Magnetism and Matter - Class 12

A conducting rod of length $L$ is pivoted at one end $O$ and rotates with a constant angular velocity $\omega$ in a horizontal plane. Its free end makes sliding contact with a stationary, perfectly conducting circular ring of radius $L$ centered at $O$. A stationary straight conducting wire connects $O$ to a fixed point $P$ on the ring. The entire system is placed in a magnetic field perpendicular to the plane of the ring. The magnetic field is a superposition of two components:

1. A uniform, constant magnetic field of magnitude $B_0$ present everywhere.
2. A time-varying magnetic field of magnitude $B_1(t) = \alpha t$ (where $\alpha > 0$) present only within a cylindrical region of radius $R$ ($R < L$) centered at $O$. Both magnetic fields point in the same direction. At $t=0$, the rotating rod coincides with the fixed wire $OP$. What is the magnitude of the total induced electromotive force (EMF) in the closed loop formed by the rod, the fixed wire $OP$, and the arc of the ring swept by the rod at time $t$?

An infinitely long cylindrical hard magnetic core of radius $R$ (in meters) has a frozen-in, non-uniform radial magnetization given by $\vec{M} = M_0 \left(\frac{r}{R}\right) \hat{r}$, where $r$ is the radial distance from the cylinder's central axis, $\hat{r}$ is the radial unit vector, and $M_0$ is a positive constant (in $\text{A/m}$). A straight, thin wire carrying a steady free current $I$ (in Amperes) is placed along the central axis of the cylinder. Which of the following represents the magnitude of the magnetic induction $\vec{B}$ (in Teslas) at a radial distance $r = \frac{R}{2}$ inside the core? (Assume vacuum permeability is $\mu_0$ in $\text{T}\cdot\text{m/A}$)

A parallel plate capacitor consists of two solid circular conducting plates of radius $R$ separated by a small gap $d$ ($d \ll R$). The space between the plates is filled with a solid dielectric material of dielectric constant $K$, except for a cylindrical cavity of radius $a$ ($a < R$). The axis of the cavity is parallel to the central axis of the capacitor but displaced by a distance $c$ (such that $c + a < R$). A time-varying potential difference $V(t)$ is applied across the plates such that $\frac{dV}{dt} = \alpha$, where $\alpha$ is a positive constant. Neglecting edge effects, what is the magnitude of the magnetic field at the center of the cylindrical cavity?

A cylindrical bar magnet of length $L$ and radius $R$ ($R \ll L$) has a non-uniform magnetization. It is modeled as a tightly wound solenoid with an azimuthal effective surface current density $K(x) = c x^2$, where $x$ is the distance from the center of the magnet along its axis ($-L/2 \le x \le L/2$) and $c$ is a positive constant. The total magnetic moment of the magnet is $M$. The magnetic field at a point on the axis at a distance $r$ ($r \gg L$) from the center can be expressed as $B = \frac{\mu_0 M}{2\pi r^3} \left( 1 + \beta \frac{L^2}{r^2} + \mathcal{O}\left(\frac{L^4}{r^4}\right) \right)$ Find the value of the dimensionless constant $\beta$.

A rigid, non-conducting rotor with moment of inertia $I_0 = 2mR^2$ about its axis of rotation is constrained to rotate freely about the $z$-axis in the horizontal plane $z = R$. A fixed magnetic dipole of magnetic moment $\vec{M} = M_0 \hat{k}$ is located at the origin. Two small beads, each of mass $m$ and charge $+q$, are initially held at the center of the rotor (on the $z$-axis). The entire system is initially rotating with an angular velocity $\omega_0 \hat{k}$. The beads are then released and slide symmetrically outward along the rotor. They come to rest relative to the rotor at its ends, at a radial distance $R$ from the $z$-axis. Assuming no mechanical friction opposes the rotation of the rotor, find the final angular velocity $\omega_f$ of the system.

A bar magnet has a magnetic length $2a$ and a constant magnetic moment $m$. The magnetic field magnitudes at an axial point and an equatorial point, both at a distance $r$ from the center of the magnet, are $B_{\text{axial}}$ and $B_{\text{eq}}$ respectively. A student evaluates the ratio $K = \frac{B_{\text{axial}}}{B_{\text{eq}}}$ for a specific distance $r = 2a$. If the maximum percentage errors in the measurement of $r$ and $a$ are both $1.5\%$, what is the maximum percentage error in the calculated value of $K$?

A short magnetic dipole of magnetic moment $\vec{m} = m_0 \hat{k}$ (where $m_0 > 0$ is in $\text{A}\cdot\text{m}^2$) is placed at the origin of a Cartesian coordinate system. In the same region, an electrostatic field is established in free space such that its electric potential is given by $V(x,y,z) = V_0 (x^2 + y^2 - 2z^2)$, where $V_0 > 0$ is a constant in $\text{V}\cdot\text{m}^{-2}$ and $x, y, z$ are in meters.

Consider a point $P$ in space, with a position vector making an angle $\theta$ ($0 < \theta < \pi/2$) with the positive $z$-axis. It is observed that at point $P$, the magnetic field vector $\vec{B}$ is strictly normal to the electrostatic equipotential surface passing through $P$.

What is the value of $\cos\theta$?

Region 1 ($z > 0$) is vacuum with magnetic permeability $\mu_0$. Region 2 ($z < 0$) is filled with a linear, isotropic magnetic material having a magnetic susceptibility $\chi_m = 3$. A uniform magnetic field exists in both regions. In Region 1, the magnetic field is given by $\mathbf{B}_1 = B_0 (\hat{i} + \sqrt{3}\hat{k})$, where $B_0$ is a positive constant in Tesla. Assuming there are no free surface currents on the interface at $z = 0$, what is the ratio of the magnetic energy density in Region 2 to the magnetic energy density in Region 1, $\frac{u_2}{u_1}$?

A toroidal ring of mean circumference $L = 0.50 \text{ m}$ and uniform cross-sectional area is made of a ferromagnetic material with relative permeability $\mu_r = 2000$. A narrow air gap of width $d = 1.0 \text{ mm}$ is cut into the ring. A coil of $N = 1000$ turns is wound uniformly around the ring and carries a steady current $I = 1.0 \text{ A}$. Assuming the magnetic flux is entirely confined within the core and the gap (ignoring fringing effects), what is the magnitude of the magnetic field $B$ in the air gap?

(Given: Permeability of free space $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$)

An AC voltage source $V(t) = V_0 \sin(\omega t)$ is connected in series with a capacitor of capacitance $C$ and an inductor of inductance $L$ having a ferromagnetic core. The ohmic resistance of the entire circuit is negligible. Due to hysteresis, the ferromagnetic core dissipates an energy $W_h = 2\pi \beta I_{\text{rms}}^2$ per cycle, where $\beta$ is a positive constant with appropriate SI units and $I_{\text{rms}}$ is the root-mean-square current. The angular frequency $\omega$ of the source is continuously varied. It is observed that for a specific fixed value of $C$, the current amplitude in the circuit attains its maximum possible value exactly at the angular frequency $\omega = \omega_0$. Which of the following expressions correctly gives the required capacitance $C$?

A ferromagnetic core of volume $V = 2 \times 10^{-3} \text{ m}^3$ is subjected to an alternating magnetic field of frequency $f = 40 \text{ Hz}$. The area of the hysteresis loop for the material is $A_{BH} = 816 \text{ J m}^{-3}$ per cycle. Assume eddy current losses are negligible. The entire thermal power generated due to the hysteresis loss is harnessed to continuously excite a sample of $\text{He}^+$ ions from their ground state to the first excited state. Calculate the number of $\text{He}^+$ ions excited per second.

(Given: $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$, Ground state energy of a Hydrogen atom $E_1 = -13.6 \text{ eV}$)

A magnetic core of mass $m$ (in $\text{kg}$) and volume $V$ (in $\text{m}^3$) is attached to a horizontal spring of force constant $k$ (in $\text{N/m}$) and oscillates on a frictionless surface. During its motion, the core passes through the equilibrium position where the external magnetic field reverses its direction spatially. This causes the core material to undergo exactly one full major hysteresis loop per mechanical cycle. The hysteresis loop of the core material is approximated as a rectangle with coercive force $H_c$ (in $\text{A/m}$) and remanent magnetic field $B_r$ (in $\text{T}$). Assume the energy dissipated per cycle is small compared to the total mechanical energy, and the core is always driven to saturation. If the initial amplitude of oscillation is $A_0$ (in $\text{m}$), which of the following expressions correctly represents the amplitude $A(t)$ (in $\text{m}$) as a function of time $t$ (in $\text{s}$) before the core comes to rest?

A thin toroidal coil of mean circumference $l = 1.0 \text{ m}$ and cross-sectional area $A = 1.0 \times 10^{-3} \text{ m}^2$ has $N = 1000$ turns of wire uniformly wound on a ferromagnetic core. The core material exhibits a non-linear magnetic behavior (ignoring hysteresis) where the magnetic field $B$ is related to the magnetic intensity $H$ by the equation $B = \frac{B_s H}{H_c + H}$ for $H > 0$. Here, the saturation magnetic field is $B_s = 2.0 \text{ T}$ and the characteristic magnetic intensity is $H_c = 4000 \text{ A/m}$. A time-varying current $I(t) = \alpha t$ is passed through the coil, where $\alpha = 2.0 \text{ A/s}$ and $t$ is the time in seconds. Calculate the magnitude of the induced EMF in the coil at $t = 2.0 \text{ s}$.

A large circular wire loop of radius $R$ lies in the $xy$-plane, centered at the origin, and carries a steady current $I$ in the counter-clockwise direction (when viewed from the positive $z$-axis). A small hard ferromagnetic sphere is fixed at the origin. It possesses a residual magnetic dipole moment $\vec{M}_r = \frac{\pi I R^2}{\sqrt{2}} \hat{i}$. A small test loop of area $A$ carrying a steady current $i$ is placed at the point $(0, 0, R)$. The test loop is free to rotate in 3D space. Assuming the test loop is small enough that the magnetic field over its area is uniform, what is the unit normal vector $\hat{n}$ of the test loop (indicating the direction of its magnetic moment) when it is in stable equilibrium?

A small solid diamagnetic sphere of mass $M$ and volume $V$ is attached to one end of a massless rigid rod of length $L$. The other end of the rod is pivoted at the origin $O$, allowing the sphere to rotate freely in the $xy$-plane. The system is placed in a static, non-uniform magnetic field given by $\vec{B} = B_0 (1 + \alpha x) \hat{i} - B_0 \alpha y \hat{j}$, where $B_0$ and $\alpha$ are positive constants. The induced magnetic dipole moment of the sphere is $\vec{m} = -\frac{\chi V}{\mu_0} \vec{B}$, where $\chi > 0$ is the magnitude of its diamagnetic susceptibility. If the sphere is projected from the position $(L, 0)$ with an initial angular velocity $\omega_0 \hat{k}$, what is its angular velocity $\omega$ when it crosses the position $(-L, 0)$? Assume no gravity and no other resistive forces.

A parallel plate capacitor consists of two square plates of area $A$ separated by a distance $d$. The plates carry constant free surface charge densities $+\sigma_0$ at $x=0$ and $-\sigma_0$ at $x=d$. The space between the plates is completely filled with a solid linear material that is both dielectric and diamagnetic. The relative permittivity of the material varies spatially as $\epsilon_r(x) = \frac{1}{1 - \alpha x}$ (where $\alpha d < 1$). The material has a uniform diamagnetic susceptibility $\chi_m = -\chi_0$ (where $0 < \chi_0 \ll 1$). An external non-uniform magnetic field $\vec{B}_{ext} = B_0 \sqrt{1 - \alpha x} \hat{k}$ is applied parallel to the plates. Calculate the total interaction potential energy $U_{int}$ of the induced electric and magnetic dipoles with the local fields, defined as $U_{int} = -\frac{1}{2} \int (\vec{P} \cdot \vec{E}) dV - \frac{1}{2} \int (\vec{M} \cdot \vec{B}) dV$, where $\vec{E}$ and $\vec{B}$ are the total local fields inside the material. (Assume $\chi_0$ is small enough that $1 - \chi_0 \approx 1$).

A uniform cylindrical rod of an isotropic diamagnetic material is perfectly constrained between two rigid, immovable walls at its flat ends. The material has Young's modulus $Y$, Poisson's ratio $\nu$, and coefficient of linear thermal expansion $\alpha$. The initial volume magnetic susceptibility of the rod at temperature $T$ is $\chi_0$ (where $\chi_0 < 0$). The temperature of the rod is increased by a small amount $\Delta T$. According to a theoretical model, the mean square atomic radius $\langle r^2 \rangle$ of the atoms in the material scales with the macroscopic volume $V$ of the rod as $\langle r^2 \rangle \propto V^{2/3}$. Find the change in the volume susceptibility $\Delta \chi$ of the rod. (Assume $\alpha \Delta T \ll 1$).

A rectangular container of length $L$ is placed along the $x$-axis from $x = 0$ to $x = L$. It is filled with an incompressible diamagnetic fluid of density $\rho$ and magnetic susceptibility $\chi$ (where $\chi < 0$). The system is permeated by a non-uniform magnetic field $\vec{B} = B_0 x \hat{k}$, where $B_0$ is a positive constant and $\hat{k}$ is the unit vector in the vertically upward $z$-direction. The container is given a constant horizontal acceleration $\vec{a} = -a_0 \hat{i}$ ($a_0 > 0$). In the steady state, the free surface of the fluid is observed to be perfectly horizontal at the edge $x = L$. Assuming the fluid does not spill, what is the difference in height of the fluid surface between the edges $x = L$ and $x = 0$, given by $\Delta h = h(L) - h(0)$? (Here $g$ is the acceleration due to gravity and $\mu_0$ is the vacuum permeability).

A perfectly rigid, macroscopic insulating ring of mass $M$ and radius $R$ is suspended by a torsionless thread in a region of zero initial magnetic field. The ring contains $N$ identical, non-interacting atoms. Each atom is modeled as a fixed nucleus of charge $+e$ and a single electron of mass $m_e$ and charge $-e$ in a spherically symmetric orbit with mean square radius $\langle r^2 \rangle = a^2$ (where $a \ll R$). A uniform magnetic field $\vec{B} = B_0 \hat{k}$ is slowly turned on. What is the net angular momentum $\Delta \vec{L}_{\text{total}}$ acquired by the entire ring system (lattice + electrons) due to the induced electric field?

Consider a hydrogen-like atom consisting of a nucleus of mass $M$ and charge $+Ze$, and an electron of mass $m$ and charge $-e$. In the Bohr model, both the nucleus and the electron revolve in circular orbits about their common center of mass. Let $L$ be the magnitude of the total orbital angular momentum of the system and $\mu$ be the magnitude of the total magnetic dipole moment of the system. Assuming $M > \sqrt{Z}m$, which of the following expressions correctly represents the gyromagnetic ratio $\gamma = \frac{\mu}{L}$ of this atom?

An electron of mass $m$ (in kg) and charge $-e$ (in C) orbits a massive nucleus in an elliptical orbit. The orbit has a semi-major axis $a$ (in m), eccentricity $\epsilon$ (dimensionless), and time period $T$ (in s). The nucleus is located at one of the foci of the ellipse. What is the magnitude of the instantaneous magnetic field $B$ (in T) produced by the electron at the nucleus when the electron is at its apocenter (farthest point from the nucleus)? Assume non-relativistic motion and vacuum permeability $\mu_0$ (in $\text{T}\cdot\text{m/A}$).

A solid non-conducting sphere of radius $R$ has a uniform positive volume charge density $\rho$. The sphere rotates with a constant angular velocity $\omega_s \hat{k}$ about the z-axis. An electron (mass $m$, charge $-e$) orbits in a stable circular path of radius $r$ ($r \ll R$) in the equatorial $xy$-plane, centered at the origin, with an orbital angular velocity $\omega \hat{k}$ (where $\omega > 0$). Assume the magnetic field produced by the rotating sphere is uniform inside and equal to its value at the center. It is given that the parameters satisfy the relations $\frac{e \rho}{3 m \epsilon_0} = 2 \Omega^2$ and $\frac{e \mu_0 \rho \omega_s R^2}{3 m} = \Omega$, where $\Omega$ is a positive constant. What is the magnitude of the magnetic dipole moment of the orbiting electron?

A uniform solid sphere of mass $m \text{ (kg)}$ and radius $R \text{ (m)}$ has a total charge $q \text{ (C)}$ uniformly distributed throughout its volume. It is initially spinning with an angular velocity $\omega_0 \text{ (rad/s)}$ about an axis passing through its center. Uncharged cosmic dust falls radially towards the center of the sphere and sticks to its surface. This process continues slowly until the dust forms a uniform thin spherical shell of mass $m \text{ (kg)}$ and radius $R \text{ (m)}$ (assuming the thickness of the dust layer is negligible). No external torques act on the system during this process. If the initial magnetic dipole moment of the sphere is $\mu_0 \text{ (A}\cdot\text{m}^2\text{)}$, what is the final magnetic dipole moment $\mu_f$ of the system after the dust has settled?

A microscopic magnetic dipole with magnetic moment $\mu$ and gyromagnetic ratio $\gamma$ is located at the origin. A uniform static magnetic field $\vec{B} = B_0 \hat{k}$ is present in the region. The dipole's axis makes a constant angle $\theta$ with the $z$-axis and precesses around it. A stationary circular coil of radius $R$ ($R$ is much larger than the dipole's dimensions), having $N$ turns and self-inductance $L_0$, is placed in the $xz$-plane, centered at the origin. The coil is connected to a capacitor of capacitance $C$ to form a closed LC circuit with negligible resistance. The capacitance $C$ is perfectly tuned such that the natural frequency of the LC circuit matches the Larmor precession frequency of the dipole. Assuming the circuit is completely unenergized at $t=0$, which of the following expressions represents the total electromagnetic energy $U(t)$ accumulated in the LC circuit as a function of time $t$? (Assume $t \gg \frac{1}{\gamma B_0}$ and the dipole's precession angle $\theta$ remains constant).

A rigid, non-conducting rod of length $L$ is pivoted at one end ($x=0$) and is free to rotate in a horizontal plane. The rod has a non-uniform mass density $\lambda_m(x) = \lambda_0 x$ and a non-uniform charge density $\lambda_q(x) = \alpha x$, where $x$ is the distance from the pivot. The total mass of the rod is $M$ and its total charge is $Q$. Initially, the rod is at rest. A uniform magnetic field, perpendicular to the plane of rotation, is switched on and slowly increased from $0$ to a steady value $B$. Assuming no friction, no radiation losses, and that all quantities are in SI units, find the final kinetic energy $K$ acquired by the rod and the motional electromotive force $\mathcal{E}$ developed across its ends in the steady state.

A thin non-conducting disk of radius $R$ lies in the $xy$-plane and rotates with a constant angular velocity $\vec{\omega} = \omega \hat{k}$ about the $z$-axis. The disk has a non-uniform mass surface density $\sigma_m = \alpha r$ and a non-uniform charge surface density $\sigma_q = \beta r^2$, where $r$ is the radial distance from the centre, and $\alpha, \beta$ are positive dimensional constants. If $Q$ is the total charge and $M$ is the total mass of the disk, determine the effective gyromagnetic ratio $\gamma = \frac{|\vec{\mu}|}{|\vec{L}|}$ of the disk, where $\vec{\mu}$ is the magnetic dipole moment and $\vec{L}$ is the angular momentum about the $z$-axis. Assume all quantities are in standard SI units.

A rigid, non-conducting thin rod of mass $M$ and length $L$ is hinged frictionlessly at the origin $(0,0)$ and is free to rotate in the $xy$-plane. The rod has a non-uniform linear mass density $\lambda(x) = \frac{2M}{L^2}x$, where $x$ is the distance from the hinge. It possesses a permanent magnetic dipole moment of magnitude $\mu$, directed along its length from the hinge to the free end. The rod initially hangs at rest vertically downwards along the $-\hat{j}$ axis. A uniform, constant magnetic field $\vec{B}_0 = B_0 \hat{j}$ exists in the region, and gravity acts along $-\hat{j}$. A short magnetic field pulse $\vec{B}_{pulse}(t) = B_p(t) \hat{i}$ is applied for a negligible duration $\Delta t$, such that $\int_0^{\Delta t} B_p(t) dt = \beta$. If the rod swings upwards and comes to instantaneous rest exactly at the horizontal position ($+\hat{i}$ axis), which of the following gives the value of $\beta^2$? (Assume $\frac{2}{3} M g L > \mu B_0$)

A small magnetic compass needle of magnetic moment $M = 0.01 \text{ A m}^2$ and moment of inertia $I = 2.0 \times 10^{-6} \text{ kg m}^2$ is suspended in a uniform magnetic field of magnitude $B$. The needle undergoes angular simple harmonic motion in a plane parallel to the magnetic field. The oscillations are observed through a telescope with an objective lens of diameter $D = 1.22 \text{ cm}$ using light of wavelength $\lambda = 600 \text{ nm}$. The amplitude of oscillation $\theta_0$ is such that the extreme position of the needle is just resolvable from its mean equilibrium position according to the Rayleigh criterion. If the maximum angular speed of the needle during the oscillation is $\Omega_{max} = 3.0 \times 10^{-3} \text{ rad s}^{-1}$, what is the magnitude of the magnetic field $B$?

A turntable rotates with a constant angular velocity $\Omega$ about the vertical $Z$-axis. A vertical frame is rigidly fixed to the turntable. A uniform thin rod of mass $M$, length $2L$, and magnetic dipole moment $\mu$ (directed along its length) is pivoted at its center to this frame, such that it can only rotate within the vertical plane of the frame. A uniform magnetic field $\vec{B}$ exists in the vertically upward direction. Let $I$ be the moment of inertia of the rod about its pivot. If the turntable's angular velocity is set to $\Omega = \sqrt{\frac{2\mu B}{I}}$, what is the angular frequency of small oscillations of the rod about its stable equilibrium position in the rotating frame?

A spherical ferromagnetic rotor of volume $V = 1.0 \times 10^{-3} \text{ m}^3$ and moment of inertia $I = 5.0 \times 10^{-3} \text{ kg m}^2$ is free to rotate about a fixed vertical axis. A uniform horizontal magnetic field of magnitude $B = 2.0 \text{ T}$ is applied and forced to rotate about the same vertical axis at a constant angular velocity $\omega_B = 150 \text{ rad/s}$. Due to magnetic hysteresis, the rotor's magnetization vector $\vec{M}$ maintains a constant magnitude $M = 1.0 \times 10^4 \text{ A/m}$ but lags behind the rotating magnetic field $\vec{B}$ by a constant angle $\delta = 30^\circ$. The rotor experiences a viscous drag torque $\tau_d = -b \omega$, where $b = 0.2 \text{ N m s}$ and $\omega$ is the rotor's angular velocity. Once the rotor reaches its steady-state terminal angular velocity $\omega_0$, what is $\omega_0$ and the rate of internal heat generation $P_h$ within the rotor strictly due to magnetic hysteresis?

A solid sphere of radius $R$ centered at the origin possesses a non-uniform permanent magnetization $\vec{M} = M_0 \left( \frac{r}{R} \right) \hat{k}$, where $r = \sqrt{x^2+y^2+z^2}$ is the radial distance from the origin, $M_0$ is a positive constant, and $\hat{k}$ is the unit vector along the $z$-axis. There are no free currents in the region. What is the magnetic induction $\vec{B}$ at the point $(0, 0, R/2)$?

Three identical perfectly conducting circular loops, each having a self-inductance $L$, are fixed in space with their planes parallel and their centers forming an equilateral triangle. The mutual inductance between any pair of loops is $M$. Loop 1 contains a capacitor of capacitance $C$ initially charged to a voltage $V_0$, and an open switch. Loops 2 and 3 are continuous, short-circuited loops with no active components. At $t=0$, the switch in Loop 1 is closed. Assuming the ohmic resistance of all loops is strictly zero, what is the maximum magnitude of the induced current in Loop 2 during the subsequent LC oscillations?

A steady magnetic field in a region of space is established by a specific current distribution and is given by the expression $\vec{B} = B_0 \left[ \left(\frac{x}{a}\right)^2 \hat{j} - \left(\frac{y}{a}\right)^2 \hat{i} \right]$ where $B_0$ is a positive constant in Teslas and $a$ is a positive constant in meters. A square loop of side $a$ is placed in the $xy$-plane with its vertices at $(0,0,0)$, $(a,0,0)$, $(a,a,0)$, and $(0,a,0)$ (coordinates in meters). What is the magnitude of the net electric current passing through the surface bounded by this loop?

A particle of mass $m$ and charge $+q$ enters the origin $(0,0,0)$ at time $t=0$ with a velocity $\vec{v} = 0.8c \hat{i}$, where $c$ is the speed of light in vacuum. The region $x > 0$ contains a static uniform magnetic field $\vec{B}_{\text{static}} = B_0 \hat{k}$ and a plane electromagnetic wave propagating in the $+x$ direction. The magnetic field component of the electromagnetic wave is given by $\vec{B}_{\text{wave}} = B_1 \cos(kx - \omega t) \hat{k}$. If the particle experiences zero initial acceleration at $t=0$, what is the required value of the static magnetic field $B_0$?

A thin non-conducting rod of length $L$ is placed along the positive $x$-axis with its ends at $x = 0$ and $x = L$. The rod possesses a non-uniform magnetic moment density (magnetic moment per unit length) given by $\vec{\lambda}(x) = \alpha x \hat{i}$, where $\alpha$ is a positive constant with units of Ampere. Find the magnetic field $\vec{B}$ at a point $P$ located on the $x$-axis at $x = d$ (where $d > L$). Assume the surrounding medium is vacuum with permeability $\mu_0$.

A small magnetic dipole of magnetic moment $m$ and moment of inertia $I$ is pivoted at the origin and is free to rotate in the $xy$-plane. A uniform external magnetic field $\vec{B} = B_0 \hat{j}$ is present in the region. A stationary circular conducting ring of radius $R$, electrical resistance $r$, and negligible self-inductance is placed parallel to the $xz$-plane, with its center fixed at $(0, d, 0)$. The dipole is given a small initial angular displacement $\theta_0 \ll 1$ from its equilibrium position (where $\vec{m}$ is aligned with $\vec{B}$) and released. Assuming the energy dissipated per cycle is very small (so the amplitude decreases negligibly during a single cycle), what is the approximate mechanical energy dissipated in the ring during the first complete oscillation cycle?

A short magnetic dipole of magnetic moment $M_1$ is fixed at the origin of a coordinate system. Another short magnetic dipole of magnetic moment $M_2$ and mass $m$ is placed on the axis of the first dipole at a distance $r_0$ from it. Both dipoles are aligned in the same direction along the axis. If the second dipole is released from rest, what is its speed $v$ when it reaches a distance of $\frac{r_0}{2}$ from the first dipole? (Assume no other forces act on the system and $\mu_0$ is the permeability of free space).

A magnetic scalar potential $V_m(r, \theta)$ is established in the vacuum region $R \le r \le 3R$ between two concentric spherical shells. The potential is given by $V_m(r, \theta) = B_0 \left( \frac{R^3}{r^2} + r \right) \cos\theta$ where $B_0$ is a positive constant with units of Tesla, $r$ is the radial distance from the origin, and $\theta$ is the polar angle measured from the z-axis. The magnetic field is defined as $\vec{B} = -\nabla V_m$. Calculate the magnitude of the magnetic field $\vec{B}$ at a point $P$ located at $r = R$ and $\theta = \pi/3$.

An electron (charge $-e$, effective mass $m^*$) is confined within a bulk semiconductor crystal and moves in a circular Bohr orbit corresponding to the principal quantum number $n=1$. The semiconductor medium is isotropic and linear, with a relative magnetic permeability $\mu_r$ and relative permittivity $\epsilon_r$. Assuming the Bohr model quantization of angular momentum applies, what is the magnitude of the effective macroscopic magnetic dipole moment of this system (including the contribution from the medium's induced magnetization)? Here, $h$ is Planck's constant.

A small, thin disk of volume $V_0$ made of a linear dielectric material with relative permittivity $\epsilon_r$ is placed in a non-uniform electric field $\vec{E} = E(x) \hat{i}$. Let $F_{e1}$ be the net electric force on the disk when its normal vector $\hat{n}$ is parallel to the $x$-axis, and $F_{e2}$ be the force when $\hat{n}$ is perpendicular to the $x$-axis.

Similarly, an identical disk made of a linear paramagnetic material with relative permeability $\mu_r$ is placed in a non-uniform magnetic field $\vec{B} = B(x) \hat{i}$. Let $F_{m1}$ be the net magnetic force on the disk when its normal vector $\hat{n}$ is parallel to the $x$-axis, and $F_{m2}$ be the force when $\hat{n}$ is perpendicular to the $x$-axis.

Assume the disk is small enough that the external fields and their gradients are uniform over its volume. Which of the following gives the correct ratios $\frac{F_{e1}}{F_{e2}}$ and $\frac{F_{m1}}{F_{m2}}$?

A cylindrical ferromagnetic core of length $L$ and cross-sectional area $A$ is rigidly clamped between two immovable walls. The core is subjected to an alternating magnetic field of frequency $f$. The area of the hysteresis loop (B-H curve) for the material is $W_h$ (in $\text{J m}^{-3}$). The material has density $\rho$, specific heat capacity $c$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$. Assuming all hysteresis loss is converted into internal heat, no heat is lost to the surroundings, and eddy current losses are negligible, what is the time $t$ required for the compressive thermal stress in the core to reach a magnitude of $\sigma_0$?

A series RLC circuit is driven by an AC voltage source of frequency $f = 50 \text{ Hz}$ such that the RMS current in the circuit is $I_{rms} = 2 \text{ A}$. The circuit consists of a capacitor of capacitance $C = \frac{1}{600\pi} \text{ F}$ and a real inductor with a ferromagnetic core. The inductor coil has a DC wire resistance $R_w = 2 \ \Omega$ and an inductance $L = \frac{0.14}{\pi} \text{ H}$. The ferromagnetic core has a volume $V = 10^{-3} \text{ m}^3$. At this operating current, the area of the B-H hysteresis loop for the core is $A_h = 480 \text{ J m}^{-3} \text{ cycle}^{-1}$. What is the power factor of the complete circuit?

A cylindrical ferromagnetic rotor of moment of inertia $I = 0.05 \text{ kg m}^2$ and volume $V = 2 \times 10^{-4} \text{ m}^3$ is spinning freely at an initial angular velocity $\omega_0 = 120 \text{ rad/s}$ inside a stator. The stator produces a constant-magnitude magnetic field that causes the rotor material to undergo exactly one complete B-H hysteresis cycle per mechanical revolution. The area of the B-H hysteresis loop for the rotor material is $A_H = 500\pi \text{ J/m}^3$. Assuming all other losses (such as bearing friction and eddy currents) are negligible, what is the time $t$ taken for the rotor to come to a complete stop?

An infinite composite structure consists of alternating planar sheets of a ferromagnetic material (thickness $d_1 = 3 \text{ mm}$) and a non-magnetic insulator (thickness $d_2 = 2 \text{ mm}$). The structure is placed inside a long solenoid carrying a slowly varying alternating current, which generates a nominal macroscopic external magnetic field intensity $H_{\text{ext}} = nI$ perpendicular to the sheets. The uniform magnetic induction $B$ inside the structure is measured, and a hysteresis loop of $B$ versus $H_{\text{ext}}$ is plotted for one complete cycle. If the area of this apparent hysteresis loop is $A_{\text{app}} = 150 \text{ J/m}^3$, what is the actual energy dissipated as heat per unit volume of the ferromagnetic material per cycle?

An LC oscillator circuit consists of an ideal capacitor of capacitance $C$ and an inductor of inductance $L$. The inductor is a solenoid of length $l$, cross-sectional area $A$, and $N$ turns, completely filled with a perfectly laminated magnetic core of permeability $\mu$ (such that $L = \frac{\mu N^2 A}{l}$). The core material exhibits a temperature-dependent hysteresis B-H loop, transitioning from a soft to a hard magnetic material as the absolute temperature $T$ increases. The area of the B-H loop (representing the energy dissipated per unit volume per cycle) is given by $A_H = \frac{\alpha T}{\mu} B_{max}^2$, where $B_{max}$ is the peak magnetic field in the core during the cycle and $\alpha$ is a positive constant with units of $K^{-1}$. Assuming the damping is weak ($\alpha T \ll 1$), which of the following correctly gives the equivalent series resistance $R_{eq}$ of the inductor due to hysteresis loss, and the time $t_0$ taken for the current amplitude to decrease to $1/e$ of its initial value?

A long straight wire carries a time-varying current $I(t) = k t^2$, where $k$ is a positive constant. A stationary rectangular frame of height $h$ is placed coplanar with the wire. The inner and outer edges of the frame are at distances $R$ and $4R$ from the wire, respectively. The frame is completely filled with a composite magnetic core consisting of two distinct regions:

Region 1 ($R \le r \le 2R$): A soft magnetic material with a non-linear magnetization response given by $\vec{M} = \gamma |\vec{H}| \vec{H}$, where $\gamma$ is a positive constant and $\vec{H}$ is the auxiliary magnetic field.

Region 2 ($2R \le r \le 4R$): A hard magnetic material with a frozen-in, permanent azimuthal magnetization $\vec{M} = M_0 \hat{\phi}$ (where $\hat{\phi}$ is the azimuthal unit vector). The hard material exhibits no additional magnetization due to the external field.

Assuming edge effects are negligible, what is the magnitude of the induced electromotive force (EMF) in the rectangular frame at time $t = t_0$?

A uniform rigid rod of mass $M = 0.6 \text{ kg}$ and length $L = 1.0 \text{ m}$ is freely pivoted at its upper end. A small soft iron core of mass $m = 0.2 \text{ kg}$ and volume $V = 5 \times 10^{-4} \text{ m}^3$ is attached to the lower end. The pendulum is released from rest in the horizontal position ($\theta = 90^\circ$). At the lowest point of its swing, the iron core passes through a narrow region of localized magnetic field. During this brief passage, the external magnetic field exerts a total retarding angular impulse of $J = 0.4 \text{ kg m}^2/\text{s}$ on the pendulum about the pivot. The soft iron core undergoes exactly one complete hysteresis cycle during the passage. The area of the hysteresis loop for the soft iron material is $A_h = 2000 \text{ J/m}^3$. Assuming the rod is non-magnetic and non-conducting, what is the energy dissipated as heat due to eddy currents in the iron core? (Take $g = 10 \text{ m/s}^2$)

A long straight wire carrying a steady current $I = 50 \text{ A}$ lies along the $z$-axis. It is surrounded by a non-homogeneous paramagnetic cylindrical shell of inner radius $a = 0.1 \text{ m}$ and outer radius $b = 0.2 \text{ m}$. The magnetic susceptibility of the shell varies radially as $\chi_m(r) = k r$, where $k = 20 \text{ m}^{-1}$ and $r$ is the perpendicular distance from the $z$-axis. Calculate the total magnetic flux $\Phi$ passing through a rectangular cross-section of the shell of length $L = 2.0 \text{ m}$ parallel to the $z$-axis. (Assume the permeability of free space is $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$).

Two infinitely long straight parallel wires, Wire 1 and Wire 2, carry equal currents $I$ in the $+y$ direction. Wire 1 is located at $x = -d, z = 0$ and Wire 2 is located at $x = d, z = 0$. The $y-z$ plane ($x=0$) separates two semi-infinite regions. The region $x < 0$ is filled with a linear magnetic material of relative permeability $\mu_r = 3$. The region $x > 0$ is vacuum ($\mu_r = 1$). A small magnetic dipole of magnetic moment $\vec{m} = m_0 \hat{k}$ is placed on the x-axis at $x = d/2$. Find the magnetic force $\vec{F}$ experienced by the dipole.

A thin rectangular plate of mass $2m$, length $L$ (spanning $0 \le x \le L$), width $2a$ (spanning $-a \le y \le a$), and uniform thickness $t$ has a non-uniform magnetization given by $\vec{M} = M_0 \left( \frac{x}{L} \hat{i} + \frac{y}{a} \hat{j} \right)$. The plate is cut into two pieces along the diagonal line connecting $(0, -a)$ and $(L, a)$. The upper fragment (the piece containing the point $(0, a)$) is taken and reshaped into a uniform solid sphere of mass $m$ and radius $R$. During this reshaping, its total magnetic dipole moment vector $\vec{\mu}$ is preserved and becomes permanently fixed relative to the sphere's material. The sphere is placed in free space and is initially at rest. A short magnetic field pulse $\vec{B}(t) = B_p \delta(t) \hat{k}$ is applied, where $\delta(t)$ is the Dirac delta function. Find the magnitude of the angular velocity of the sphere immediately after the pulse.