Moving Charges and Magnetism - Class 12

A proton of mass $m$ and charge $q$ enters a region of non-uniform static magnetic field. At the origin ($z=0$), the magnetic field is $\vec{B} = B_0 \hat{k}$ and the proton has a velocity $\vec{v} = v_0 \cos(30^\circ) \hat{k} + v_0 \sin(30^\circ) \hat{i}$. The magnetic field points primarily in the $+z$ direction, and its magnitude slowly increases along the $z$-axis. Assume the spatial variation of the magnetic field is gradual enough that the magnetic moment $\mu = \frac{m v_{\perp}^2}{2B}$ remains an adiabatic invariant. Find the ratio of the local pitch of the proton's helical path at $z=0$ to its local pitch at a point where the magnetic field magnitude becomes $2B_0$. (The local pitch is defined as $p = v_{\parallel} T$, where $v_{\parallel}$ is the longitudinal velocity and $T$ is the local time period of gyration).

A smooth, non-conducting circular ring of radius $R$ is fixed in a vertical plane. A uniform magnetic field is confined to a cylindrical region of radius $R/2$ whose axis passes through the center of the ring and is perpendicular to its plane. The magnetic field is zero outside this region. The magnetic field is directed into the page and its magnitude varies with time $t$ as $B(t) = \alpha t$, where $\alpha$ is a positive constant. A small bead of mass $m$ and positive charge $q$ is released from rest at the highest point of the ring. Assuming the induced electric field is the only non-conservative field present, what is the speed of the bead when it reaches the lowest point of the ring?

A particle of mass $m$ and positive charge $q$ is projected from an initial position $\vec{r}_0$ (with magnitude $r_0$) with an initial velocity $\vec{v}_0$ perpendicular to $\vec{r}_0$. The space contains a magnetic field $\vec{B} = \frac{k}{r^2} \hat{r}$, where $\hat{r}$ is the radial unit vector from the origin and $k$ is a positive constant. It is observed that the particle's trajectory is confined to the surface of a cone with its vertex at the origin. Identify the correct expression for the magnitude of the particle's areal velocity about the origin, and the semi-vertical angle $\alpha$ of this cone.

A solid conducting sphere of radius $R$ and total charge $Q > 0$ is rigidly fixed at the origin of a reference frame attached to a spacecraft. The spacecraft is accelerating in deep space with a constant acceleration $\vec{a} = \frac{qQ}{16\pi\epsilon_0 m R^2} \hat{i}$, where $\epsilon_0$ is the vacuum permittivity. A small particle of mass $m$ and charge $q > 0$ is projected from the position $x = 3R$ on the x-axis towards the origin, with an initial speed $v_0$ relative to the spacecraft. Assume the charge $Q$ remains uniformly distributed on the sphere. What is the minimum value of $v_0$ required for the particle to strike the surface of the sphere?

A particle of mass $m$ and charge $+q$ is projected from the origin $(0,0,0)$ with an initial velocity $\vec{v} = \sqrt{3}v_0 \hat{i} + v_0 \hat{j}$. A non-uniform magnetic field $\vec{B} = B_0 x \hat{k}$ exists in the region $x \ge 0$, where $B_0$ is a positive constant. Neglecting gravity, the maximum x-coordinate reached by the particle is:

A particle of rest mass $m_0$ and charge $q$ is accelerated from rest through a potential difference $V$. It then enters a region of uniform magnetic field $B$ perpendicular to its velocity. Considering relativistic effects, what is the frequency of revolution of the particle in the magnetic field? (Here $c$ is the speed of light in vacuum)

A particle of mass $m$ and charge $q > 0$ is initially moving in a circular orbit of radius $r_0$ in the $xy$-plane, centered at the origin. A uniform magnetic field $\vec{B} = B_0 \hat{k}$ exists throughout the region. At $t=0$, the magnetic field is abruptly decreased to $\vec{B} = \frac{B_0}{2} \hat{k}$ over a negligibly small time interval. What is the maximum distance of the particle from the origin during its subsequent motion?

Two charged particles, $P_1$ and $P_2$, are moving in the $xy$-plane under the influence of a uniform magnetic field $\vec{B} = B_0 \hat{k}$. Particle $P_1$ has mass $m$, charge $q$ (where $q > 0$), and at $t=0$ is at the origin $(0,0)$ moving with velocity $\vec{v}_1 = v_0 \hat{i}$. Particle $P_2$ has mass $2m$, charge $2q$, and at $t=0$ is also at the origin moving with velocity $\vec{v}_2 = v_0 \hat{j}$. The two particles undergo a perfectly inelastic collision at the origin and merge into a single particle. What is the magnitude of the change in the total orbital magnetic dipole moment of the system due to this collision?

An infinitely long solenoid of radius $a$ has $n$ turns per unit length and carries a steady current $I$. The interior of the solenoid ($r < a$) is filled with a non-conducting magnetic material whose relative permeability varies radially as $\mu_r(r) = 1 + \alpha r^2$, where $r$ is the distance from the solenoid axis and $\alpha$ is a positive constant. A particle of mass $m$ and charge $q$ ($q > 0$) is injected into the medium such that it moves in a steady circular orbit of radius $R$ ($R < a$) centered on the solenoid axis, in a plane perpendicular to the axis. Neglecting any energy losses, what is the kinetic energy of the particle required to maintain this orbit?

A long, straight, thick cylindrical conductor of radius $R$ carries a steady DC current. The current density inside this main conductor varies radially as $J(r) = J_0 \left(1 - \frac{r}{R}\right)$, where $r$ is the distance from the central axis and $J_0$ is a positive constant. At a certain junction, this main conductor splits into $4$ identical, long, straight cylindrical branches, each of radius $\frac{R}{2}$. After a sufficient distance from the junction, the current in each branch redistributes such that its current density becomes uniform across its cross-section. Let $B_1$ be the magnitude of the magnetic field inside the main conductor at a radial distance $r = \frac{R}{2}$ from its axis. Let $B_2$ be the magnitude of the magnetic field inside one of the branches at a radial distance $r' = \frac{R}{4}$ from its own central axis. Find the ratio $\frac{B_1}{B_2}$.

A long solenoid of length $L$ has a total of $N$ closely and uniformly wound turns. It carries a steady current $I$. The cross-section of the solenoid is circular, but its radius varies linearly from $r_1$ at one end ($x=0$) to $r_2$ at the other end ($x=L$). Assume that $r_1, r_2 \ll L$, such that the magnetic field inside the solenoid at any cross-section can be approximated by the field of an ideal infinite solenoid. What is the total magnetic energy stored inside the solenoid?

An infinite current slab of thickness $2d$ (spanning $-d \le x \le d$) carries a uniform volume current density $\vec{J} = J_0 \hat{k}$. A conducting wheel of radius $R$ ($R < d$) is placed in the $xz$-plane, centered at the origin. The wheel consists of $N$ ($N \ge 3$) thin radial conducting spokes, each of resistance $r_0$, connected at the center to a perfectly conducting axle and at the edge to a perfectly conducting rim. The wheel is forced to rotate about the $y$-axis with a constant angular velocity $\vec{\omega} = \omega \hat{j}$. Find the external mechanical power $P_{ext}$ required to maintain this rotation. (Ignore self-inductance and friction).

Consider a hydrogen-like exotic atom consisting of a nucleus of mass $M$ and charge $+e$, and a negatively charged particle of mass $m$ and charge $-e$. The two particles orbit their common center of mass under mutual Coulombic attraction. If the system is in the $n^{\text{th}}$ Bohr quantized state (where the total orbital angular momentum of the system is $n\hbar$), what is the magnitude of the total orbital magnetic dipole moment of the atom? Assume $M > m$.

A fixed large circular coil of radius $R$ carries a steady current $I_1$. A small rigid circular wire loop of mass $m$ and radius $r$ ($r \ll R$) carries a steady current $I_2$. The small loop is suspended from a frictionless horizontal axis passing through its highest point and lying in its own plane. The center of the small loop is located on the central axis of the large coil at a distance $x = \sqrt{3}R$ from the center of the large coil. The central axis of the large coil is horizontal. In the equilibrium position, the small loop hangs vertically such that the magnetic field of the large coil is perpendicular to the plane of the small loop. The currents are directed such that the magnetic torque opposes any angular deflection. Find the time period $T$ of small oscillations of the small loop.

A thin spherical shell of radius $R$ carries a non-uniform surface current density $\vec{K} = K_0 \sin\theta \hat{\phi}$, where $\theta$ is the polar angle measured from the $z$-axis, $\hat{\phi}$ is the azimuthal unit vector, and $K_0$ is a positive constant. Which of the following represents the net magnetic field vector $\vec{B}$ at the center of the spherical shell?

A particle of mass $m$ and charge $q$ moves in an elliptical orbit around a fixed heavy nucleus located at one of the foci. The semi-major axis of the orbit is $a$, its eccentricity is $e$, and the time period of revolution is $T$. What is the time-averaged magnitude of the magnetic field $\langle B \rangle$ produced by the particle at the location of the nucleus? (Assume all parameters are in standard SI units and $\mu_0$ is the vacuum permeability)

A perfectly conducting U-shaped rail is fixed on a horizontal plane. A perfectly conducting rod of initial mass $m_0$ is placed at $x=0$, closing the circuit. The self-inductance of the closed loop is given by $L(x) = L_0 + \alpha x$, where $\alpha$ is a positive constant. The region $x > 0$ is covered with a uniform layer of stationary non-conducting dust. As the rod moves, it sweeps up and completely accumulates the dust. The linear mass density of the dust is $\rho = \frac{m_0 \alpha}{L_0}$. At $t=0$, an initial current $I_0$ is established in the loop, and the rod is released from rest. Assuming electrical resistance is zero and there is no mechanical friction other than the accretion of dust, what is the velocity of the rod when it reaches a position where the self-inductance of the loop has doubled ($L = 2L_0$)?

A flexible, straight wire of length $L$, cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$ is rigidly fixed at its two ends. It is positioned exactly midway between two infinite, rigid parallel wires. The distance between the middle wire and each outer wire is $d$. All three wires lie in the same horizontal plane. The middle wire carries a steady current $I_1$, and the two outer wires each carry a steady current $I_2$ in the same direction as $I_1$. Initially, at temperature $T_0$, the middle wire is perfectly straight and has zero tension. The temperature of the middle wire is then uniformly lowered. Neglecting gravity and the magnetic field of the Earth, what is the minimum temperature drop $\Delta T$ required to keep the middle wire in stable equilibrium in its straight configuration?

Two infinitely long parallel wires, each of radius $a$, carry constant currents $I$ in opposite directions. The distance between their central axes is $x$ (where $x \gg a$). The magnetic force per unit length $F$ between the wires can be derived using the principle of virtual work by considering the variation of the magnetic interaction energy per unit length $U_{int}(x)$ with respect to $x$. The interaction energy per unit length is defined as $U_{int} = \int \frac{\vec{B}_1 \cdot \vec{B}_2}{\mu_0} dA$, where the integral is over the entire 2D plane perpendicular to the wires. Which of the following expressions correctly represents the $x$-dependent part of $U_{int}(x)$ and the correct thermodynamic relation used to find the force $F$ (where $F > 0$ denotes a repulsive force)? Let $C$ be a constant independent of $x$.

An infinitely long straight wire carrying a constant current $I_1$ in the $+z$ direction is fixed along the $z$-axis. A uniform rigid rectangular wire loop of total mass $m$, width $a$ (along the $x$-axis), and height $h$ (along the $z$-axis) is placed in the $x$-$z$ plane. The loop rests on a horizontal rough surface ($x$-$y$ plane) and is constrained to remain in the $x$-$z$ plane. The inner vertical edge of the loop is at a distance $d$ from the infinite wire. A current $I_2$ is gradually increased in the loop such that the current in its inner vertical edge flows in the $+z$ direction. What is the minimum coefficient of static friction $\mu$ between the loop and the surface required such that the loop topples before it begins to slide?

Consider a composite thermodynamic system consisting of $n$ moles of an ideal monoatomic gas and a macroscopic magnetic dipole of magnetic moment magnitude $\mu$. The system is confined in a perfectly insulated cylinder with a movable frictionless piston. The total internal energy of the system is $U = U_{\text{gas}} + U_{\text{mag}}$, where $U_{\text{gas}}$ is the thermal internal energy of the gas and $U_{\text{mag}} = -\vec{\mu} \cdot \vec{B}$ is the magnetic potential energy of the dipole in a uniform external magnetic field of magnitude $B$. Due to an internal reversible mechanism, the angle $\theta$ between $\vec{\mu}$ and $\vec{B}$ is constrained by the volume $V$ of the gas such that $\cos\theta = \left(\frac{V_0}{V}\right)^2$, where $V_0$ is the initial volume. The system expands quasi-statically, doing external work $P dV$. If the initial pressure of the gas is $P_0$ and the initial angle is $\theta = 0$, what is the gas pressure $P$ when the volume reaches $2V_0$?

A solid sphere of radius $R$ is centered at the origin. It has a non-uniform volume charge density given by $\rho(r, \theta) = \rho_0 \left( \frac{r}{R} \right) \sin\theta$, where $\rho_0$ is a positive constant, $r$ is the radial distance from the origin, and $\theta$ is the polar angle from the $z$-axis. The sphere rotates with a constant angular velocity $\boldsymbol{\omega} = \omega \hat{k}$ about the $z$-axis. Find the magnetic dipole moment $\boldsymbol{\mu}$ of the sphere and the net magnetic flux $\Phi_B$ through the entire closed surface of the sphere.

A cylindrical sample of a high-resistivity ferrimagnetic material of volume $V = 5.0 \times 10^{-4} \text{ m}^3$ is suspended vertically by a light torsional wire of torsional constant $C = \frac{2.0}{\pi^2} \text{ N}\cdot\text{m/rad}$. A uniform horizontal magnetic field $\vec{B}$ is applied. Initially, the wire is untwisted, and the sample is in equilibrium. An external agent slowly rotates the cylinder around the vertical axis by exactly one full turn ($2\pi \text{ rad}$) and holds it in this new position. During this $360^\circ$ rotation, the magnetization of the sample undergoes exactly one complete hysteresis cycle. Assume the energy dissipated per unit volume during this rotation is equal to the area of the material's $B-H$ hysteresis loop, which is given as $A_H = 4000 \text{ J/m}^3$. Assuming eddy current losses are negligible, what is the total work done by the external agent?

A rigid circular wire loop of radius $R$ and mass $m$ carries a constant steady current $i$. The loop is suspended vertically by a torsionless, insulating thread passing through its vertical diameter. A uniform horizontal magnetic field of magnitude $B$ is applied perpendicular to the initial plane of the loop. The loop is immersed in a viscous fluid that exerts a retarding torque $\tau_d = -b \omega$, where $\omega$ is the angular velocity of the loop and $b$ is a positive damping constant. The loop is given a small angular displacement about the vertical axis and released. Assuming the damping is very small, what is the fractional increase in the time period of its oscillation, $\frac{\Delta T}{T_0}$, compared to the undamped case?

A small rigid circular loop of radius $R$ and mass $M$ carries a steady current $I$. It is placed with its center at the origin $(0,0,0)$ of a Cartesian coordinate system, lying in the $y$-$z$ plane such that its magnetic dipole moment $\vec{\mu}$ is directed along the $+x$-axis. The region contains a static, non-uniform magnetic field given by $\vec{B} = B_0 (\alpha z \hat{i} + \alpha x \hat{k} + \hat{k})$, where $B_0$ and $\alpha$ are positive constants. If the loop is released from rest, what are its initial translational acceleration $\vec{a}$ and initial angular acceleration $\vec{\alpha}_{ang}$? (Assume gravity is negligible).

A rigid planar conducting loop of arbitrary shape has area $A$ and moment of inertia $J$ about a fixed axis of rotation in its plane. The loop is placed in a radial magnetic field of constant magnitude $B$, such that the magnetic field lines are always parallel to the plane of the loop and exert a magnetic torque $\tau = i A B$ when a current $i$ flows through it. At $t=0$, the stationary loop is connected to a circuit containing a DC voltage source $V_0$, a resistor of resistance $R$, and an inductor of inductance $L$ in series. What is the minimum value of the resistance $R$ such that the angular velocity of the loop approaches its steady-state value without any oscillations?

A galvanometer coil is modeled as a physical pendulum. It has a moment of inertia $I = 2 \times 10^{-4} \text{ kg m}^2$, a torsional spring constant $k = 10^{-2} \text{ N m/rad}$, $N = 100$ turns, cross-sectional area $A = 10^{-3} \text{ m}^2$, and total electrical resistance $R = 12 \text{ }\Omega$. The coil is placed in a uniform radial magnetic field of unknown magnitude $B$.

At $t=0$, the coil is at its equilibrium position ($\theta = 0$) and is given an initial angular velocity $\omega_0 = 10 \text{ rad/s}$. It swings and comes to instantaneous rest for the first time at an angle $\theta_f = 1 \text{ rad}$. The phase space trajectory of the coil is recorded, and the area under the angular velocity versus angle curve ($\omega$ vs $\theta$) from $\theta = 0$ to $\theta = 1 \text{ rad}$ is measured to be $S = 6 \text{ rad}^2/\text{s}$.

Assuming the only non-conservative force acting on the system is the magnetic damping due to the induced current in the closed coil circuit, what is the magnitude of the magnetic field $B$?

A moving coil galvanometer has a coil of $N$ turns and area $A$, suspended by a wire of torsional constant $k$. Unlike a standard galvanometer, it is placed in a non-radial magnetic field. When the coil deflects by an angle $\theta$ from its zero-current equilibrium position, the angle between the magnetic field vector and the plane of the coil is exactly $\theta$. Additionally, the magnitude of the magnetic field at the coil's location varies with the deflection angle as $B(\theta) = B_0 \cos\theta$, where $B_0$ is a positive constant. A steady current $I$ flows through the coil, producing a steady deflection $\theta = \frac{\pi}{4} \text{ rad}$. If the uncertainty in measuring the deflection angle is $\Delta\theta$, what is the relative error $\frac{\Delta I}{I}$ in the determination of the current?

A moving coil galvanometer consists of a coil of $N$ turns and area $A$, suspended such that it can rotate freely about the $z$-axis. It is placed in a permanent radial magnetic field of constant magnitude $B_0$. The torsional constant of the suspension is $C$. When no current flows, the coil is in equilibrium and its area vector points along the positive $x$-axis. When a steady current $I_0$ is passed through the coil, its area vector deflects by an angle $\theta_0 = 60^\circ$ towards the positive $y$-axis to reach a new equilibrium.

An infinite flat sheet carrying a uniform surface current density $\vec{K} = \frac{4 B_0}{\mu_0} \hat{k}$ is now placed in the $yz$-plane at $x = -d$ (where $d > 0$). The galvanometer is located at the origin. Assume the galvanometer components do not distort the magnetic field produced by the sheet.

A new steady current $I'$ is passed through the galvanometer, and it is observed that the coil deflects by the exact same angle $\theta_0 = 60^\circ$ towards the positive $y$-axis. Find the ratio $\frac{I'}{I_0}$.

A moving coil galvanometer consists of a coil with moment of inertia $I$, torsional constant $k$, $N$ turns, and cross-sectional area $A$. It is placed in a uniform radial magnetic field of magnitude $B$. The galvanometer is connected in a closed circuit to an ideal AC voltage source given by $V(t) = V_0 \sin(\omega t)$. The total electrical resistance of the circuit is $R$. Assume mechanical friction and air drag are completely absent. If the angular frequency $\omega$ of the voltage source is tuned exactly to the natural angular frequency of the galvanometer ($\omega = \sqrt{k/I}$), what is the steady-state amplitude of the angular deflection of the coil?

A moving coil galvanometer consists of a coil with $N = 100$ turns, each of area $A = 2 \times 10^{-4} \text{ m}^2$, and has a moment of inertia $I = 10^{-6} \text{ kg m}^2$ about its axis of rotation. The coil is placed in a uniform radial magnetic field $B = 0.5 \text{ T}$. The torsional constant of the suspension spring is $C = 10^{-6} \text{ N m rad}^{-1}$. The resistance of the galvanometer coil is $R_g = 40 \ \Omega$. The galvanometer is disconnected from any external circuit, and its terminals are shorted using a shunt resistance $R_s$. What should be the value of $R_s$ so that if the coil is slightly displaced and released, it returns to its equilibrium position without oscillating (i.e., the system is critically damped)? Ignore mechanical friction and air drag.

An AC generator consists of a flat coil of $N$ turns and area $A$, rotating at a constant angular velocity $\omega$ in a uniform magnetic field of induction $B$. The axis of rotation is perpendicular to the magnetic field. The internal resistance and self-inductance of the generator coil are negligible. The output of the generator is connected across a galvanometer having a resistance $R$ and self-inductance $L$. To protect the galvanometer, a pure resistor of resistance $R_S$ is connected in parallel with it as a shunt. The value of $R_S$ is chosen such that the average electrical power dissipated in the shunt is exactly equal to the average electrical power dissipated in the galvanometer. Determine the average mechanical torque $\tau$ that an external agent must apply to the generator coil to maintain its constant angular velocity $\omega$. (Assume no mechanical friction).

A cylindrical shunt resistor of radius $a$ (in $\text{m}$), length $l$ (in $\text{m}$), electrical conductivity $\sigma$ (in $\text{S/m}$), and electrical permittivity $\epsilon$ (in $\text{F/m}$) is connected across an AC voltage source $V(t) = V_0 \sin(\omega t)$, where $V_0$ is the voltage amplitude (in $\text{V}$) and $\omega$ is the angular frequency (in $\text{rad/s}$). Assuming the electric field is uniform throughout the volume of the resistor and edge effects are negligible, what is the maximum instantaneous magnitude of the Poynting vector, $S_{max}$ (in $\text{W/m}^2$), evaluated at the curved surface of the cylinder?

A modified moving coil galvanometer has a coil resistance $R_g = 18 \ \Omega$. Instead of a standard torsional spring, its restoring torque is provided by a small pendulum mass attached to the coil's pivot, making the restoring torque directly proportional to the effective acceleration due to gravity. When stationary in an inertial frame on Earth, it is converted into an ammeter of maximum range $I_0$ using a shunt resistance $S_1 = 2 \ \Omega$. The ammeter is subsequently mounted inside an elevator accelerating vertically upwards at $a = g$ (where $g$ is the acceleration due to gravity). To ensure the ammeter still measures the exact same maximum full-scale current $I_0$ in this accelerating frame, a new shunt resistance $S_2$ is required. Calculate the value of $S_2$.

A particle of mass $m$ and charge $+q$ is released from rest at the origin of a Cartesian coordinate system. The region $x > 0$ contains a uniform electric field $\vec{E} = E_0 \hat{i}$ and a non-uniform magnetic field $\vec{B} = B_0 x \hat{k}$, where $E_0$ and $B_0$ are positive constants. Which of the following represents the maximum x-coordinate ($x_{max}$) reached by the particle?

A particle of mass $m$ and charge $q > 0$ is initially at rest at a radial distance $r_0$ from the z-axis. At $t=0$, a time-varying, spatially non-uniform magnetic field $\vec{B} = \beta r t \hat{k}$ is switched on, where $\beta$ is a positive constant and $r$ is the distance from the z-axis. What is the magnitude of the angular velocity of the particle about the z-axis at a later time $t$, in terms of its instantaneous radial distance $r$?

A particle of mass $m$ and charge $q$ is projected with a velocity $\vec{v} = v_0 \hat{j}$ from the position $\vec{r} = x_0 \hat{i}$ in an inertial laboratory frame $S$. A non-uniform magnetic field $\vec{B} = B_0(1 + \alpha x) \hat{k}$ exists in the region, where $B_0$ and $\alpha$ are positive constants. A non-inertial frame $S'$ rotates with a constant angular velocity $\vec{\omega} = \omega \hat{k}$ relative to $S$. Assume the coordinate axes of $S$ and $S'$ perfectly coincide at the instant of projection ($t=0$). What is the initial acceleration $\vec{a}'$ of the particle as measured by an observer in the rotating frame $S'$ at $t=0$?

A point charge $Q$ is fixed at the origin of a cylindrical coordinate system $(r, \theta, z)$. A magnetic field exists in the region, given by $\mathbf{B} = B_0 \hat{k}$ for $r < a$, and $\mathbf{B} = \frac{B_0 a}{r} \hat{k}$ for $r \ge a$, where $B_0$ and $a$ are positive constants. A particle of mass $m$ and charge $q$ ($q > 0$) is released from rest at the position $(r=a, \theta=0, z=0)$. Assuming the particle's motion is naturally confined to the $z=0$ plane, what is the minimum value of the charge $Q$ such that the particle can escape to infinity?

A toroidal solenoid consists of $N$ turns of wire wound uniformly around a non-magnetic, isotropic solid core. The core has an initial major radius $R$ and an initial cross-sectional area $A$ (where $\sqrt{A} \ll R$). The material of the core has a coefficient of linear thermal expansion $\alpha$. A constant direct current $I$ is maintained in the wire. The temperature of the core is uniformly increased at a constant rate $\frac{dT}{dt} = \beta$. Assuming the wire perfectly accommodates the core's expansion without changing the total number of turns $N$, what is the magnitude of the induced electromotive force (EMF) in the solenoid? (Assume $\alpha \Delta T \ll 1$)

A toroid of mean radius $R$ has $N$ turns of wire carrying a steady current $I_1$. The wire is wound with a uniform pitch such that the current path completes exactly one full revolution around the toroid's axis of symmetry (the $z$-axis). The toroid lies in the $xy$-plane with its geometric center at the origin. A small rigid square loop of side length $L$ ($L \ll R$) carrying a steady current $I_2$ is placed at the origin. The small loop lies in the $xz$-plane and is rigidly mounted to a massless axle aligned with the $x$-axis. Which of the following represents the magnitude of the magnetic torque acting on the small loop about the $x$-axis?

A long solid non-magnetic cylinder of radius $R$ is oriented with its axis along the $z$-axis. It carries a steady current with a non-uniform current density vector given by $\mathbf{J} = J_0 (1 + \alpha \cos \theta) \hat{k}$, where $J_0$ and $\alpha$ are positive constants, $\theta$ is the azimuthal angle measured from the positive $x$-axis, and $\hat{k}$ is the unit vector along the $z$-axis. What is the magnetic field vector $\mathbf{B}$ at the central axis of the cylinder ($x=0, y=0$)?

An electron (rest mass $m_0$, charge $-e$) is injected through a microscopic gap into the vacuum core of a toroidal solenoid. The toroid has $N$ turns carrying a steady current $I$. The electron enters at a radial distance $R_1$ from the toroid's central axis of symmetry, with an initial velocity $\vec{v} = v_0 \hat{z}$ (parallel to the central axis), where $v_0$ is a relativistic speed. Inside the toroid, the electron experiences the azimuthal magnetic field $\vec{B}$. Assuming the toroid's outer radius is sufficiently large so the electron does not collide with the outer wall, which of the following expressions gives the exact maximum radial distance $\rho_{max}$ reached by the electron? (Let $\gamma = \frac{1}{\sqrt{1 - v_0^2/c^2}}$ and assume the magnetic field inside the core follows Ampere's Law exactly).

A hydrogen-like heavy ion of atomic number $Z$ has an electron (rest mass $m_0$, charge $-e$) in a circular orbit corresponding to the principal quantum number $n$. The Bohr quantization condition applies to the relativistic angular momentum, i.e., $L = \gamma m_0 v r = n\hbar$, where $\gamma = (1-v^2/c^2)^{-1/2}$ and $c$ is the speed of light. If the binding energy of the electron in this orbit is $B$ (defined such that the total relativistic energy including rest mass is $E = m_0 c^2 - B$), which of the following expressions correctly represents the orbital magnetic dipole moment $\mu$ of the electron? (Here $\mu_B = \frac{e\hbar}{2m_0}$ is the classical Bohr magneton).

A solid sphere of radius $R$ has a uniform bare volume charge density $\rho_0$. The sphere is composed of a theoretical non-uniform dielectric material whose relative permittivity varies with the radial distance $r$ from the center as $\epsilon_r(r) = \alpha r$, where $\alpha$ is a positive constant. The sphere is rotated with a constant angular velocity $\omega$ about one of its diameters. Assuming that the bound polarization charges rotate along with the sphere, what is the total magnetic dipole moment of the rotating sphere?

In a semi-classical model, an electron cloud of total charge $-e$ is treated as a continuous fluid flowing steadily in a fixed circular ring of radius $R$. Due to an external perturbation, the speed of the fluid elements varies with angular position $\theta$ as $v(\theta) = v_0 \left(1 + \frac{3}{5} \cos \theta\right)$. The unperturbed speed $v_0$ corresponds to a uniform circular Keplerian orbit of radius $R$ and period $T_0$. Find the magnitude of the orbital magnetic moment of this electron cloud.

A classical, non-relativistic electron plasma of uniform number density $n$ is confined in a region permeated by a uniform external magnetic field $\vec{B} = B_0 \hat{k}$. The plasma is initially in thermal equilibrium at temperature $T_0$ with an isotropic Maxwell-Boltzmann velocity distribution. Ignoring boundary effects, the bulk electrons undergo cyclotron motion, creating microscopic current loops. The plasma cools by radiating heat to its surroundings (maintained at a constant temperature $T_s = T_0 / 3$) according to Newton's law of cooling, with a cooling constant $\alpha$. Which of the following expressions correctly represents the magnitude of the effective bulk magnetization $M(t)$ of the plasma at time $t$? (Assume the electron mass is $m$, charge magnitude is $e$, and Boltzmann constant is $k_B$.)

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a distance $d$. The capacitor is connected to an ideal DC battery that maintains a constant potential difference $V_0$ across the plates. The space between the plates is completely filled with a dielectric material whose relative permittivity (dielectric constant) varies with time $t$ as $K(t) = K_0 + \alpha t$, where $K_0$ is a dimensionless positive constant and $\alpha$ is a positive constant with units of $\text{s}^{-1}$. Neglecting edge effects, what is the magnitude of the magnetic field $B$ at a radial distance $r$ ($r < R$) from the central axis of the capacitor?

A parallel plate capacitor with circular plates of radius $R = 5 \text{ cm}$ is filled with a lossy dielectric material. The material has a relative permittivity (dielectric constant) $K = 9$ and a uniform electrical conductivity $\sigma = \sqrt{3} \times 10^{-3} \text{ S/m}$. An alternating electric field $E(t) = E_0 \sin(\omega t)$ is established inside the dielectric, where $E_0 = 5 \times 10^4 \text{ V/m}$ and $\omega = 4\pi \times 10^6 \text{ rad/s}$. Neglecting edge effects, what is the amplitude of the induced magnetic field $B_{max}$ at a radial distance $r = 2 \text{ cm}$ from the central axis of the capacitor? (Take $\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ N m}^2/\text{C}^2$ and $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$)

An ideal parallel plate capacitor of plate area $A$ is at rest in an inertial frame $S$, with its plates parallel to the $x-z$ plane. It is being charged by a constant conduction current $I_0$. An observer in an inertial frame $S'$ moves with a constant velocity $v \hat{i}$ relative to $S$. Neglecting the induced magnetic field between the plates, the observer in $S'$ calculates the total displacement current $I'_d = \int \epsilon_0 \frac{\partial \vec{E}'}{\partial t'} \cdot d\vec{A}'$ through a surface fixed in $S'$ that exactly coincides with the electric field region between the plates at a given instant $t'$. Which of the following expressions gives the correct value of $I'_d$? (Here $c$ is the speed of light in vacuum)

A parallel plate capacitor consists of two circular plates of radius $a$ separated by a distance $d$. The space between the plates is completely filled with a lossy dielectric material of uniform permittivity $\epsilon$ and uniform conductivity $\sigma$. An alternating voltage $V(t) = V_0 \sin(\omega t)$ is applied across the plates. At a specific angular frequency $\omega_0$, the amplitude of the conduction current density in the dielectric equals the amplitude of the displacement current density. What is the amplitude of the magnetic field $B_0$ at a radial distance $r$ ($r < a$) from the central axis of the plates when the applied voltage has an angular frequency $\omega = \sqrt{3}\omega_0$?