Electromagnetic Waves - Class 12

An ideal LC circuit consists of an inductor of inductance $L$ and a parallel plate capacitor with circular plates of radius $R$ and separation $d$. The space between the capacitor plates is filled with a non-uniform dielectric material whose relative permittivity varies radially as $\epsilon_r(r) = \alpha r^2$, where $\alpha$ is a positive constant and $r$ is the radial distance from the central axis of the plates. The circuit sustains undamped electromagnetic oscillations. If the maximum electrical energy stored in the capacitor during the oscillation cycle is $U_0$, what is the maximum magnitude of the induced magnetic field $B_{max}$ at a radial distance $r = R/2$ from the central axis, inside the capacitor? (Assume edge effects are negligible and all quantities are in SI units)

In a source-free region of vacuum, an electromagnetic standing wave has an electric field given by $\vec{E}(x,t) = E_0 \cos(kx) \sin(\omega t) \hat{j}$, where $E_0$, $k$, and $\omega$ are positive constants. Consider a rectangular loop $C$ in the $xz$-plane with vertices traversed in the order $(0, 0, 0) \rightarrow (0, 0, L) \rightarrow (x_0, 0, L) \rightarrow (x_0, 0, 0) \rightarrow (0, 0, 0)$. Which of the following expressions correctly represents the circulation of the magnetic field, $\oint_C \vec{B} \cdot d\vec{l}$, around this loop at time $t$?

A parallel-plate capacitor consists of two circular plates of radius $R$ separated by a very small vacuum gap $d$ ($d \ll R$). The plates are initially uncharged. Starting at $t=0$, a radial surface current flows inward on the positive plate (located at $z=0$) and outward on the negative plate (located at $z=d$). The surface current density on the positive plate is given by $\vec{K}(r,t) = - \beta r^2 t \hat{r}$, where $\beta$ is a positive constant, $r$ is the radial distance from the center, and $\hat{r}$ is the outward radial unit vector. Assuming edge effects are negligible, what is the magnitude of the magnetic field $B$ at a radial distance $r$ ($r < R$) inside the gap between the plates?

A parallel plate vacuum capacitor consists of two circular plates of radius $R$ separated by a distance $d$. It is connected to an alternating voltage source $V(t) = V_0 \sin(\omega t)$. Let $X_C$ be the capacitive reactance of the capacitor and $\mu_0$ be the permeability of free space. Neglecting edge effects, which of the following expressions represents the azimuthal magnetic field $B(t)$ at a radial distance $r = R/2$ from the central axis of the capacitor, in the region between the plates?

A thin, infinite sheet lying in the $xy$-plane ($z=0$) carries a radial surface current density $\vec{K} = -\alpha r \hat{r}$, where $\alpha$ is a positive constant, $r = \sqrt{x^2+y^2}$ is the radial distance from the origin, and $\hat{r}$ is the radial unit vector in the $xy$-plane. This surface current causes a time-varying surface charge density $\sigma(t)$ to accumulate on the sheet. Assuming the sheet is initially uncharged at $t=0$, what is the induced magnetic field vector $\vec{B}$ at a point $(x, y, z)$ in the region $z > 0$?

Consider an ideal LC circuit consisting of an inductor of inductance $L$ and a parallel-plate capacitor of capacitance $C$ with plate separation $d$. The physical dimensions of the circuit are much smaller than the wavelength of the emitted electromagnetic waves. Due to the time-varying displacement current between the capacitor plates, the system acts as a Hertzian dipole and radiates electromagnetic energy into space. The power radiated by a dipole of moment $p$ is given by Larmor's formula $P = \frac{\mu_0 \ddot{p}^2}{6 \pi c}$, where $c$ is the speed of light in vacuum and $\mu_0$ is the vacuum permeability. Assuming the radiative damping is very small, what is the time $t$ taken for the total electromagnetic energy stored in the LC circuit to decrease to $1/e$ of its initial value?

A uniform, perfectly absorbing semi-circular disc of mass $M$ and radius $R$ lies in the $xy$-plane, occupying the region $x \ge 0$. Its straight edge is hinged along the $y$-axis, allowing it to rotate freely about the $y$-axis. The disc has a total permanent electric dipole moment $\vec{p} = p_0 \hat{k}$ uniformly distributed over its surface.

A linearly polarized plane electromagnetic wave pulse travels in the $+z$-direction, illuminating the entire semi-disc. The electric field of the pulse at $z=0$ is given by $\vec{E}(t) = E_0 \sin(\omega t) \hat{i}$ for $0 \le t \le \pi/\omega$, and $\vec{E}(t) = 0$ otherwise.

Assuming the pulse duration is extremely short such that the disc does not rotate significantly during the pulse, what is the magnitude of the angular velocity acquired by the disc immediately after the pulse passes?

A plane electromagnetic wave of wavelength $\lambda$ and electric field amplitude $E_0$ is incident from vacuum ($z < 0$) onto a perfectly conducting flat surface at $z=0$. The wave vector $\vec{k}_i$ lies in the $xz$-plane and makes an angle $\theta$ ($0 < \theta < \pi/2$) with the $z$-axis. The electric field is polarized parallel to the $y$-axis. A small circular wire loop of area $A$ and negligible self-inductance, having resistance $R$, is placed such that its center is at a distance $d$ from the conducting surface ($z = -d$). The loop lies entirely in a plane parallel to the $xy$-plane. Assume the dimensions of the loop are much smaller than $\lambda$. Let $\omega$ be the angular frequency of the wave and $c$ be the speed of light. Determine the maximum possible magnitude of the time-averaged torque $\langle \tau \rangle_{max}$ acting on the loop as $d$ is varied, and the minimum positive distance $d_{max}$ at which this maximum torque occurs.

A particle of mass $m$ and charge $q$ is embedded in a viscous medium that exerts a drag force $\vec{F}_d = -b\vec{v}$, where $b$ is a positive constant and $\vec{v}$ is the particle's velocity. A plane electromagnetic wave propagates through the medium in the $+z$ direction. At the particle's location, the electric and magnetic fields are given by $\vec{E} = E_0 \cos(\omega t) \hat{i}$ and $\vec{B} = \frac{E_0}{c} \cos(\omega t) \hat{j}$, where $c$ is the speed of light in the medium. Assume the particle's displacement is much smaller than the wavelength of the EM wave, so the spatial variation of the fields can be neglected. Furthermore, assume the velocity induced in the $z$-direction is negligible compared to that in the $x$-direction. After a long time, the particle reaches a steady state. What is the time-averaged magnetic Lorentz force $\langle F_z \rangle$ acting on the particle in the $+z$ direction?

A uniform plane electromagnetic wave of angular frequency $\omega$ and electric field amplitude $E_0$ is traveling in vacuum ($x < 0$) in the $+x$ direction. It is normally incident on a semi-infinite, non-magnetic ($\mu = \mu_0$), lossless dielectric medium ($x \ge 0$) with relative permittivity $\epsilon_r = 4$. Let $\langle u_e \rangle$ and $\langle u_m \rangle$ denote the time-averaged electric and magnetic energy densities, respectively, in the vacuum region. Calculate the ratio $\frac{\langle u_e \rangle}{\langle u_m \rangle}$ at a distance $d = \frac{\pi c}{2\omega}$ from the boundary (i.e., at $x = -d$), where $c$ is the speed of light in vacuum.

A monochromatic light beam propagating in the $+z$ direction has an electric field given by $\vec{E}(z,t) = \sqrt{5} E_0 \cos(kz - \omega t) \hat{i} + \sqrt{3} E_0 \cos\left(kz - \omega t + \frac{\pi}{6}\right) \hat{j}$ where $E_0$ is a positive constant in $\text{V/m}$. The total time-averaged intensity of this incident beam is $I_0$. The beam passes through an ideal linear polarizer whose transmission axis lies in the $x$-$y$ plane and can be rotated to any angle $\theta$ with respect to the positive $x$-axis. As the polarizer is rotated, let $I_{max}$ and $I_{min}$ be the maximum and minimum possible values of the transmitted intensity, respectively. Which of the following gives the correct values of $I_{max}$ and $I_{min}$?

A small circular loop of radius $a$ is placed at the origin in the $xy$-plane and carries an alternating current $I(t) = I_0 \cos(\omega t)$. This loop acts as an oscillating magnetic dipole. Consider a point $P$ in the $xy$-plane at a distance $r$ from the origin ($r \gg a$). The displacement current density $\vec{J}_d$ at point $P$ due to the electromagnetic fields of the dipole can be expressed in the form $\vec{J}_d = J_{\text{max}} \cos(\omega t + \phi) \hat{\phi}$, where $\hat{\phi}$ is the azimuthal unit vector, $J_{\text{max}} > 0$ is a time-independent scalar amplitude, and $\phi$ is the phase difference relative to the source current $I(t)$. Given that the speed of light in vacuum is $c$, which of the following expressions correctly represents the phase difference $\phi$?

A long ideal solenoid is completely filled with a paramagnetic fluid of uniform magnetic susceptibility $\chi$. A steady current flows through the solenoid's windings, establishing a magnetic field. A particle of mass $m$ and charge $q$ is projected inside the fluid with a speed $v$ ($v \ll c$) in a plane perpendicular to the solenoid's axis. The particle undergoes uniform circular motion of radius $R$. Assuming the fluid does not exert any viscous drag on the particle, what is the magnitude of the magnetization vector $|\vec{M}|$ of the paramagnetic fluid? (Let $\mu_0$ be the permeability of free space)

A non-magnetic semiconductor has a temperature-dependent energy band gap given by $E_g(T) = E_0 - \alpha T$, where $E_0$ and $\alpha$ are positive constants and $T$ is the absolute temperature. The semiconductor is placed inside a blackbody cavity. At a specific equilibrium temperature $T_0$, the peak wavelength of the cavity's thermal radiation, given by Wien's displacement law ($\lambda_{max} T = b$), exactly matches the band-gap wavelength $\lambda_g = \frac{hc}{E_g(T)}$ of the semiconductor. For an electromagnetic wave of this peak wavelength propagating inside the semiconductor, the refractive index is given by $n = n_0 \frac{E_0}{E_g(T_0)}$, where $n_0$ is a dimensionless constant. Let $E_{0}$ and $B_{0}$ be the amplitudes of the electric and magnetic fields of this wave inside the semiconductor. Which of the following expressions correctly represents the ratio $\frac{E_{0}}{B_{0}}$ at temperature $T_0$? (Here $c$ is the speed of light in vacuum, $h$ is Planck's constant, and $b$ is Wien's constant).

A plane electromagnetic wave of angular frequency $\omega$ propagates in the $+x$ direction through a non-magnetic ($\mu = \mu_0$) inhomogeneous lossless dielectric medium. The permittivity of the medium varies spatially as $\epsilon(x) = \epsilon_0 e^{\alpha x}$, where $\alpha > 0$ is a small constant such that the WKB approximation is valid ($\alpha \ll \omega \sqrt{\mu_0 \epsilon_0}$). Let $\langle u_E \rangle$ and $\langle u_B \rangle$ denote the time-averaged electric and magnetic energy densities, respectively, at a position $x$. Which of the following expressions correctly represents the fractional difference $\frac{\langle u_B \rangle - \langle u_E \rangle}{\langle u_E \rangle}$ at position $x$?

Two coherent plane electromagnetic waves, Wave 1 and Wave 2, propagate in a vacuum. Wave 1 propagates along the positive $z$-axis with its electric field given by $\vec{E}_1 = E_0 \cos(kz - \omega t) \hat{i}$. Wave 2 propagates along the positive $y$-axis with its electric field given by $\vec{E}_2 = E_0 \cos(ky - \omega t) \hat{i}$. Let $\langle u \rangle$ represent the time-averaged total energy density of the superposed electromagnetic field at a given point in space. What is the ratio of the maximum time-averaged energy density to the minimum time-averaged energy density, $\frac{\langle u \rangle_{\text{max}}}{\langle u \rangle_{\text{min}}}$, in the region of superposition?

A long straight solid cylindrical conductor of radius $a$ carries a slowly varying current $I(t)$. The material of the conductor exhibits non-Ohmic behavior such that the local electric field $E$ required to drive a current density $J$ is given by $E = \beta J^2$, where $\beta$ is a positive constant. Assume the rate of change of current is small enough that the current density $J$ can be considered uniform across the cross-section at any instant. By considering the macroscopic energy balance (Poynting's theorem), which of the following expressions gives the magnitude of the Poynting vector at the surface of the conductor?

An electromagnetic wave propagates in the $+x$ direction through a non-magnetic inhomogeneous dielectric medium ($\mu = \mu_0$). The medium has a spatially varying permittivity $\epsilon(x)$. At a specific instant $t=0$, the electric and magnetic fields of the wave are given by: $\mathbf{E}(x,0) = E_0 e^{-\alpha x} \cos(kx) \hat{j}$ $\mathbf{B}(x,0) = B_0 e^{-\alpha x} \cos(kx) \hat{k}$ where $E_0, B_0, \alpha,$ and $k$ are positive constants with appropriate SI units. Calculate the instantaneous rate of change of the total electromagnetic energy density, $\frac{\partial u}{\partial t}$, at the position $x = \frac{\pi}{4k}$ at $t=0$.

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a distance $d$ ($d \ll R$). It is being charged by a constant current $I_0$ starting from an uncharged state at $t = 0$. Assume the electric field $\vec{E}$ is uniform between the plates and zero outside. Which of the following expressions correctly gives the magnitude of the Poynting vector $S$ at a radial distance $r$ ($r < R$) from the central axis between the plates, and the total electromagnetic power $P_{in}$ flowing into the cylindrical volume of radius $r$ and height $d$ between the plates at time $t$?

A solid right circular cone of base radius $R$ and height $H = \sqrt{3}R$ is made of a material of uniform density $\sigma$. The cone is completely submerged in a transparent liquid of density $\rho$ (where $\rho < \sigma$). The curved surface of the cone is perfectly reflecting. The cone is oriented with its axis vertical and its apex pointing downwards. A uniform, parallel laser beam of intensity $I$ is directed vertically upwards from below, illuminating the entire curved surface of the cone. If the cone remains in stationary equilibrium, what is the required intensity $I$ of the laser beam? (Assume the momentum flux of the laser beam in the liquid is given by $I/c$, where $c$ is the speed of light).

Two identical dust particles, A and B, each of rest mass $m_0$, are initially at rest in free space. Particle A is located at the origin ($x=0$) and particle B is located at $x=L$. At $t = 0$, particle A begins to emit a perfectly collimated laser beam of constant power $P$ directed along the positive x-axis towards particle B. Particle B perfectly absorbs all the radiation incident on it. Assuming $P t \ll m_0 c^2$ (so that the rest masses of the particles can be considered constant), what is the velocity vector $\vec{v}_{cm}$ of the center of mass of the two-particle system (A and B) for time $t > L/c$? (Here $c$ is the speed of light in vacuum).

A uniform electric field $\mathbf{E}(t) = E_0 \cos(\omega_0 t) \hat{k}$ is maintained in a cylindrical region of space of radius $R$. The region is filled with a non-magnetic, non-conducting dielectric medium whose permittivity is externally modulated such that it varies with time as $\epsilon(t) = \epsilon_0 (1 + \beta \cos(\Omega t))$. Here, $E_0 = 100 \text{ V m}^{-1}$, $\omega_0 = 10^6 \text{ rad s}^{-1}$, $\beta = 0.1$, and $\Omega = 2 \times 10^5 \text{ rad s}^{-1}$. Due to the Ampere-Maxwell law, an induced magnetic field $\mathbf{B}$ is generated inside the cylinder, containing components oscillating at multiple angular frequencies. What is the ratio of the amplitude of the magnetic field component oscillating at angular frequency $\omega_0 + \Omega$ to the amplitude of the component oscillating at angular frequency $\omega_0$?

A stationary space station emits two electromagnetic waves simultaneously in the same direction through a vacuum. The waves have frequencies $f_1 = 8.0 \times 10^{14} \text{ Hz}$ and $f_2 = 4.0 \times 10^{14} \text{ Hz}$. A spacecraft moves away from the station along the direction of wave propagation at a constant speed $v = \frac{3}{5}c$, where $c$ is the speed of light in vacuum. An onboard detector measures the superposition of the two waves. What is the beat frequency (frequency of the intensity maxima) detected by the spacecraft?

A graded-index optical fiber is modeled as a 2D dielectric slab occupying the region $0 \le y \le d$, where $d = 5 \text{ cm}$. The refractive index of the core depends on the transverse coordinate $y$ and the wavelength $\lambda$ as: $n(y, \lambda) = A + \frac{B}{\lambda^2} - k y$ where $A = 1.6$, $B = 2.5 \times 10^{-14} \text{ m}^2$, and $k = 4 \text{ m}^{-1}$. The region $y > d$ is a uniform cladding with refractive index $n_c = 1.5$. A narrow beam of white light enters the core at the origin $(0,0)$ from air ($n_{air} = 1$). The beam is incident on the entry face $x=0$ at an angle $\theta_i = \sin^{-1}(0.8)$ with respect to the normal (x-axis), directed towards the $+y$ region. Which of the following represents the range of wavelengths $\lambda$ that are completely confined within the core (i.e., do not leak into the cladding)?

A perfectly reflecting flat mirror moves with a constant velocity $v$ directly towards a stationary laser source in the laboratory frame. The laser emits a continuous, parallel beam of electromagnetic radiation with intensity $I$ and frequency $\nu$, as measured in the laboratory frame. What is the radiation pressure $P$ exerted by the light beam on the mirror, measured in the laboratory frame? (Assume $c$ is the speed of light in vacuum)

An electromagnetic wave propagates along the $+z$-direction in a linear, isotropic, but non-homogeneous dielectric medium. The medium has a constant electric permittivity $\epsilon = \epsilon_0$ and a spatially varying magnetic susceptibility given by $\chi_m(z) = \alpha z$, where $\alpha > 0$ is a constant with units of $\text{m}^{-1}$. The electric field of the wave is linearly polarized along the $x$-axis and is given by $\vec{E}(z,t) = E(z) e^{-i\omega t} \hat{i}$, where $\omega$ is the angular frequency. If $c$ is the speed of light in vacuum and $k_0 = \omega/c$, which of the following differential equations correctly describes the spatial amplitude $E(z)$?

A transparent liquid is contained in a large tank. Let $y$ represent the depth measured downwards from the top surface ($y=0$). The temperature $T$ of the liquid varies with depth as $T(y) = T_0 - \beta y$, where $\beta > 0$ is a constant. The density $\rho$ of the liquid depends on temperature as $\rho(T) = \rho_0 [1 - \gamma (T - T_0)]$, where $\gamma$ is the coefficient of volume expansion and $\rho_0$ is the density at the surface. The refractive index $n$ of the liquid is related to its density by the Gladstone-Dale relation $n - 1 = C \rho$, where $C$ is a positive constant. A narrow laser beam is injected horizontally into the liquid at the origin $(x=0, y=0)$ in the $+x$ direction. Assuming $\gamma \beta y \ll 1$, which of the following represents the equation of the trajectory $y(x)$ of the beam? (Let $n_0 = 1 + C\rho_0$ be the refractive index at $y=0$)

A point charge $+Q$ is placed at the exact center of a spherical cavity of radius $R$ located inside an uncharged, isolated solid conducting sphere of outer radius $3R$. The cavity is completely filled with two different linear, isotropic dielectric materials: the upper hemisphere ($z > 0$) has a relative permittivity $\epsilon_1 = 2$, and the lower hemisphere ($z < 0$) has a relative permittivity $\epsilon_2 = 4$. Let $u_1$ be the electrostatic energy density in the upper hemisphere at a radial distance $r = R/2$, and $u_2$ be the electrostatic energy density in the lower hemisphere at the same radial distance $r = R/2$. Let $u_3$ be the electrostatic energy density at a radial distance $r = 2R$ (inside the solid conductor). Which of the following represents the correct ratio $u_1 : u_2 : u_3$?

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a distance $d$. The space between the plates is completely filled with a lossy dielectric medium of uniform permittivity $\epsilon$ and uniform conductivity $\sigma$. An alternating voltage $V(t) = V_0 \sin(\omega t)$ is applied across the plates. Neglecting edge effects, what is the amplitude of the magnetic field $B_0$ at a radial distance $r$ ($r < R$) from the central axis of the capacitor?

A plane electromagnetic wave of angular frequency $\omega$ is incident normally from vacuum onto the flat surface of a semiconductor device. The semiconductor lattice has a static relative permittivity $\epsilon_r = 4$ and relative permeability $\mu_r = 1$. When the device is operated in reverse bias (as a photo-diode), the active region is fully depleted of free charge carriers. When operated in forward bias (as an LED), charge carriers are injected into the active region, giving it a uniform conductivity $\sigma = 3 \epsilon_0 \omega$. Assuming the Drude-Lorentz model for the complex permittivity $\tilde{\epsilon} = \epsilon_0 \epsilon_r + i \frac{\sigma}{\omega}$, what is the ratio of the power reflection coefficient under forward bias ($R_F$) to that under reverse bias ($R_R$)?

A small bob of mass $m$ and charge $q$ is suspended by a massless string of length $L$ in a uniform downward gravitational field $g$. The bob is subjected to a weak viscous drag force $\vec{F}_d = -b\vec{v}$, where $\vec{v}$ is its velocity and $b$ is a positive constant. The bob is displaced by a small angle $\theta_0 \ll 1 \text{ rad}$ and released from rest. As the bob oscillates, it emits electromagnetic radiation due to its acceleration. Assuming non-relativistic motion and that the instantaneous power radiated is given by the Larmor formula $P = \frac{q^2 a^2}{6\pi\epsilon_0 c^3}$ (where $a$ is the acceleration, $c$ is the speed of light, and $\epsilon_0$ is the vacuum permittivity), find the time $t$ after which the amplitude of oscillation reduces to $\theta_0 / e$. (Assume the energy loss per cycle is very small compared to the total energy).

An RLC circuit with resistance $R$ and inductance $L$ drives an antenna, emitting a plane electromagnetic pulse. Due to the circuit's damping, the emitted pulse is not monochromatic but has a Lorentzian energy fluence spectrum (energy per unit area per unit angular frequency) given by: $J(\omega) = \frac{J_0}{\pi} \frac{\Gamma}{(\omega - \omega_0)^2 + \Gamma^2}$ where $J_0$ is the total energy fluence, $\omega_0$ is the central frequency, and $\Gamma = \frac{R}{2L}$ is the spectral half-width. This pulse is normally incident on a resonant microscopic solar sail. The sail's effective absorption area depends on the frequency of the incident radiation as: $A(\omega) = A_0 \frac{\gamma^2}{(\omega - \omega_0)^2 + \gamma^2}$ where $A_0$ is the peak area and $\gamma$ is the resonance half-width of the sail. Assuming the sail absorbs the radiation completely according to this profile and reflects nothing, what is the total momentum imparted to the sail? (Assume $\omega_0 \gg \Gamma, \gamma$ so the integration limits can be approximated as $-\infty$ to $\infty$).

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a small distance $d$. The space between the plates is completely filled with a non-uniform dielectric whose permittivity varies radially as $\epsilon(r) = \epsilon_0 \left(1 + 2\frac{r^2}{R^2}\right)$, where $r$ is the distance from the central axis of the plates. A constant conduction current $I_0$ is supplied to the capacitor, charging it. Neglecting edge effects, what is the magnitude of the induced magnetic field $B$ at a radial distance $r = \frac{R}{2}$ from the central axis between the plates? (Assume $I_0$ is in Amperes, $R$ is in meters, and $\mu_0$ is the vacuum permeability in $\text{T}\cdot\text{m/A}$)

An ideal oscillating electric dipole with dipole moment $\vec{p}(t) = p_0 \sin(\omega t) \hat{k}$ is placed at the origin. The surrounding region also contains a uniform time-varying electric field $\vec{E}_c(t) = E_0 \sin(\omega t) \hat{k}$ produced by external sources. Consider a circular Amperian loop of radius $R$ in the $xy$-plane, centered at the origin. To apply the Ampere-Maxwell law without encountering the conduction current singularity of the point dipole at the origin, a student chooses the open surface $S$ bounded by the loop to be the upper hemisphere ($x^2 + y^2 + z^2 = R^2$, $z > 0$). What is the total displacement current $I_d$ passing through the surface $S$ in the outward radial direction $\hat{r}$?

A time-varying electric field $\vec{E}(\rho, t)$ is confined to a cylindrical region of radius $R$ and is directed parallel to the axis of the cylinder. The magnitude of the electric field at a radial distance $\rho$ ($\rho \le R$) from the axis is given by: $E(\rho, t) = E_0 \left( 1 - \frac{\rho}{R} \right) \sin(\omega t) + E_0 \left( \frac{\rho}{R} \right) \sin(2\omega t)$ where $E_0$ and $\omega$ are positive constants. What is the root-mean-square (RMS) value of the induced magnetic field, $B_{rms}$, at a distance $r$ ($r > R$) from the axis of the cylinder?

A region of space ($x > 0$) is filled with a non-uniform, time-varying dielectric medium. The permittivity of the medium is given by $\epsilon(x,t) = \epsilon_0 (1 + \beta x)(1 + \alpha t)$, where $\epsilon_0$ is the vacuum permittivity, and $\alpha$ and $\beta$ are positive constants with units of $\text{s}^{-1}$ and $\text{m}^{-1}$ respectively. The electric field in this region is maintained as $\mathbf{E}(x,t) = E_0 t \hat{i}$, where $E_0$ is a positive constant with units of $\text{V} \cdot \text{m}^{-1} \cdot \text{s}^{-1}$ and $t > 0$ is time. According to the macroscopic Maxwell's equations, which of the following represents the correct expressions for the free volume charge density $\rho_f(x,t)$ and the magnitude of the displacement current density $J_d(x,t)$ in this medium?

An AC voltage source $V(t) = V_0 \sin(\omega t)$ with internal resistance $R$ is connected across the input terminals of an infinite ladder network of capacitors. The ladder consists of series capacitors $C_1 = 2C$ and shunt capacitors $C_2 = C$. The first element connected to the source is a series capacitor $C_1$, whose plates are circular with radius $a$. Given that $R = \frac{1}{\omega C}$, what is the magnetic field $B$ induced by the displacement current at a radial distance $r = a/2$ from the central axis of the capacitor $C_1$ as a function of time? (Assume a uniform electric field between the plates and ignore edge effects).

A rigid non-conducting rod of length $L$ and negligible mass rotates in a horizontal plane about a frictionless vertical axis through one end (the pivot). A central hub at the pivot has a moment of inertia $I_0 = m L^2$. The system is given an initial angular velocity $\omega_0$. A uniform vertical magnetic field $B$ is present everywhere. A small conducting bead of mass $m$ slides outward along the rod at a constant speed $v_0$ relative to the rod, starting from the pivot at $t=0$. A fixed conducting plate of area $A$ is attached to the free end of the rod ($r=L$). The bead acts as the second plate of area $A$, forming a parallel-plate capacitor with a time-varying gap $L - r(t)$. To establish a potential difference, a thin conducting wire connects the pivot to the bead, and another separate thin conducting wire connects the pivot to the fixed end plate. Assume the wires have negligible mass, the electrical energy stored is negligible compared to the kinetic energy, and fringing effects are ignored. What is the magnitude of the displacement current in the gap at the instant the bead passes the midpoint of the rod ($r = L/2$)?

A plane electromagnetic wave propagates in vacuum in the $+x$ direction in the laboratory frame $S$. Its electric field is given by $\vec{E} = E_0 \cos(kx - \omega t) \hat{y}$, where $E_0$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency. An observer in frame $S'$ moves with a constant velocity $\vec{v} = \frac{\sqrt{3}}{2} c \hat{y}$ relative to $S$, where $c$ is the speed of light. Let $\alpha$ be the angle that the electric field vector $\vec{E}'$ makes with the $+x'$ axis in $S'$, and let $u'$ be the time-averaged energy density of the wave in $S'$. Which of the following correctly gives $\tan \alpha$ and the ratio $u'/u$ (where $u$ is the time-averaged energy density in $S$)?

Two electromagnetic waves propagate in a vacuum along the $+z$-axis. Their electric field vectors are given by: $\vec{E}_1(z,t) = E_0 \cos(kz - \omega t) \hat{i}$ $\vec{E}_2(z,t) = 2E_0 \cos(kz - \omega t + \Omega t) \left( \frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j} \right)$ where $E_0$ is a positive constant, $\omega = 3 \times 10^{15} \text{ rad/s}$, and $\Omega = 10^3 \text{ rad/s}$. A detector is placed at $z = 0$ to measure the intensity of the resultant wave. The detector's response time $\tau$ satisfies $\frac{2\pi}{\omega} \ll \tau \ll \frac{2\pi}{\Omega}$. What is the ratio of the maximum to the minimum intensity recorded by the detector?

A plane electromagnetic wave with electric field $\vec{E}_i = E_0 \cos(\omega t - kz) \hat{i}$ is incident normally on a thin, infinite resistive sheet located at $z=0$. The sheet has a uniform surface conductivity $\sigma_s = \frac{4}{\mu_0 c}$, such that the induced surface current density is $\vec{K} = \sigma_s \vec{E}_{sheet}$. Here $\mu_0$ is the vacuum permeability, $\epsilon_0$ is the vacuum permittivity, and $c$ is the speed of light in vacuum. Determine the time-averaged radiation pressure $\langle P \rangle$ exerted on the sheet.

An electromagnetic wave of angular frequency $\omega$ propagates in the $+x$ direction through a non-magnetic dispersive medium ($\mu = \mu_0$). The medium's dielectric response is modeled by an equivalent series LCR circuit. The applied electric field $\vec{E}$ drives a polarization current density $\vec{J}_p$ such that $\vec{J}_p = \vec{E} / Z$, where $Z$ is the complex specific impedance of the medium.

The specific impedance is given by $Z = \rho + i\left(\omega L - \frac{1}{\omega C}\right)$, where $\rho$, $L$, and $C$ are the effective resistivity, specific inductance, and specific capacitance of the medium. The wave frequency is tuned to the resonant frequency of the medium ($\omega = 1/\sqrt{LC}$). At this frequency, the effective resistivity is $\rho = \frac{1}{\sqrt{3} \epsilon_0 \omega}$, where $\epsilon_0$ is the vacuum permittivity.

If the electric field of the wave is $\vec{E}(x,t) = E_0 e^{-\alpha x} \cos(\beta x - \omega t) \hat{y}$, which of the following represents the corresponding magnetic field $\vec{B}(x,t)$? (Here $c = 1/\sqrt{\mu_0 \epsilon_0}$ is the speed of light in vacuum)

A spherical region of radius $R$ in vacuum contains a time-varying, non-uniform volumetric charge density given by $\rho(r,t) = \rho_0 \frac{R}{r} \cos(\omega t)$ for $r \le R$, where $r$ is the radial distance from the center, $\rho_0$ is a positive constant, and $\omega$ is the angular frequency. Using the standard retarded scalar potential formulation (Lorenz gauge), the electric potential at the center of the sphere can be expressed as $V(0,t) = V_0 \cos(\omega t - \delta)$. What are the exact expressions for the amplitude $V_0$ and the phase retardation $\delta$?

Two coherent monochromatic plane electromagnetic waves, Wave 1 and Wave 2, propagate in a vacuum. Wave 1 propagates along the positive $z$-axis with its electric field polarized along the $x$-axis: $\vec{E}_1(z,t) = E_0 \cos(kz - \omega t) \hat{i}$. Wave 2 propagates along the positive $y$-axis with its electric field also polarized along the $x$-axis: $\vec{E}_2(y,t) = E_0 \cos(ky - \omega t) \hat{i}$. Here, $E_0$ is the electric field amplitude in $\text{V/m}$, $k$ is the wave number in $\text{rad/m}$, and $\omega$ is the angular frequency in $\text{rad/s}$. Let $\langle u \rangle$ denote the time-averaged total electromagnetic energy density at a spatial coordinate $(x, y, z)$. What is the ratio of the maximum to the minimum value of $\langle u \rangle$ in the region where the two waves superimpose?

A uniform plane electromagnetic wave traveling in vacuum (characteristic impedance $Z_0 = 120\pi \ \Omega$) along the $+z$-direction is normally incident on a semiconductor interface at $z=0$. The incident electric field is given by $\vec{E}_i = E_0 \cos(\omega t - kz) \hat{x}$. Due to charge carrier dynamics, the semiconductor surface cannot be treated as a perfect conductor. Instead, it is modeled with a complex surface impedance $Z_s = R_d + jX_s$, where the dynamic resistance is $R_d = 60\pi \ \Omega$ and the surface reactance is $X_s = 60\pi\sqrt{3} \ \Omega$ ($j = \sqrt{-1}$). The reflected electric field at the interface is expressed as $\vec{E}_r(z=0) = E_r \cos(\omega t + \phi) \hat{x}$, with $E_r > 0$ and $\phi \in (-\pi, \pi]$. What is the value of the phase shift $\phi$?

A high-frequency electromagnetic wave propagates in the $+z$ direction from vacuum ($z < 0$) into a non-magnetic, lossless dielectric medium ($z \ge 0$). The relative permittivity of the medium varies spatially as $\epsilon_r(z) = \left(1 + \frac{z}{L}\right)^2$, where $L$ is a positive constant. The amplitude of the electric field of the wave at the boundary $z=0$ is $E_0$. Assuming the frequency of the wave is sufficiently high such that the WKB approximation is valid (i.e., internal reflections are negligible), what is the amplitude of the magnetic field at a depth $z = 3L$? (Here $c$ is the speed of light in vacuum).

An electromagnetic wave with electric field amplitude $E_0$ propagates through a non-magnetic ($\mu = \mu_0$) ideal monatomic gas at a uniform absolute temperature $T$. The gas acts as a dielectric medium with a temperature-dependent permittivity $\epsilon(T) = \epsilon_0 \left( 1 + \frac{C}{T} \right)$, where $C$ is a positive constant and $\epsilon_0$ is the vacuum permittivity. According to the thermodynamics of dielectrics, the electric internal energy density of a field $E$ in such a medium is $u_E = \frac{1}{2} \left( \epsilon + T \frac{d\epsilon}{dT} \right) E^2$. The total time-averaged thermodynamic internal energy density of the electromagnetic wave (sum of electric and magnetic energy densities) is found to be exactly equal to the thermal energy density of the gas. Given that the number density of the gas atoms is $n$ and $k_B$ is the Boltzmann constant, which of the following expressions gives the correct value of $E_0$?

A region of vacuum contains a uniform static electric field $\vec{E}_0 = E_0 \hat{i}$ and a uniform static magnetic field $\vec{B}_0 = B_0 \hat{k}$. The magnitudes of these fields are such that their energy densities in the vacuum are equal. A large flat slab of a linear diamagnetic material with relative permeability $\mu_r = 0.80$ and relative permittivity $\epsilon_r = 1.0$ is introduced into this region. The slab is oriented such that its two parallel faces are perpendicular to the $z$-axis. Let $u_E$ and $u_B$ be the electric and magnetic energy densities inside the diamagnetic slab, respectively. What is the ratio $\frac{u_B}{u_E}$?

A solid non-conducting sphere of radius $R$ has a non-uniform volume charge density given by $\rho(r) = \rho_0 \left(\frac{r}{R}\right)$ for $r \le R$, and $\rho(r) = 0$ for $r > R$, where $\rho_0$ is a positive constant and $r$ is the radial distance from the center of the sphere. Let $U_{in}$ be the total electrostatic potential energy stored in the electric field within the volume of the sphere ($r \le R$), and $U_{out}$ be the total electrostatic potential energy stored in the electric field outside the sphere ($r > R$). What is the ratio $\frac{U_{in}}{U_{out}}$?

A perfectly absorbing spherical dust particle of mass $m$ and radius $R$ is in an unperturbed elliptical orbit around a star of mass $M$. The orbit has a semi-major axis $a$ and eccentricity $e$. A uniform, parallel photon beam of constant intensity $I$ from a distant quasar illuminates the system. The beam propagates in the plane of the orbit and is perpendicular to the major axis of the ellipse. Assuming the radiation force on the particle is a very small perturbation compared to the gravitational force from the star, and that the star does not shadow the particle, find the magnitude of the change in the particle's orbital angular momentum $|\Delta L|$ over one complete revolution.

(Given: $c$ is the speed of light in vacuum, $G$ is the universal gravitational constant)