1.
A rectangular coil of 100 turns, with dimensions 20 cm x 10 cm, is placed perpendicular to a uniform magnetic field of 0.05 T. If the field drops to zero in 0.5 s, calculate the average induced EMF.
2.
A magnetic field is given by $\vec{B} = (2\hat{i} + 3\hat{j} - 4\hat{k})$ T. Calculate the magnitude of the magnetic flux through a surface of area $5 \text{ m}^2$ lying completely in the xy-plane.
3.
The magnetic flux linked with a coil varies with time as $\Phi = (5t^3 - 3t^2 + 2t - 1)$ Wb. Find the magnitude of the induced EMF exactly at $t = 2$ s.
4.
State Faraday's second law of electromagnetic induction. How does the equation dynamically connect EMF to the rate of change of magnetic flux?
5.
A circular loop of radius 10 cm is placed in a time-varying magnetic field $B(t) = 0.5 \sin(100\pi t)$ T, normal to its plane. Derive the specific expression for the induced EMF as a function of time.
6.
A closed loop of wire of area $A$ and resistance $R$ is placed in a uniform magnetic field $B$ perpendicular to its plane. If it is quickly withdrawn from the field in time $\Delta t$, derive the expression for the total electrical charge $Q$ that flows through the loop.
7.
A coil of 50 turns and area $0.1 \text{ m}^2$ is rotated from a position parallel to a magnetic field of 0.2 T to a position completely perpendicular to it in 0.1 s. Find the average induced EMF.
8.
Does the magnitude of induced EMF in a coil depend on the resistance of the coil? Does the induced current depend on it? Provide a mathematical justification.
9.
A 10 $\Omega$ coil of 200 turns is linked with an initial flux of 0.04 Wb per turn. If the external magnetic field is reversed in direction in 0.2 s, calculate the average induced current in the coil.
10.
A flexible circular loop of radius 20 cm is situated in a uniform magnetic field of 0.5 T, perpendicular to its plane. If it is stretched and pulled into a perfect square shape in 0.2 s, calculate the average induced EMF.
11.
State Lenz’s Law. Explain comprehensively how it is perfectly consistent with the fundamental principle of conservation of energy.
12.
A strong bar magnet is dropped vertically through a thick horizontal copper ring. Explain scientifically why its acceleration will be strictly less than $g$ until it fully exits the ring.
13.
Two circular loops are placed concentrically and coplanar. If the steady clockwise current in the outer loop is suddenly switched off, what will be the direction of the momentarily induced current in the inner loop?
14.
A closed rectangular copper loop is pushed completely into a uniform magnetic field region, kept moving inside, and then pulled out. Plot a qualitative graph showing the external opposing force required versus time, assuming constant velocity throughout.
15.
A metallic ring is held horizontally and a magnet is dropped through it with its North pole pointing downwards. Will the induced current be clockwise or counter-clockwise as seen from above? Justify your answer.
16.
Explain conceptually why a metallic pendulum swinging freely in a uniform magnetic field eventually comes to rest much faster than an identical wooden pendulum.
17.
An aluminum ring B faces the pole of an electromagnet A. If the current $I$ in the electromagnet is suddenly increased, will the ring B be physically attracted or repelled? State the law used.
18.
A highly conducting closed coil is kept stationary in a uniform magnetic field. If the field is slowly turned off to zero, the coil measurably heats up. Explain the primary source of this thermal energy.
19.
Can Lenz's law be used exclusively to predict the direction of a motional EMF? Briefly explain how it logically aligns with Fleming's Right-Hand Rule.
20.
A visible spark is often observed when a switch disconnecting a highly inductive circuit (like a heavy motor) is suddenly opened. Explain this physical phenomenon using Faraday's and Lenz's laws.
21.
Derive the standard expression for the motional EMF ($E = Blv$) strictly using the concept of Lorentz magnetic force acting on the free electrons inside a moving conductor.
22.
A 1.2 m long metal rod is moved at a uniform velocity of 5 m/s in a direction exactly perpendicular to a uniform magnetic field of 0.4 T. Calculate the motional EMF induced across its two ends.
23.
A conducting rod of length $L$ is rotated with uniform angular velocity $\omega$ about one of its hinged ends in a uniform perpendicular magnetic field $B$. Derive the expression for the EMF induced across its ends ($E = \frac{1}{2}B\omega L^2$).
24.
A bicycle wheel with 10 metallic spokes, each 0.5 m long, is rotated with an angular speed of 120 rpm in a plane normal to the horizontal component of Earth's magnetic field ($H_E = 0.4 \text{ G}$). Find the induced EMF between the axle and the rim. ($1 \text{ G} = 10^{-4} \text{ T}$).
25.
A straight conducting wire falls horizontally downwards under gravity, remaining perfectly parallel to the ground aligned East-West. If it falls in a region where the Earth's magnetic field has both horizontal and vertical components, which specific component induces the EMF?
26.
Calculate the mechanical power required to steadily pull a rectangular conducting loop of total resistance $R$ out of a uniform magnetic field $B$ with a constant velocity $v$, if the side length perpendicular to the motion is $l$.
27.
In the context of the previous question, prove mathematically that the mechanical power provided by the external agent is exactly equal to the Joule heating power ($I^2 R$) dissipated electrically in the loop.
28.
A rectangular loop of sides 10 cm and 5 cm with an internal resistance of 2 $\Omega$ is pulled out of a uniform magnetic field of 0.5 T perpendicular to its plane, with a velocity of 1 m/s moving along the longer side. Calculate the induced current.
29.
A train is moving from North to South at a speed of 108 km/h. If the vertical component of the Earth's magnetic field is $0.2 \times 10^{-4} \text{ T}$ and the distance between the parallel rails is 1.5 m, calculate the EMF induced across the metal axle.
30.
A straight conducting rod of length $l$ moves with velocity $v$ at an angle $\theta$ relative to a uniform magnetic field $B$. The rod itself is positioned perpendicular to both $v$ and $B$. What is the precise magnitude of the motional EMF developed?
31.
What are eddy currents? Describe the fundamental condition and mechanism required for their formation in bulk metallic pieces.
32.
Explain mathematically and physically why constructing a transformer core from laminated sheets reduces eddy current losses significantly compared to using a solid bulk core.
33.
How do electromagnetic brakes in modern high-speed trains utilize the concept of eddy currents? Describe the specific energy conversion taking place during braking.
34.
An aluminum plate is allowed to swing freely like a pendulum between the poles of a strong electromagnet. What instantly happens to the swing when the electromagnet is switched on? What is this specific physical phenomenon called?
35.
How exactly are eddy currents utilized in the operation of an industrial induction furnace to melt heavy scrap metals?
36.
Why are the metallic swinging bobbins or plates of dead-beat galvanometers often constructed with large slots cut into them?
37.
A solid conducting cylinder and a hollow conducting cylinder of the exact same material and outer dimensions are rotated in a transverse uniform magnetic field. Which one will experience greater electromagnetic damping and why?
38.
Briefly explain how a modern domestic induction cooktop boils water using the principles of electromagnetic induction and eddy currents within the cooking pan.
39.
State two practical engineering methods to minimize detrimental eddy current losses in the armature core of a heavy AC motor.
40.
Do eddy currents produce a measurable magnetic field of their own? If so, what is its orientation relative to the primary changing magnetic field that originally induced it?
41.
Define the coefficient of self-inductance ($L$) of a coil. Show mathematically that it is numerically equal to the magnetic flux linked with the coil when a unit current flows through it.
42.
Derive the formal expression for the self-inductance of a long ideal air-cored solenoid of length $l$, cross-sectional area $A$, and having $N$ total turns.
43.
A steady current of 2 A flowing through a coil of 500 turns produces a magnetic flux of $4 \times 10^{-3} \text{ Wb}$ per turn. Calculate the total self-inductance of the coil.
44.
If the current in an inductive coil uniformly changes from 10 A to 0 A in 0.1 seconds and the induced average EMF is measured as 50 V, find the self-inductance of the coil.
45.
Define mutual inductance ($M$) between two neighboring coils. Write its SI unit and derive its basic dimensional formula.
46.
Derive the formal expression for the mutual inductance of two long coaxial solenoids of length $l$, inner radius $r_1$, outer radius $r_2$, and turn densities $n_1$ and $n_2$.
47.
Two heavily coupled coils have a mutual inductance of 1.5 H. If the current in the primary circuit varies as $I = 5 \sin(100\pi t)$ Amperes, write the exact mathematical expression for the instantaneous EMF induced in the secondary coil.
48.
An air-cored solenoid has an initial self-inductance of 2.0 mH. If a solid soft iron core of relative permeability ($\mu_r$) 500 is inserted completely into it, what will be its new calculated self-inductance?
49.
Calculate the magnetic potential energy stored in a heavy inductor of 100 mH when a steady current of 4 A is flowing through it. What specific physical form does this stored energy take?
50.
Two identical coils, each of self-inductance $L$, are connected in series. Find the equivalent inductance of the combination if their mutual inductance is $M$ and they are wound in the same sense (fluxes aiding each other).
51.
Explain the fundamental physical principle and working mechanism of an AC generator. State the roles of the magnetic field and the armature.
52.
A rectangular armature coil of $N$ turns and cross-sectional area $A$ rotates with a uniform angular velocity $\omega$ in a uniform magnetic field $B$. Derive the standard expression for the instantaneous induced EMF ($e(t)$).
53.
An AC generator consists of a coil of 100 turns and cross-sectional area $0.1 \text{ m}^2$, rotating at a constant angular speed of 60 rad/s in a uniform magnetic field of 0.04 T. Calculate the theoretical peak induced EMF.
54.
In the context of the generator in Question 53, if the total electrical resistance of the closed coil circuit is 50 $\Omega$, calculate the maximum (peak) induced current.
55.
Describe the graphical variation of induced EMF with the angle of rotation ($\theta = \omega t$) for one complete cycle of the AC generator coil. At what specific angles is the EMF zero?
56.
What is the critical fundamental difference in the mechanical construction of the output sliding contacts between an AC generator and a DC generator?
57.
A circular coil of 50 turns and radius 10 cm rotates rapidly at 3000 rpm in a uniform magnetic field of 0.1 T. Calculate the maximum (peak) EMF generated across its terminals.
58.
In an operating AC generator, what happens precisely to both the magnitude (peak EMF) and the frequency of the alternating output if the physical speed of rotation of the armature is strictly doubled?
59.
Describe the essential role of the stationary brushes (usually constructed of graphite/carbon) in the continuous operation of an AC generator.
60.
An AC generator is connected to a heavy external load. Does the generator inherently supply mechanical energy or electrical energy? According to Lenz's law, why does it become significantly harder to manually turn the generator crank when this heavy electrical load is switched on?
