GATE EC Electromagnetics questions

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 $\hat{z}$ direction. The refractive indices for the electric fields along $\hat{x}$ and $\hat{y}$ directions in the fiber are $n_{x}=1.5000$ and $n_{y}=1.5001$, 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 …

The expression for an electric field in free space is , $\textbf{E}=E_{0}( \hat{x}+\hat{y}+j2 \hat{z} )e^{-j( \omega t-kx+ky)}$ where $x,y,z$ represent the special coordinates, $t$ represents time, and $\omega,k$ are constants.This electric field,

A half wavelength dipole is kept in the $x-y$ plane and oriented along $45^{\circ}$ from the $x$-axis. Determine the direction of null in the radiation pattern for $0\leq \phi \leq \pi$. Here the angle $\theta (0\leq \theta \leq \pi )$ is measured from the $z$-axis and the angle$\phi (0\leq \theta \leq 2\pi )$ is measured from the $x$-axis in …

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 $V_a$ and $E_a$ and that on S2 are $V_b$ and $E_b$, respectively. …

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 _________

An integral $I$ over a counterclockwise circle $\textbf{C}$ is given by $$I=\oint _c \frac{z^2-1}{z^2+1} e^z dz.$$ If $\textbf{C}$ is defined as$\mid z \mid=3$ , then the value of $I$ is

Standard air-filled rectangular waveguides of dimensions $a=2.29$ cm and $b=1.02$ cm are designed for radar applications. It is desired that these waveguides operate only in the dominant $TE_{10}$ mode with the operating frequency at least $25 \%$ above the cutoff frequency of the $TE_{10}$ mode but not higher than $95 \%$ of the next higher …

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 $20 \: nC/m^2$, $– 4 \: nC/m^2$ and $\rho_s$ respectively. The value of $\rho_s$ ($nC/m^2$) required to ensure that the electric flux density $\overrightarrow D$ = $\overrightarrow 0$ at radius $10$ m is ________

The propagation constant of a lossy transmission line is $(2+j5) m^{-1}$ and its characteristic impedance is $(50+j0) \Omega$ at $\omega = 10^6 rad \: s^{-1}$. The values of the line constants L,C,R,G are,respectively,

An antenna pointing in a certain direction has a noise temperature of $50 \: K$. The ambient temperature is $290 \: K$. The antenna is connected to a pre-amplifier that has a noise figure of $2$ dB and an available gain of $40$ dB over an effective bandwidth of $12$ MHz. The effective input noise temperature $T_e$ for the amplifier and the noise …

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 directives of the transmitting and receiving antennas (over the isotropic antenna) are $18$ dB and $22$ dB, respectively. …

The electric field of a uniform plane wave traveling along the negative $z$ direction is given by the following equation: $${\overrightarrow {E}^i_w= ( \hat {a}_x+ j\hat {a}_y)E_0e^{jkz}}$$ 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: …

The far-zone power density radiated by a helical antenna is approximated as: $${\overrightarrow W_{rad}= \overrightarrow W_{average} \approx \hat {a_r}C_0\frac{1}{r^2}\cos^4\theta}$$ The radiated power density is symmetrical with respect to $\phi$ and exists only in the upper hemisphere: $0\leq\theta\leq\frac{\pi}{2}; \: 0\leq\phi\leq 2\pi$; $C_0$ is a …

A uniform and constant magnetic field $\textbf{B}=\hat{z}B$ exists in the $\hat{z}$ direction in vacuum. A particle of mass $m$ with a small charge $q$ is introduced into this region with an initial velocity $\textbf{v}=\hat{x}v_{x}+\hat{z}v_{z}$. Given that $B$, $m$, $q$, $v_{x}$ and $v_{z}$ are all non-zero, which one of the following …

Let the electric field vector of a plane electromagnetic wave propagating in a homogeneous medium be expressed as $\textbf{E}=\hat{x}E_{x}e^{-j(\omega t-\beta z)},$ where the propogation 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, …

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 $n _{1}=1.5$ and $n _{2}=1.4$, respectively. The maximum value of $\theta _{i}$ (in degrees) for which the incident light will be guided in the core of the fibre is _________

A lossless microstrip transmission line consists of a trace of width $w$. It is drawn over a pratically infinite ground plane and is separated by a dielectric slab of thickness $t$ and relative permittivity $\varepsilon _{r}> 1.$ The inductance per unit length and the characteristic impedance of this line are $L$ and $Z_{0}$, respectively. Which …

A microwave circuit consisting of lossless transmission lines $T_{1}$ and $T_{2}$ is shown in the figure. The plot shows the magnitude of the input reflection coefficient $\Gamma$ as a function of frequency $f$. The phase velocity of the signal in the transmission lines is $2\times 10^{8}$ $m/s$. The length $L$ (in meters) of $T_{2}$ is …