}^{2}\mathrm{x}\right)\mathrm{cos}\left(2.618×{10}^{2}\mathrm{y}\right)\mathrm{cos}\left(6.283×{10}^{10}\mathrm{t}-\mathrm{\beta z}\right)$