1.
A stationary conducting loop of area $A$ and total resistance $R$ is placed in a spatially uniform but time-varying magnetic field $B(t) = B_0 e^{-t/\tau}$. Calculate the total net charge that flows through any cross-section of the loop from $t=0$ to $t=\infty$.
2.
A cylindrical region of radius $R$ contains a uniform magnetic field strictly parallel to its axis. If the field changes at a constant rate $\frac{dB}{dt} = \alpha$, derive the expression for the induced electric field $E(r)$ at a distance $r < R$ and $r > R$ from the axis.
3.
A flexible circular loop of initial radius $r_0$ is placed perpendicular to a steady uniform magnetic field $B_0$. The loop is pulled such that its radius shrinks at a constant rate $\frac{dr}{dt} = -v$. Find the instantaneous induced EMF as a function of time $t$.
4.
A square metallic loop of side $a$ and resistance $R$ is moved with a constant velocity $\vec{v} = v\hat{i}$ through a non-uniform magnetic field given by $\vec{B} = B_0 \left(\frac{x}{a}\right)\hat{k}$. Derive the mathematical expression for the induced current in the loop.
5.
A non-conducting thin ring of mass $M$, radius $R$, and carrying a uniform line charge density $\lambda$ is placed symmetrically in the cylindrical magnetic field of Question 2 ($\frac{dB}{dt} = \alpha$). Find the angular acceleration $\alpha_{ang}$ acquired by the ring.
6.
The magnetic flux $\Phi$ through a circuit of resistance $R$ changes according to the relation $\Phi = \alpha t^2 - \beta t$. Determine the exact time at which the induced current completely reverses its direction.
7.
A superconducting loop of strictly zero resistance and self-inductance $L$ is placed in a time-varying external magnetic field $B_{ext}(t)$. Prove mathematically using Faraday's law that the total magnetic flux linking the loop remains absolutely constant.
8.
Plot the induced EMF versus time graph for a rectangular loop of width $w$ being pulled with uniform velocity $v$ completely through a localized uniform magnetic field region of width $3w$.
9.
A circular copper coil of $N$ turns and area $A$ is rotated from a position orthogonal to a magnetic field $B$ to a position exactly parallel to it. If the coil's resistance is $R$, calculate the total heat dissipated during this rotation if it takes time $\Delta t$ at a uniform angular speed.
10.
Does the induced non-conservative electric field ($\oint \vec{E} \cdot d\vec{l} \neq 0$) created by a time-varying magnetic field have associated equipotential surfaces? Provide a rigorous conceptual justification.
11.
A small permanent magnet of mass $m$ and dipole moment $\mu$ is dropped vertically through a long, highly conducting copper pipe. Formulate the physical reasoning and condition under which the magnet will attain a constant terminal velocity.
12.
Two coplanar, concentric circular loops have radii $r$ and $R$ ($r \ll R$). A time-varying current $I(t) = I_0 \sin(\omega t)$ flows in the larger loop. Determine the instantaneous direction of the magnetic force exerted on the smaller loop, assuming its resistance is $R_{small}$.
13.
A conducting ring is held horizontally, and a small bar magnet is released from rest exactly along its central axis. Sketch a qualitative graph of the magnet's acceleration $a$ as a function of time $t$ as it falls from above, passes through, and drops below the ring.
14.
A rectangular loop is pulled out of a uniform magnetic field $B$ with constant velocity $v$. Prove definitively that the external mechanical power supplied $P_{ext}$ is equal to the rate of Joulean heat dissipation $I^2R$ using energy conservation principles.
15.
An aluminum ring is placed on a vertical solenoid projecting above an AC power supply. Explain Thomson's "Jumping Ring" experiment using Lenz's Law and the phase difference between the inducing flux and induced current.
16.
Two perfectly identical loops, one complete and one with a microscopic radial cut (gap), are oriented horizontally. Identical magnets are dropped simultaneously through both. Compare their transit times and explain the thermodynamic differences.
17.
A metal sheet is pulled at a constant velocity $\vec{v}$ through a localized, non-uniform perpendicular magnetic field. Use Lenz's law to explain the origin of "magnetic drag," and prove that the drag force is directly proportional to the velocity $v$.
18.
A circular loop of wire is placed in a uniform magnetic field directed out of the page. If the field magnitude $B$ is rapidly decreased, determine the direction of the induced magnetic dipole moment of the loop.
19.
Apply Lenz's law to a purely inductive circuit ($R=0$). If the external EMF is abruptly removed, how does the inductor attempt to conserve the system's energy state, and what prevents infinite current?
20.
A bar magnet is pushed into a coil connected to a capacitor instead of a galvanometer. Which plate of the capacitor becomes positively charged relative to the magnet's approaching North pole?
21.
A perfectly conducting rod of length $L$ and mass $m$ slides frictionlessly down two vertical, parallel conducting rails separated by $L$. The rails are connected at the top by a capacitor $C$. A uniform horizontal magnetic field $B$ exists. Find the steady acceleration of the falling rod.
22.
Replace the capacitor in Question 21 with a pure inductor $L_{ind}$. If the rod is released from rest, derive the differential equation of its motion and prove that it will execute Simple Harmonic Motion (SHM). Find its angular frequency.
23.
A conducting rod of length $L$ rotates with uniform angular velocity $\omega$ in a perpendicular uniform magnetic field $B$. The axis of rotation is perpendicular to the rod and passes through a point at a distance $x$ from one end. Calculate the EMF induced between the two ends of the rod.
24.
A semicircular conducting wire of radius $R$ is rotated with constant angular velocity $\omega$ about a vertical diameter in a uniform horizontal magnetic field $B_0$. Determine the instantaneous EMF induced across the ends of the wire.
25.
A conducting rod of mass $m$ and length $l$ is projected with an initial velocity $v_0$ along two horizontal parallel rails situated in a uniform vertical magnetic field $B$. The rails are connected by a resistor $R$. Calculate the total distance the rod travels before coming to rest.
26.
Derive the expression for the EMF induced in a Faraday Disc Dynamo: a solid conducting disc of radius $R$ rotating with angular velocity $\omega$ in a uniform perpendicular magnetic field $B_0$.
27.
A rectangular loop of mass $m$, resistance $R$, and dimensions $a \times b$ falls freely under gravity into a uniform magnetic field $B$ perpendicular to its plane. Derive the exact expression for its terminal velocity assuming the field region is infinitely deep.
28.
Using the Lorentz force formulation $\vec{F} = q(\vec{v} \times \vec{B})$, derive the steady-state radial electric field distribution inside a cylindrical rod of length $L$ rotating with $\omega$ about one end in a uniform magnetic field $B$.
29.
A straight conducting wire of length $L$ is moving with velocity $\vec{v} = v_0\hat{i}$ in a non-uniform magnetic field $\vec{B} = B_0 y \hat{k}$. If the wire is kept perfectly parallel to the y-axis (from $y=0$ to $y=L$), calculate the motional EMF induced across it.
30.
Two parallel conducting rails are placed on an incline of angle $\theta$. A rod of mass $m$ and length $l$ slides down. A uniform vertical magnetic field $B$ exists throughout. If the rails are closed by a resistance $R$ at the bottom, find the steady terminal velocity of the sliding rod.
31.
A solid metallic cylinder of radius $R$, length $L$, and resistivity $\rho$ is placed in an alternating magnetic field $B(t) = B_0 \sin(\omega t)$ parallel to its axis. Derive an approximate expression for the average power dissipated per unit volume due to eddy currents.
32.
Explain the concept of "Skin Effect" in AC power transmission. How do internal high-frequency eddy currents mathematically force the main current to flow strictly along the outer periphery of a solid conductor?
33.
A flat circular aluminum disc is allowed to oscillate as a pendulum in a strong electromagnet's gap. Prove mathematically that the electromagnetic damping force $F_d$ is directly proportional to its instantaneous velocity $v$, giving rise to exponential decay $e^{-\gamma t}$.
34.
In the design of modern transformers, state the empirical formula showing the dependence of eddy current power loss $P_e$ on the AC frequency $f$ and maximum flux density $B_{max}$. Why are ferrites superior to silicon steel at high (radio) frequencies?
35.
If a solid transformer core is replaced by a core built from $n$ electrically insulated laminations of equal total volume, prove that the eddy current loss is theoretically reduced by a factor of $1/n^2$.
36.
Analyze the operation of a magnetic levitation (Maglev) train. How do the induced eddy currents in the guideway create both a repulsive levitation force and an opposing drag force simultaneously?
37.
An induction heater operates based on an effective "depth of penetration" ($\delta$) of the magnetic field into the metal. State the dependence of $\delta$ on the AC frequency $\omega$, permeability $\mu$, and conductivity $\sigma$ of the material.
38.
A metallic coin is placed on a coil connected to an AC source. When switched on, the coin violently jumps into the air. If the AC frequency is significantly increased, does the jump height increase or decrease? Justify physically.
39.
Is it physically possible to design a magnetic brake that utilizes eddy currents to bring a spinning metal disc to an absolute, perfect dead stop in a finite time without mechanical friction? Explain the asymptotic nature of the damping.
40.
Discuss the role of eddy currents in the operation of an analog induction-type energy meter. How do two out-of-phase AC magnetic fluxes create a driving torque on the aluminum disc?
41.
Derive the expression for the mutual inductance $M$ between a very long straight wire carrying current $I$ and a coplanar rectangular loop of sides $a$ and $b$, if the nearer side $b$ is parallel to the wire at a distance $x_0$.
42.
Calculate the mutual inductance $M$ of a small circular loop of radius $r$ placed exactly at the center of a much larger coplanar circular loop of radius $R$ (where $r \ll R$).
43.
Derive the exact formula for the self-inductance per unit length of a long coaxial cable consisting of an inner solid conductor of radius $a$ and a thin outer cylindrical shell of radius $b$. (Assume high-frequency skin effect limits current strictly to the surfaces).
44.
Determine the total self-inductance $L$ of a toroid with a rectangular cross-section of height $h$, inner radius $a$, outer radius $b$, and wound with $N$ total turns.
45.
Prove the Reciprocity Theorem of mutual inductance ($M_{12} = M_{21}$) rigorously using the concept of magnetic vector potential $\vec{A}$ and Neumann's formula for two arbitrary closed loops.
46.
Two tightly wound coils have self-inductances $L_1$ and $L_2$. If they are perfectly coupled magnetically (no flux leakage), prove that their mutual inductance is precisely $M = \sqrt{L_1 L_2}$.
47.
Two inductors $L_1$ and $L_2$ are connected in parallel. If their mutual inductance is $M$ (aiding configuration), derive the equivalent inductance $L_{eq}$ of the parallel combination.
48.
An LR circuit is connected to a DC battery of EMF $E$. Derive the expression for the instantaneous current $I(t)$ and determine the exact time $t$ required for the magnetic potential energy stored in the inductor to reach 25% of its maximum steady-state value.
49.
A solenoid of length $l$ is wound with a continuously varying pitch such that the turn density varies linearly as $n(x) = n_0 + \alpha x$ from $x=0$ to $x=l$. Set up the integral for its total self-inductance.
50.
Using the energy density formula for a magnetic field ($u_B = \frac{B^2}{2\mu_0}$), integrate over all space to independently verify the self-inductance $L = \mu_0 n^2 A l$ of an ideal long solenoid.
51.
An AC generator armature rotates with a non-uniform angular acceleration $\alpha$. If $\theta(t) = \frac{1}{2}\alpha t^2$, derive the expression for the instantaneous induced EMF. How does the frequency and amplitude vary with time?
52.
A generator coil is connected to a purely resistive load $R$. Derive the expression for the instantaneous mechanical torque $\tau_{mech}(t)$ that the external prime mover must supply to keep the coil rotating at a strictly constant angular velocity $\omega$.
53.
For a standard sinusoidal AC generator producing $e(t) = E_0 \sin(\omega t)$, prove via integration that the average EMF over one complete cycle is rigorously zero, but the average over a positive half-cycle is $\frac{2}{\pi} E_0$.
54.
Explain the fundamental principle of a Three-Phase AC Generator. Write the three simultaneous mathematical equations for the EMFs induced in the three symmetrically spaced armature coils.
55.
The armature coil of an AC generator possesses its own internal self-inductance $L$ and resistance $R_a$. If the terminals are suddenly short-circuited, set up the differential equation for the short-circuit current and determine its steady-state amplitude.
56.
Define the "Form Factor" and "Peak Factor" for an AC waveform. Calculate their exact numerical values for the pure sinusoidal EMF produced by an ideal AC generator.
57.
A highly unusual generator has its armature rotating at angular speed $\omega$, but the external magnetic field providing the flux is also oscillating independently as $B(t) = B_0 \cos(\omega_0 t)$. Derive the resulting EMF waveform and identify the beat frequencies generated.
58.
Discuss the phenomenon of "Armature Reaction" in a heavily loaded practical AC generator. How does the magnetic field produced by the induced load current physically distort the main pole magnetic flux?
59.
Compute the average mechanical power $\langle P \rangle$ supplied by the external turbine to an AC generator connected to a load $R$ over one full rotation cycle, showing it equals the average electrical power dissipated.
60.
If the physical shape of the rotating armature is changed from rectangular to perfectly circular (keeping total area $A$, turns $N$, and field $B$ identical), mathematically prove whether the peak EMF generated changes or remains exactly invariant.
