GATE EC Maxwell's Equations questions

A square metal sheet of $4 \mathrm{~m} \times 4 \mathrm{~m}$ is placed on the $x-y$ plane as shown in the figure below. If the surface charge density (in $\mu \mathrm{C} / \mathrm{m}^{2}$ ) on the sheet is $\rho_{s}(x, y)=4\mid y \mid$, then the total charge (in $\mu \mathrm{C}$, rounded off to the nearest integer) on the sheet is $\_\_\_\_\_\_$

An electric field of $0.01 \mathrm{~V} / \mathrm{m}$ is applied along the length of a copper wire of circular cross-section with diameter $1 \: \mathrm{mm}$. Copper has a conductivity of $5.8 \times 10^{7} \mathrm{~S} / \mathrm{m}$. The current (in Amperes, rounded off to two decimal places) flowing through the wire is $\_\_\_\_\_\_$.

In the table shown, List I and List II, respectively, contain terms appearing on the left-hand side and the right-hand side of Maxwell’s equations (in their standard form). Match the left-hand side with the corresponding right-hand side. …

In the circuit shown, $V_{s}$ is square wave of period $T$ with maximum and minimum values of $8\: V$ and $-10\: V$, respectively. Assume that the diode is ideal and $R_{1}=R_{2}=50\: \Omega.$ The average value of $V_{L}$ is _______ volts (rounded off to $1$ decimal place).

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. …

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

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 ________

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, …

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

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

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 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 $\epsilon\:$ is

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 $120 \pi \: \Omega$, the magnitude of average power density due to this antenna at a distance of $2$ km from the antenna (in $nW/m^2$) is _________.

A $16$ – bit ripple carry adder is realized using $16$ identical full adders (FA) as shown in the figure. The carry-propagation delay of each FA is $12$ ns and the sum-propagation delay of each FA is $15$ ns. The worst case delay (in ns) of this $16$-bit adder will be _______

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 ____________

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

The average power delivered to an impedance $(4-j3)\:\Omega$ by a current $5\cos(100\pi t+100)\:A$ is

A medium is divided into regions I and II about $x=0$ plane, as shown in the figure below. An electromagnetic wave with electric field $\mathrm{E}_{1}=4 \hat{a}_{x}+3 \hat{a}_{y}+5 \hat{a}_{z}$ is incident normally on the interface from region-I. The electric field $E_{2}$ in region-II at the interface is

A parallel plate air-filled capacitor has plate area of $10^{-1} \mathrm{~m}^{2}$ and plate separation of $10^{-3} \mathrm{~m}$. It is connected to a $0.5 \mathrm{~V}, 3.6 \; \mathrm{GHz}$ source. The magnitude of the displacement current is $\left(\varepsilon_{0}=1 / 36 \pi \times 10^{-9} \mathrm{~F} / \mathrm{m}\right)$