# GATE Questions & Answers of Electromagnetics Electronics and Communication Engg

#### Electromagnetics 117 Question(s) | Weightage 10 (Marks)

The points $P$, $Q$, and $R$ shown on the Smith chart (normalized impedance chart) in the following figure represent:

A lossy transmission line has resistance per unit length $R=0.05\Omega$/m . The line is distortionless and has characteristic impedance of $50\Omega$ . The attenuation constant (in Np/m, correct to three decimal places) of the line is _______.

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

The cutoff frequency of $TE_{01}$ mode of an air filled rectangular waveguide having inner dimensions $a$ cm $\times$ $b$ cm $\left(a>b\right)$ is twice that of the dominant $TE_{10}$ mode. When the waveguide is operated at a frequency which is 25% higher than the cutoff frequency of the dominant mode, the guide wavelength is found to be 4 cm. The value of $b$ (in cm, correct to two decimal places) is _______.

A uniform plane wave traveling in free space and having the electric field

$\overrightarrow E=\left(\sqrt2{\widehat a}_x-{\widehat a}_z\right)\cos\left[6\sqrt3\pi\times10^8t-2\pi\left(x+\sqrt2z\right)\right]\;V/m$

is incident on a dielectric medium (relative permittivity $>$ 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 ___________.

The voltage of an electromagnetic wave propagating in a coaxial cable with uniform characteristic impedance is $V\left(l\right)={e}^{-\gamma l+j\omega t}$ Volts, where l is the distance along the length of the cable in meters,$\gamma =\left(0.1+j40\right){m}^{-1}$ is the complex propagation constant, and $\omega =2\pi ×{10}^{9}$ rad/s is the angular frequency. The absolute value of the attenuation in the cable in dB/metre is__________.

Consider a wireless communication link between a transmitter and a receiver located in free space, with finite and strictly positive capacity. If the effective areas of the transmitter and the receiver antennas, and the distance between them are all doubled, and everything else remains unchanged, the maximum capacity of the wireless link

An optical fiber is kept along the $\stackrel{^}{z}$ direction. The refractive indicates for the electric fields along $\stackrel{^}{x}$ and $\stackrel{^}{y}$ directions in the fiber are nx=1.5000 and ny=1.5001, respectively ($n_x\neq n_y$ due to the imperfection in the fiber cross-section). The free space wavelength of a light wave propagating in the fiber is 1.5 µm. If the lightwave is circularly polarized at the input of the fiber, the minimum propagation distance after which it becomes linearly polarized, in centimeters, is____________.

The expression for an electric field in free space is , where x, y, z represent the special coordinates, t represent time, and ω, k are constant. This electric field

A half wavelength dipole is kept in the x-y plane and oriented along 45° from the x –axis. Determine the direction of null in the radiation pattern for $0\leq\mathrm\phi\leq\pi$, Here the angle $\theta\left(0\leq\mathrm\theta\leq\pi\right)$ is measured from the z-axis, and the angle $\mathrm\phi\left(0\leq\mathrm\phi\leq2\pi\right)$ is measured from the x-axis in the x-y plane.

Two conducting spheres S1 and S2 of radii $a$ and $b (b > a)$ respectively, are placed far apart and connected by a long, thin conducting wire, as shown in the figure.

For some charge placed on this structure, the potential and surface electric field on S1 are Va and Ea and  that on S2 are Vb and Eb, respectively. Then, which of the following is CORRECT?

A two-wire transmission line terminates in a television set. The VSWR measured on the line is 5.8. The percentage of power that is reflected from the television set is___________

If the vector function $\stackrel{\to }{F}=\stackrel{^}{{a}_{x}}\left(3y-{k}_{1}z\right)+\stackrel{^}{{a}_{y}}\left({k}_{2}x-2z\right)-\stackrel{^}{{a}_{z}}\left({k}_{3}y+z\right)$ is irrotational, then the values of the constant k1, k2 and k3 , respectively, are

An electron (q1) is moving in free space with velocity 105 m/s towards a stationary electron (q2) far away. The closest distance that this moving electron gets to the stationary electron before the repulsive force diverts its path is___________ ×10-8 m.

[Given, mass of electron m = 9.11×10-31 kg, charge of electron e = 1.6 ×10-19 C, and permittivity $\varepsilon_0=\left(1/36\pi\right)\times10^{-9\;}\mathrm F/\mathrm m$]

Standard air-filled rectangular waveguides of dimension = 2.29 cm and = 1.02 cm are designed for radar application. It is desired that these waveguides operate only in the dominant TE10 mode but not higher than 95% of the next cutoff frequency. The range of the allowable operating frequency f is

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 5 m, as shown in the figure. The critical angle is $\theta_c$ .

The value of d (in meter) is___________

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 _____.