The current density in a medium is given by
$\overrightarrow J=\frac{400\;\sin\theta}{2\mathrm\pi\left(\mathrm r^2+4\right)}{\widehat a}_r\mathrm{Am}^{-2}$
${\overrightarrow{E}}_{w}^{i}=\left({\hat{a}}_{x}+j{\hat{a}}_{y}\right){E}_{0}{e}^{jkz}$
${\overrightarrow{E}}_{a}=\left({\hat{a}}_{x}+2{\hat{a}}_{y}\right){E}_{I}\frac{1}{r}{e}^{-jkr}$
The far-zone power density radiated by a helical antenna is approximated as:
${\overrightarrow W}_{rad}={\overrightarrow W}_{average}\approx\widehat{a_r}C_0\frac1{r^2}\cos^4\theta$
At this new plate separation, what is the energy stored in the capacitor, neglecting fringing effects?
Which one of the following inequalities is always satisfied?
${Z}_{0}\sqrt{\frac{Lt}{{\epsilon}_{0}{\epsilon}_{r}w}}$
The length L (in meters) of T_{2} 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 electric field component of a plane wave traveling in a lossless dielectric medium is given by $\overrightarrow{E}\left(z,t\right)={\hat{a}}_{y}2\mathrm{cos}\left({10}^{8}t-\frac{z}{\sqrt{2}}\right)\mathrm{V}/\mathrm{m}$. Thewavelength (in m) for the wave is_______.
A vector $\overrightarrow{P}$ is given by $\overrightarrow{P}={x}^{3}y{\overrightarrow{a}}_{x}-{x}^{2}{y}^{2}{\overrightarrow{a}}_{y}-{x}^{2}yz{\overrightarrow{a}}_{z}$. 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
${H}_{z}\left(x,y,z,t\right)=0.1\mathrm{cos}\left(25\pi \mathrm{x}\right)\mathrm{cos}\left(30.3\pi y\right)\mathrm{cos}\left(12\pi \times {10}^{9}t-\beta z\right)\left(A/m\right)$
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 (E_{0}) 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
$\overrightarrow{E}=\left({\overrightarrow{a}}_{x}+j4{\overrightarrow{a}}_{y}\right)\mathrm{exp}\left[j\left(2\mathrm{\pi}\times {10}^{7}t-0.2z\right)\right]$
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 TE_{10} mode. For the same waveguide, if the cutoff frequency of the TM_{11} mode is 15 GHz, the cutoff frequency of the TE_{01} 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 60^{o}, 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
$\mathrm{E}\left(z,t\right)={\mathrm{a}}_{x}5\mathrm{cos}\left(2\pi \times {10}^{9}t+\beta z\right)+{\mathrm{a}}_{y}3\mathrm{cos}\left(2\pi \times {10}^{9}t+\beta z-\frac{\pi}{2}\right)$
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 $\mathbf{D}=2{\rho}^{2}{\mathbf{a}}_{\rho}+z{\mathbf{a}}_{z}$ 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 $\left({\u222f}_{s}\mathbf{D}\mathbf{\xb7}\mathbf{ds}\right)$ 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 (Z_{in})?
A 200 m long transmission line having parameters shown in the figure is terminated into a load R_{L}. The line is connected to a 400 V source having source resistance R_{S} 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 R_{L} (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 $r=1km,0\le \theta \le \raisebox{1ex}{$\mathrm{\pi}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ 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 Z_{in} (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 TE_{11 }mode is the arithmetic mean of the cut-off frequencies for TE_{10} mode and TE_{20} 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 TE_{21} 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 Z_{T} of Tx, the characteristic impedance Z_{0} of the cable and the input impedance Z_{R} 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/m^{2} ) is ____.
Match column A with column B.
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 ${x}^{2}+{y}^{2}+{z}^{2}=1$ 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 $m is propagating in the direction as shown in the figure below.$\overrightarrow{{E}_{i}},\overrightarrow{{E}_{r}}$ and $\overrightarrow{{E}_{t}}$ denote incident, reflected, and transmitted electric field vectors associated with the wave.
The angle of incidence ${\theta}_{i}$ and the expression for $\overrightarrow{{E}_{i}}$ are
The expression for $\overrightarrow{{E}_{r}}$ is
A plane wave propagating in air with $\overrightarrow{E}=\left(8{\hat{a}}_{x}+6{\hat{a}}_{y}+5{\hat{a}}_{z}\right){e}^{j\left(wt+3x-4y\right)}\mathrm{V}/\mathrm{m}$ is incident on a perfectly conducting slab positioned at $x\le 0$.The $\overrightarrow{E}$ 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 $\overrightarrow{E}=10\left({\hat{a}}_{y}+j{\hat{a}}_{z}\right){e}^{-j25x}$.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 ${\mu}_{0}=4\pi \times {10}^{-7}H/m,{\epsilon}_{0}=\frac{{10}^{-9}}{36\pi}F/m$, 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 r^{2} = x^{2} +y^{2} +z^{2} and k is a constant. The value of n for which $\nabla \xb7\mathrm{A}=0$ is
The magnetic field along the propagation direction inside a rectangular waveguide with the crosssection shown in the figure is
${\mathrm{H}}_{\mathrm{z}}=3\mathrm{cos}\left(2.094\times {10}^{2}\mathrm{x}\right)\mathrm{cos}\left(2.618\times {10}^{2}\mathrm{y}\right)\mathrm{cos}\left(6.283\times {10}^{10}\mathrm{t}-\mathrm{\beta z}\right)$
The phase velocity v_{p} of the wave inside the waveguide satisfies