Electromagnetic Induction - Class 12

A very long dielectric slab of thickness $d$ (in meters) along the $y$-axis and relative permittivity $\epsilon_r$ (dimensionless) is moving with a constant velocity $\vec{v} = v_0 \hat{i}$ (in m/s) along its length (the $x$-axis). The slab is infinite in the $z$-direction. It passes through a region of uniform external magnetic field $\vec{B}_{ext} = B_0 \hat{k}$ (in Tesla). Assume $v_0 \ll c$, where $c$ (in m/s) is the speed of light in vacuum. Due to the motion of the polarized dielectric, an induced magnetic field $\vec{B}_{ind}$ is generated inside the slab. What is the magnitude and direction of $\vec{B}_{ind}$ far from the edges of the slab?

A conducting rod of mass $m$ (in kg) and length $L$ (in m) slides down a rough inclined plane of angle $\theta$ with the horizontal. The plane has a coefficient of kinetic friction $\mu$. The rod is in contact with two parallel, perfectly conducting rails fixed to the incline, which are connected at the bottom by a resistor of resistance $R$ (in $\Omega$). A uniform magnetic field of magnetic induction $B$ (in T) exists vertically downwards throughout the region. The rod is released from rest. Assuming the rails are very long, what is the terminal velocity $v_0$ (in m/s) acquired by the rod? (Neglect the self-inductance of the loop and electrical resistance of the rod and rails).

A small magnetic dipole of constant magnitude $m_0$ is fixed at the center of a stationary solid sphere of radius $R$. The sphere is made of a linear paramagnetic material with magnetic susceptibility $\chi$. The dipole is forced by an external agent to rotate with a constant angular velocity $\omega$ in the $xy$-plane, such that its magnetic moment vector is $\vec{m}(t) = m_0(\cos\omega t \hat{i} + \sin\omega t \hat{j})$. A stationary circular wire loop of radius $a$ ($a \gg R$) and electrical resistance $r_0$ is fixed in the $xz$-plane, centered at the origin. The medium outside the sphere is vacuum. Assuming the self-inductance of the loop is negligible and the system is in the quasi-static regime, what is the time-averaged external torque $\langle \tau_{ext} \rangle$ required to maintain the constant angular velocity of the dipole?

The primary circuit of a potentiometer lies in the plane of the paper and consists of a DC battery of constant EMF $V_0 = 10 \text{ V}$ and negligible internal resistance. It is connected in series with a conducting circular loop of area $A = 0.1 \text{ m}^2$ and resistance $R_L = 5 \ \Omega$, and a uniform potentiometer wire $AB$ of length $L = 1 \text{ m}$ and resistance $R_w = 5 \ \Omega$. The battery drives a clockwise current through the loop. A uniform magnetic field $B$, localized to the region of the loop and directed into the paper, varies with time $t$ as $B(t) = 5 t^2 \text{ T}$. A secondary circuit containing a standard cell of EMF $E_s = 2.88 \text{ V}$ and a galvanometer is connected between point $A$ and a jockey to find the null point $x$ on the wire $AB$. Taking the direction from $A$ to $B$ as positive, what is the velocity of the null point at $t = 2 \text{ s}$?

Two long, parallel conducting rails of zero resistance, separated by a distance $L$, are fixed on an inclined plane at an angle $\theta$ to the horizontal. The rails are connected at the bottom by a conducting wire of zero resistance. A uniform magnetic field $\vec{B}$ points vertically downwards. A conducting rod of mass $m$ and resistance $R$ is placed horizontally across the rails. The coefficient of kinetic friction between the rod and the rails is $\mu$. A constant external force $F$ pulls the rod up the incline, parallel to the rails. The rod starts from rest. Which of the following expressions gives the distance $x$ moved by the rod along the incline when it attains a velocity $v$? (Assume the rails are long enough and $F$ is sufficiently large to move the rod upwards. Let $F_{eff} = F - mg(\sin\theta + \mu \cos\theta)$).

A small square conducting loop of mass $m$ (in kg), side length $a$ (in m), and electrical resistance $R$ (in $\Omega$) is attached to the lower end of a massless rigid rod of length $L$ (in m). The upper end of the rod is pivoted so the loop can oscillate as a pendulum in the vertical $xy$-plane ($x$ is horizontal, $y$ is vertical). A time-independent magnetic field $\vec{B} = (B_0 + \alpha x) \hat{k}$ (in T) exists in the region, where $B_0$ and $\alpha$ are positive constants. An external horizontal driving force $F(t) = F_0 \cos(\omega t)$ (in N) is applied to the loop. Assuming small amplitude oscillations, negligible self-inductance, and no air drag, what is the average electrical power (in W) dissipated in the loop when the system is driven at its velocity resonance frequency?

A rigid conducting rod of length $L$ and uniform electrical resistance $R$ rotates in a horizontal plane with a constant angular velocity $\omega$ about a vertical axis passing through one of its ends ($r=0$). The other end of the rod slides without friction on a stationary, perfectly conducting circular ring of radius $L$. A stationary, perfectly conducting wire connects the pivot to the ring, completing the electrical circuit. A non-uniform, time-independent magnetic field is applied perpendicular to the plane of rotation. The magnitude of the magnetic field varies with the radial distance $r$ from the pivot as $B(r) = B_0 \left(1 - \frac{r}{L}\right)$, where $B_0$ is a positive constant. Neglecting the self-inductance of the loop, what is the magnitude of the external mechanical torque $\tau_{\text{ext}}$ that must be applied to the rod to maintain its constant angular velocity $\omega$?

A parallel plate capacitor with circular plates of radius $R$ is filled completely with a linear dielectric material of relative permittivity $\epsilon_r$. The electric field $\vec{E}$ between the plates is uniform, directed along the central axis, and its magnitude is increasing at a constant rate $\frac{dE}{dt} = \alpha$. What is the magnitude of the magnetic field $B$ at a radial distance $r$ ($r < R$) from the central axis inside the dielectric? (Assume edge effects are negligible, and $\mu_0, \epsilon_0$ are the permeability and permittivity of free space, respectively).

An electron of mass $m$ and charge $-e$ is moving in a circular orbit of radius $r_0$ with speed $v_0$ around a stationary nucleus of charge $+e$. A uniform magnetic field $\vec{B}$, perpendicular to the plane of the orbit, is suddenly switched on. The direction of $\vec{B}$ is parallel to the initial orbital angular momentum of the electron. Assuming the duration of the switch-on process is extremely short such that the orbital radius remains approximately $r_0$ during this time, what is the minimum final magnitude of the magnetic field $B$ required for the electron to just escape the Coulombic binding of the nucleus (i.e., its total mechanical energy becomes zero)?

A plane electromagnetic wave with electric field $\vec{E}_{\text{inc}} = E_0 \cos(\omega t - kx) \hat{j}$ is normally incident on a thin, infinite conducting grid located at $x=0$. The grid has zero electrical resistance but possesses a constant surface self-inductance $L_s$ (in Henry). The electric field at the grid induces a surface current density $\vec{K}(t) = K(t) \hat{j}$ (in A/m). The dynamics of this current are governed by $E(0,t) = L_s \frac{dK}{dt}$, where $E(0,t)$ is the net electric field at $x=0$.

The induced current generates secondary electromagnetic waves propagating away from the grid in both directions. Assuming the surrounding medium is vacuum (permeability $\mu_0$, speed of light $c$), the reflected electric field for $x < 0$ is given by $\vec{E}_{\text{ref}} = E_r \cos(\omega t + kx + \theta) \hat{j}$, where $E_r > 0$.

For low frequencies such that $\omega L_s \ll \mu_0 c$, what is the approximate phase shift $\theta$ of the reflected wave?

A stationary loop of inductance $L = 2.0 \text{ H}$ and zero resistance is placed in a uniform magnetic field. The initial magnetic flux through the loop is $\Phi_0 = 20 \text{ Wb}$. A thermally insulated resistor is connected in series with the loop, but is initially short-circuited by a superconducting switch. At $t=0$, the external magnetic field is abruptly reduced to zero. Immediately after, the switch is opened so that the induced current flows through the resistor. The resistor has an initial resistance $R_0 = 1.0 \ \Omega$ and a constant heat capacity $C = 5.0 \text{ J/K}$. Its resistance varies with its temperature $T$ as $R(T) = R_0 [1 + \alpha (T - T_0)]$, where $T_0$ is the initial temperature and $\alpha = 0.05 \text{ K}^{-1}$. Assuming no heat is lost to the surroundings, find the time $t$ it takes for the current to drop to half of its initial value.

A perfectly conducting rigid wire is bent into the shape of a parabola $y = \frac{x^2}{2L}$ and is bounded between $x = -L$ and $x = L$. The wire is placed in a time-independent, non-uniform magnetic field given by $\vec{B} = \beta x \hat{k}$, where $\beta$ is a positive constant. At a specific instant $t = t_0$, the wire is positioned such that its vertex is at the origin and its axis of symmetry is aligned with the positive y-axis. At this exact instant, the wire is undergoing a combined motion: its vertex translates with a velocity $\vec{v}_{v} = \gamma L t_0 \hat{i}$, and the entire wire rotates about its vertex with an angular velocity $\vec{\omega} = \gamma t_0 \hat{k}$, where $\gamma$ is a positive constant. Determine the magnitude of the instantaneous motional electromotive force (EMF) induced across the ends of the wire at $t = t_0$.

A rigid conducting wire of arbitrary shape and unknown mass connects the center $O$ of a stationary vertical circular conducting ring of radius $a = 2.0 \text{ m}$ to a point on its rim. The ring lies in the vertical $xy$-plane, with uniform gravity acting in the $-y$ direction. The center $O$ and the rim are connected via ideal slip rings to an external stationary resistor $R = 10 \ \Omega$. The electrical resistance of the wire and the ring are negligible. A magnetic field $\vec{B} = B_0 \left( \frac{r}{a} \right)^2 \hat{k}$ exists in the region, where $r$ is the radial distance from $O$, $\hat{k}$ is the unit vector normal to the $xy$-plane, and $B_0 = 4.0 \text{ T}$. An external motor rotates the wire about the $z$-axis with a constant angular velocity $\omega = 5.0 \text{ rad s}^{-1}$. Calculate the total work done by the motor during exactly one complete revolution of the wire.

A large rectangular dielectric slab of thickness $d$ (spanning from $y=0$ to $y=d$) moves with a constant velocity $\vec{v} = v_0 \hat{i}$ through a uniform, time-independent magnetic field $\vec{B} = B_0 \hat{k}$. The relative permittivity of the dielectric material varies non-uniformly along the $y$-axis as $\epsilon_r(y) = K_0 e^{y/d}$, where $K_0 > 1$ is a constant. Assuming there are no free charges present anywhere and the magnetic Reynolds number is negligibly small, what is the magnitude of the steady-state potential difference $\Delta V$ between the surfaces $y=0$ and $y=d$ in the laboratory frame?

A uniform conducting rod of mass $M$ and length $3l$ is pivoted at a point $P$ which is at a distance $l$ from one of its ends. The rod can rotate freely in a horizontal plane. A uniform vertical magnetic field $B$ exists throughout the region. The longer segment of the rod (of length $2l$) has its free end sliding on a fixed, perfectly conducting circular ring. A resistor of resistance $R$ connects the pivot $P$ and the circular ring. The shorter segment of the rod (of length $l$) is free and does not touch any ring.

Initially, the rod is rotating with an angular velocity $\omega_0$ and the resistor circuit is open. At $t=0$, a small piece of putty of mass $m = M/3$ is dropped gently onto the free end of the shorter segment and sticks to it. Simultaneously, the resistor circuit is closed.

Assume the self-inductance of the circuit is negligible. Neglecting the resistance of the rod and friction at the pivot, find the total energy dissipated as Joule heating in the resistor $R$ until the rod comes to a complete stop.

A uniform conducting rod of length $L = 1.0 \text{ m}$ and resistance $R = 5.0 \ \Omega$ is pivoted at one end and rotates with a constant angular velocity $\omega = 10 \text{ rad/s}$ in a horizontal plane. A uniform vertical magnetic field $B = 2.0 \text{ T}$ exists throughout the region. The free end of the rod slides on a fixed, perfectly conducting circular ring. The pivot and the ring are connected by a stationary wire containing an ideal battery of EMF $V_0 = 5.0 \text{ V}$. The battery is oriented such that it opposes the motional EMF generated by the rod. Let $P_B$ be the net rate of work done by the magnetic force on all charge carriers in the rod, $P_{emf}$ be the rate at which energy is converted from mechanical to electrical form by the motional EMF, and $P_{ext}$ be the mechanical power supplied by the external agent to maintain the constant angular velocity. Which of the following sets of values is correct?

A uniform conducting rod of initial length $L_0$, mass $m$, and specific heat capacity $c$ is pivoted at one end and rotates with a constant angular velocity $\omega$ in a horizontal plane. A uniform vertical magnetic field $B$ exists throughout the region. The free end of the rod makes sliding contact with a fixed, perfectly conducting circular ring. A stationary circuit is completed by connecting a wire of negligible resistance between the pivot and the ring. Assume the rod is thermally insulated from its surroundings, so all the electrical power dissipated in the rod is entirely used to heat it. The coefficient of linear thermal expansion of the rod is $\alpha$. To isolate the effect of length change, assume the cross-sectional area and electrical resistivity of the rod remain strictly constant as it heats up. If the initial electrical power dissipated in the rod at $t=0$ is $P_0$, what is the initial rate of change of the power dissipated, $\left. \frac{dP}{dt} \right|_{t=0}$?

A self-supporting, single-layer, tightly wound solenoid is made of a uniform conducting wire of circular cross-section. The material of the wire has a coefficient of linear thermal expansion $\alpha$ and a temperature coefficient of resistivity $\alpha_\rho$. If the temperature of the entire solenoid is increased by a small amount $\Delta T$, what is the fractional change in the inductive time constant $\left(\frac{\Delta \tau}{\tau}\right)$ of the solenoid? (Assume $\alpha \Delta T \ll 1$ and $\alpha_\rho \Delta T \ll 1$)

A highly flexible, tightly wound solenoid of natural length $L_0$, total number of turns $N$, cross-sectional area $A$, and total mass $M$ is suspended vertically from its upper end in a uniform downward gravitational field $g$. The solenoid behaves elastically like a helical spring with an overall equivalent spring constant $k$. Assuming the cross-sectional area $A$ remains constant and the solenoid reaches mechanical equilibrium under its own weight, what is the total self-inductance $L$ of the suspended solenoid? (Ignore end effects and assume $\mu_0$ is the permeability of free space)

A long elastic solenoid has length $l$, $N$ total turns, un-stretched radius $r_0$, and total mass $M$. The solenoid is mounted on a platform and rotated about its longitudinal axis with a time-varying angular velocity $\omega(t) = \beta t$, where $\beta$ is a positive constant. A constant direct current $I$ is maintained in the solenoid by an external source. The wire has an effective elastic force constant $F_0$ such that the tension $T$ in the wire produces a fractional change in its local radius given by $\frac{\Delta r}{r_0} = \frac{T}{F_0}$. Assuming the radial expansion is small ($\Delta r \ll r_0$) and ignoring any tangential deformation or self-inductance changes due to the wire's finite thickness, what is the magnitude of the induced electromotive force (EMF) across the solenoid at time $t$?

A real solenoid of inductance $L$ (in $\text{H}$) and internal resistance $r$ (in $\Omega$) is connected in series with a variable load resistance $R$ (in $\Omega$) and an ideal DC voltage source of electromotive force $V$ (in $\text{V}$). The circuit is closed at $t=0$ and allowed to reach a steady state. Once the steady state is reached, the voltage source is instantaneously bypassed by a short circuit, allowing the current to decay through the resistors. Let $E_R$ (in $\text{J}$) be the total energy dissipated in the load resistance $R$ during the entire decay process. Which of the following gives the value of $R$ that maximizes $E_R$, and the corresponding maximum energy $(E_R)_{\max}$?

An ideal battery of electromotive force $V_0$ is connected in series with a resistor of resistance $R$, a switch, and a variable inductor. At $t = 0$, the switch is closed. A mechanical system alters the inductor such that its inductance increases linearly with time as $L(t) = L_0 + \alpha t$, where $L_0$ and $\alpha$ are positive constants. Which of the following expressions correctly represents the current $i(t)$ in the circuit for $t > 0$?

An ideal capacitor of capacitance $C$ is connected in a closed series loop with a long ideal solenoid of self-inductance $L$. A small conducting ring of resistance $R$ and negligible self-inductance is placed coaxially inside the solenoid. The mutual inductance between the solenoid and the ring is $M$. The capacitor is initially charged, and at $t=0$, the circuit is allowed to oscillate. Let $I$ be the current in the solenoid and $Q$ be the charge on the capacitor such that $I = \frac{dQ}{dt}$. If $E_0 = \frac{Q^2}{2C} + \frac{1}{2} L I^2$ represents the sum of the energy stored in the capacitor and the self-magnetic energy of the solenoid, which of the following expressions correctly represents the rate of change of $E_0$ with respect to time?

Two rigid coils, 1 and 2, have self-inductances $L_1$ and $L_2$. Their mutual inductance depends on the separation $x$ between them as $M(x) = M_0 \left(1 - \frac{x}{d}\right)$ for $0 \le x \le d$, where $M_0$ and $d$ are positive constants. Coil 1 is held fixed at $x = 0$. Coil 2 is moved slowly from $x = 0$ to $x = d$ by an external mechanical agent. During this displacement, the currents in the coils are maintained at constant positive values $I_1$ and $I_2$ respectively by ideal batteries. Let $W_{ext}$ be the mechanical work done by the external agent and $W_{elec}$ be the net extra electrical work done by the batteries (excluding the work done to compensate for Joule heating). Which of the following represents the pair $(W_{ext}, W_{elec})$?

A non-linear inductor with a ferromagnetic core operates in a regime where magnetic saturation begins to occur. The total magnetic flux linkage $\Phi$ (in $\text{Wb}$) through the inductor varies with the current $I$ (in $\text{A}$) according to the relation $\Phi(I) = 2I - \frac{1}{4}I^2$ for $0 \le I \le 4 \text{ A}$. Calculate the total magnetic energy stored in the inductor when the current is slowly increased from $0 \text{ A}$ to $4 \text{ A}$.

Two LC circuits are coupled via mutual inductance $M$. Circuit 1 consists of an inductor of inductance $L_1 = L$ and a capacitor of capacitance $C_1 = C$. Circuit 2 consists of an inductor of inductance $L_2 = L$ and a capacitor of capacitance $C_2 = C/4$. The mutual inductance between the inductors is $M = \frac{\sqrt{7}}{4} L$. At time $t=0$, the capacitor in Circuit 1 has an initial charge $Q_0$, while the capacitor in Circuit 2 is uncharged. The initial currents in both circuits are zero. Let $E_0$ be the initial total energy of the system. What is the maximum energy stored in the capacitor of Circuit 2 during the subsequent oscillations? (Assume ideal components and no resistance)

A large stationary circular wire loop of radius $R$ lies in the $xy$-plane, centered at the origin. A small planar coil of area $A$ ($A \ll R^2$) is placed at the origin. The small coil rotates about the $x$-axis with a constant angular velocity $\omega$. At $t=0$, the area vector $\hat{n}$ of the small coil is directed along the $+z$-axis. The small coil carries a time-varying current $I(t) = I_0 t$ (where $I_0$ is a positive constant), such that its magnetic moment is $\vec{m}(t) = I(t) A \hat{n}$. Additionally, a uniform external magnetic field $\vec{B}_{ext} = \beta t \hat{k}$ (where $\beta$ is a positive constant) exists in the entire region. What is the induced electromotive force (EMF) in the large loop at time $t = \frac{\pi}{2\omega}$? Take the counter-clockwise direction in the $xy$-plane (viewed from the $+z$-axis) as positive.

A long straight wire and a coplanar square loop of side $a$ lie on a smooth horizontal table. The wire carries a time-varying current $I(t) = I_0(1 + \alpha t)$, where $I_0$ and $\alpha$ are positive constants. At time $t = 0$, the near edge of the loop is parallel to the wire and at a distance $a$ from it. If the loop is being pulled away from the wire with a constant velocity $v$ perpendicular to the wire, which of the following expressions represents the magnitude of the total induced electromotive force (EMF) in the loop at $t = 0$?

A single-turn rectangular conducting loop is driven to rotate about its central axis of symmetry at a constant angular velocity $\omega$. The loop consists of two rigid wires, each of length $l$ and mass $m$, parallel to the rotation axis. The wires are connected by flexible, massless conducting cords of negligible tension. The two wires are mechanically constrained by a spring mechanism that exerts a radial restoring force of magnitude $k(r - r_0)$ on each wire directed towards the axis, where $r$ is the distance of the wire from the axis and $r_0$ is a positive constant. The loop has a total electrical resistance $R$ and self-inductance $L$. It rotates in a uniform magnetic field of magnitude $B$ perpendicular to the rotation axis. The entire system is immersed in a viscous fluid that exerts a drag torque $\tau_d = b\omega$ on the loop. Assuming $k > m\omega^2$, what is the average external mechanical power $P$ required to maintain the constant angular velocity $\omega$?

A conducting rod of mass $m$ and length $l$ is given an initial velocity $v_0$ at $x=0$ on two long, frictionless, horizontal parallel conducting rails. The rails are connected at $x=0$ by a resistor of resistance $R$. A spatially varying magnetic field $\vec{B} = \alpha x \hat{k}$ exists in the region, where $\alpha$ is a positive constant and $\hat{k}$ is perpendicular to the plane of the rails. The self-inductance of the circuit is negligible. What fraction of the rod's initial kinetic energy is dissipated as heat in the resistor by the time the rod has traveled half of its maximum stopping distance?

An ideal LC circuit contains a capacitor of capacitance $C$ and an inductor whose inductance varies with time as $L(t) = L_0 + \alpha t$, where $L_0$ and $\alpha$ are positive constants. The variation in inductance is achieved by an external agent mechanically moving a magnetic core. Let $Q(t)$ be the charge on the capacitor and $I(t) = \frac{dQ}{dt}$ be the current in the circuit. The differential equation governing the charge can be written in the form $\frac{d^2Q}{dt^2} + \gamma(t) \frac{dQ}{dt} + \omega^2(t) Q = 0$. Which of the following correctly gives the effective damping coefficient $\gamma(t)$ and the rate of change of the total electromagnetic energy $\frac{dE}{dt}$ of the circuit at an instant when the current is $I$?

A capacitor of capacitance $C$ is initially charged to a voltage $V_0$. At time $t=0$, it is connected in series with an inductor of inductance $L$ and a resistor of resistance $R$ to form a closed loop. The system is weakly damped such that $R \ll \sqrt{\frac{L}{C}}$. Which of the following expressions correctly represents the average power dissipated in the resistor over the time interval from $t=0$ to $t = \frac{L}{R}$?

A perfectly conducting (superconducting) closed loop of wire has a variable self-inductance given by $L(\theta) = L_0 (3 + 2\sin\theta)$, where $\theta$ is the angular position of a movable segment of the loop. The movable segment has a moment of inertia $J$ about its fixed axis of rotation. Initially, the segment is held at rest at $\theta = 0$, and a steady current $I_0$ flows through the loop. The segment is then released. Assuming no mechanical friction and no external magnetic field, what is the maximum angular speed $\omega_{\text{max}}$ achieved by the movable segment?

An ideal inductor of inductance $L$ is connected in a closed loop with a non-linear capacitor. The voltage $V$ across the capacitor depends on the charge $q$ stored in it according to the relation $V(q) = \frac{q}{C_0} + \beta q^3$, where $C_0$ and $\beta$ are positive constants. The circuit is set into oscillation such that the maximum charge on the capacitor is $Q_0$. Assuming the non-linear term is small ($\beta Q_0^2 \ll \frac{1}{C_0}$), what is the approximate fundamental angular frequency $\omega$ of the oscillation?

Two identical LC circuits, each consisting of a capacitor of capacitance $C$ and an inductor of inductance $L$, are placed near each other. The total energy of the system is given by $E = \frac{q_1^2}{2C} + \frac{q_2^2}{2C} + \frac{1}{2} L i_1^2 + \frac{1}{2} L i_2^2 + M i_1 i_2$, where $q_1, q_2$ are the instantaneous charges on the capacitors, $i_1 = \frac{dq_1}{dt}, i_2 = \frac{dq_2}{dt}$ are the currents, and $M = 0.8 L$ is the mutual inductance between the inductors. At $t=0$, the currents in both circuits are zero, the charge on the first capacitor is $q_1(0) = Q_0$, and the charge on the second capacitor is $q_2(0) = Q_0/2$. What is the maximum charge that will appear on the second capacitor during the subsequent oscillations?

A rigid wire frame consists of two identical square loops, Loop 1 (left) and Loop 2 (right), each of side length $L$, sharing a common vertical branch. The common branch contains an ideal parallel-plate capacitor of capacitance $C$. The left vertical branch of Loop 1 has a resistance $R$, and the right vertical branch of Loop 2 has a resistance $R$. All other wire segments have negligible resistance. The frame lies in the $x-y$ plane. A spatially uniform magnetic field $\vec{B}_1 = 4\alpha t \hat{k}$ is confined to the region of Loop 1, and $\vec{B}_2 = \alpha t \hat{k}$ is confined to the region of Loop 2, where $\alpha > 0$ is a constant and $t \ge 0$ is time. At $t=0$, the capacitor is uncharged. Assuming the self-inductance of the loops is negligible, what is the magnitude of the displacement current $I_d$ through the capacitor as a function of time $t$?

A rectangular conducting loop of length $L$ and width $w = \frac{\pi}{2k}$ is placed in the $x-y$ plane, bounded by the lines $x = 0$, $x = \frac{\pi}{2k}$, $y = 0$, and $y = L$. A spatially varying magnetic field $\vec{B}(x,t) = B_0 \cos(kx - \omega t) \hat{k}$ is applied across the region, where $B_0, k, \omega$ are positive constants. The induced EMF $\mathcal{E}(t)$ in the loop is measured with the counter-clockwise direction (viewed from the $+z$ axis) taken as positive. If the induced EMF is expressed in the form $\mathcal{E}(t) = \mathcal{E}_0 \cos(\omega t + \phi)$ with $\mathcal{E}_0 > 0$ and $\phi \in (-\pi, \pi]$, find the ordered pair $(\mathcal{E}_0, \phi)$.

A conducting rod of mass $m$ and length $L$ rests on two long, frictionless, horizontal parallel rails. The rod is attached to a fixed wall by a light spring of force constant $k$, parallel to the rails. A uniform vertical magnetic field $B$ exists throughout the region. The rails are connected at the end near the wall by a stationary circuit consisting of an uncharged capacitor of capacitance $C$, an ideal diode, and an open switch $S$ in series. The diode is oriented to allow current to flow when the rod moves away from the wall. At $t=0$, the rod is at its natural equilibrium position, moving away from the wall with velocity $v_0$, and the switch $S$ is instantaneously closed. The resistance of the entire circuit is negligible. What is the maximum extension of the spring?

A rigid, rectangular superconducting loop of mass $m$, self-inductance $L$, and area $S$, with its sides parallel to the $x$ and $y$ axes, is attached to a horizontal ideal spring of spring constant $k$. The loop is constrained to oscillate on a frictionless horizontal surface along the $x$-axis. A non-uniform, time-independent magnetic field is present in the region, given by $\vec{B} = B_0 (1 + \alpha x) \hat{k}$, where $B_0$ and $\alpha$ are positive constants. The plane of the loop is parallel to the $xy$-plane. At $t = 0$, the loop is at the equilibrium position of the spring ($x = 0$) with velocity $v_0$, and the current in the loop is exactly zero. Which of the following expressions represents the angular frequency $\omega$ of the subsequent harmonic oscillations?

A rigid square loop of side $L$ is hinged along the $y$-axis. At $t=0$, the loop lies in the $xy$-plane with its vertices at $(0,0,0)$, $(L,0,0)$, $(L,L,0)$, and $(0,L,0)$. The loop rotates about the $y$-axis with a constant angular velocity $\omega$ such that it moves into the region $z > 0$. A static, non-uniform magnetic field exists in the region, given by $\vec{B} = \beta x^2 \hat{i} - 2\beta x y \hat{j}$, where $\beta$ is a positive constant. Find the magnitude of the induced electromotive force (EMF) in the loop at time $t = \frac{\pi}{3\omega}$.

A square conducting loop of mass $m$, side length $L$, and electrical resistance $R$ is released from rest at $y=0$ and falls vertically downward in the $xy$-plane. The loop falls through a viscous fluid that exerts a drag force $\vec{F}_v = -k\vec{v}$, where $k$ is a positive constant and $\vec{v}$ is the velocity of the loop. A non-uniform horizontal magnetic field exists in the region, given by $\vec{B} = \beta y \hat{k}$, where $\beta$ is a positive constant and $\hat{k}$ is perpendicular to the plane of the loop. Assuming the loop remains perfectly vertical and reaches its terminal velocity before falling a total distance $H$, what is the total Joule heat dissipated in the loop during this fall? (Acceleration due to gravity is $g$ in the $+y$ direction. Neglect the self-inductance of the loop).

A small conducting ring of mass $m$, radius $a$, resistance $R$, and self-inductance $L$ is attached to a horizontal spring of force constant $K$. The ring is constrained to move along the $x$-axis such that its plane remains parallel to the $y$-$z$ plane. A time-independent, non-uniform magnetic field $\vec{B} = (B_0 + \beta x) \hat{i}$ exists in the region, where $B_0$ and $\beta$ are positive constants. The ring is forced to oscillate along the $x$-axis with a displacement $x(t) = x_0 \sin(\omega t)$. Due to electromagnetic induction, a force $F_{em}$ acts on the ring. This force can be decomposed into a damping component proportional to velocity $v$ and a restoring component proportional to displacement $x$, such that $F_{em} = - b v - \Delta K x$. Determine the effective additional stiffness $\Delta K$ provided by the electromagnetic interaction.

A flexible conducting wire is anchored at two fixed points $(0, a)$ and $(0, -a)$ in the $xy$-plane, where $a$ is a positive constant in meters. The wire is pulled in the $+x$ direction such that its shape at any time $t \ge 0$ (in seconds) is a parabola given by the equation $x(y, t) = \frac{v_0 t}{a^2} (a^2 - y^2)$, where $v_0$ is a positive constant in $\text{m/s}$. A non-uniform, time-independent magnetic field $\vec{B} = B_0 x^2 \hat{k}$ exists in the region, where $B_0$ is a positive constant in $\text{T/m}^2$. The fixed points are connected by a straight stationary wire along the $y$-axis to form a closed loop. What is the magnitude of the induced EMF in the loop at time $t$?

A variable inductor has an inductance given by $L(x) = L_0 e^{-\alpha x}$, where $x$ is the position of a movable magnetic core, $L_0$ is a positive constant in Henrys ($\text{H}$), and $\alpha$ is a positive constant in $\text{m}^{-1}$. The inductor is connected in series with a resistor of resistance $R$ ($\Omega$) and a programmable voltage source that maintains a strictly constant current $I_0$ ($\text{A}$) in the circuit. The core is pulled mechanically in the $+x$ direction at a constant speed $v$ ($\text{m/s}$). Let $P_{ext}$ be the external mechanical power required to maintain the core's motion, and $P_{mag} = \frac{dU_B}{dt}$ be the rate of change of the magnetic energy stored in the inductor. Which of the following correctly gives the ordered pair $(P_{ext}, P_{mag})$ at any position $x$?

A horizontal metallic rod of mass $m$ and length $L$ is released from rest at $t=0$ and falls under gravity (acceleration $g$) along two vertical, frictionless, perfectly conducting rails. The rails are connected at the top ($y=0$) by a resistor of resistance $R$. The entire system is immersed in a uniform, time-varying magnetic field $\vec{B} = B_0 t \hat{k}$ perpendicular to the plane of the rails, where $B_0$ is a positive constant. The rod experiences a viscous drag force of magnitude $F_d = b v$, where $v$ is its downward speed and $b$ is a positive constant. Let $y$ be the downward displacement of the rod and $I$ be the magnitude of the instantaneous induced current. Which of the following expressions gives the instantaneous rate of loss of mechanical energy of the rod?

A small conducting loop of mass $m$, resistance $R$, and area $A$ is attached to a light rigid rod of length $L$ to form a simple pendulum. The pendulum oscillates in a vertical $xy$-plane, with $x$ being the horizontal displacement from equilibrium. A non-uniform magnetic field $\vec{B}$ is perpendicular to the oscillation plane, varying as $B(x) = B_0 + \beta x^2$ ($B_0, \beta > 0$). The loop is released from a small initial horizontal displacement $x_0$ ($x_0 \ll L$). Assume the loop's dimensions are small enough that $B$ is uniform over its area at any position $x$, its plane remains perpendicular to $\vec{B}$, and self-inductance is negligible. Due to electromagnetic damping, the mechanical energy decreases. Assuming the energy loss per cycle is very small, find the time $t_{1/2}$ required for the total mechanical energy to reduce to half of its initial value.

A ferromagnetic solid sphere of moment of inertia $I = 0.1 \text{ kg m}^2$ is driven by a motor providing a constant external torque $\tau_0 = 0.53 \text{ N m}$ in a uniform magnetic field. The rotation induces eddy currents, dissipating power $P_e = k \omega^2$, where $k = 0.02 \text{ N m s}$ and $\omega$ is the angular velocity. Additionally, magnetic hysteresis dissipates an energy $E_h = 0.06\pi \text{ J}$ per complete revolution. If the sphere starts from rest, what is its steady-state angular velocity $\omega_0$ and the time $t$ taken to reach $\omega = \frac{\omega_0}{2}$?

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down a rough inclined plane of angle $\alpha$. Its center is connected to a fixed support at the top of the incline by a uniform heavy spring of mass $m$ and stiffness $k$, which is parallel to the incline. The cylinder is also coupled to a stationary superconducting coil of inductance $L$ via an ideal, massless electromechanical transducer. The transducer ensures that the magnetic flux $\Phi$ linked to the inductor is directly proportional to the displacement $x$ of the cylinder's center from the spring's unstretched position, given by $\Phi = \beta x$ (where $\beta$ is a positive constant). The circuit has zero electrical resistance and the initial current is zero when $x=0$. If the cylinder is released from rest at $x=0$, what is the angular frequency $\omega$ of its oscillations?

A toroidal inductor is constructed with a non-magnetic core of uniform height $h$. The core is densely wound with $N$ turns of wire carrying a current $I$. The inner boundary of the core is a cylinder of constant radius $a$. The outer boundary of the core has a variable radius $b(\theta)$ that depends on the azimuthal angle $\theta$ as: $b(\theta) = \begin{cases} a e^{k \theta} & \text{for } 0 \leq \theta \leq \pi \\ a e^{k (2\pi - \theta)} & \text{for } \pi < \theta \leq 2\pi \end{cases}$ where $k$ is a positive dimensionless constant. Assuming the magnetic field inside the core at a distance $r$ from the central axis is given by $B = \frac{\mu_0 N I}{2\pi r}$, what is the self-inductance $L$ of this toroid?

A small circular coil of radius $a$ and $N$ turns is placed parallel to the $xy$-plane at a height $z = d$ above a perfectly conducting infinite plane located at $z = 0$. An alternating current $I(t) = I_0 \cos(\omega t)$ flows through the coil. The frequency $\omega$ is such that $d = \lambda/3$, where $\lambda$ is the wavelength of the emitted electromagnetic radiation. A small secondary receiver is placed at a distance $R$ ($R \gg d, \lambda$) from the origin, such that its position vector makes an angle $\theta = \pi/3$ with the positive $z$-axis. Assuming the coil acts as an ideal magnetic dipole and the space above the plane is vacuum, what is the time-averaged intensity of the electromagnetic radiation at the receiver's location?