1.Derive the expression for the magnetic potential energy of a magnetic dipole of moment $\vec{m}$ placed in a uniform magnetic field $\vec{B}$.
2.A magnetic needle suspended parallel to a uniform magnetic field requires $\sqrt{3} \text{ J}$ of work to turn it through $60^\circ$. Calculate the torque needed to maintain the needle in this position.
3.Establish the formula for the time period of oscillation of a magnetic dipole placed in a uniform magnetic field: $T = 2\pi\sqrt{I/mB}$.
4.A short bar magnet of magnetic moment $0.48 \text{ J/T}$ is placed with its axis making an angle of $30^\circ$ with a uniform magnetic field. If it experiences a torque of $0.024 \text{ J}$, calculate the magnitude of the magnetic field.
5.Calculate the magnitude of the magnetic field of the bar magnet in the previous question at a distance of $10 \text{ cm}$ from the center of the magnet on (a) its axis, and (b) its equatorial lines.
6.Two identical short bar magnets of magnetic moment $M$ are placed forming a cross (at right angles to each other) such that their centers coincide. Find the net magnetic field at a point on the bisector of the angle between them, at a distance $d$ from the center.
7.A closely wound solenoid of $2000$ turns and area of cross-section $1.6 \times 10^{-4} \text{ m}^2$ carries a current of $4.0 \text{ A}$. It is suspended freely and a uniform horizontal magnetic field of $7.5 \times 10^{-2} \text{ T}$ is applied at an angle of $30^\circ$ to its axis. Find the torque on the solenoid.
8.A bar magnet of magnetic moment $M$ and magnetic length $L$ is cut into two equal halves (a) transverse to its length, and (b) along its length. What is the new magnetic moment of each part in both cases?
9.Show that the axial magnetic field of a finite solenoid of radius $a$ and length $2l$ carrying current $I$ is equivalent to that of a bar magnet.
10.State Gauss's Law in magnetism. Explain mathematically why the magnetic flux through any closed surface enclosing a bar magnet is strictly zero.
11.A Gaussian surface encloses a tightly wound coil of $N$ turns carrying current $I$. What is the net magnetic flux leaving the surface? Prove it logically.
12.Explain the fundamental difference between an electric dipole and a magnetic dipole based on Gauss's laws for electrostatics and magnetism.
13.Does Gauss's law for magnetism hold true for a non-uniform magnetic field? Justify your answer.
14.If we find a closed surface over which the integral $\oint \vec{B} \cdot d\vec{S}$ is non-zero, what revolutionary discovery in physics would this indicate?
15.Define magnetic declination, angle of dip, and the horizontal component of Earth's magnetic field. Establish the relation $B = \sqrt{B_H^2 + B_V^2}$.
16.A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip pointing down at $22^\circ$ with the horizontal. The horizontal component of Earth's magnetic field at that place is $0.35 \text{ G}$. Determine the magnitude of Earth's total magnetic field at that place. ($\cos 22^\circ \approx 0.927$)
17.At a certain location, the horizontal component of Earth's magnetic field is $\sqrt{3}$ times its vertical component. Calculate the angle of dip. If the total magnetic field is $0.4 \text{ G}$, find the horizontal component.
18.A ship has to reach a port located $15^\circ$ South of West. The magnetic declination at the starting point is $17^\circ$ West. What should be the steering direction on the magnetic compass to reach the port?
19.If the Earth's magnetic field is $0.4 \text{ G}$ at the magnetic equator, estimate the magnetic dipole moment of the Earth. Assume the Earth's radius to be $6.4 \times 10^6 \text{ m}$.
20.Explain why a compass needle does not show the correct geographic direction near the Earth's geographic poles.
21.An airplane is flying uniformly along the magnetic equator. What is the vertical component of Earth's magnetic field cutting across its metallic wings? Will there be an induced emf across the wings?
22.A dip circle is initially at the magnetic meridian and shows a true dip of $\delta$. If it is rotated by an angle $\theta$ in the horizontal plane, derive the expression for the apparent dip $\delta'$.
23.Differentiate between diamagnetic, paramagnetic, and ferromagnetic materials on the basis of their (i) behavior in a non-uniform magnetic field, (ii) magnetic susceptibility ($\chi$), and (iii) relative permeability ($\mu_r$).
24.State Curie's law for paramagnetic materials. Draw a graph showing the variation of magnetic susceptibility ($\chi$) with absolute temperature ($T$) for a paramagnetic material.
25.Explain the atomic cause of diamagnetism. Why is the diamagnetic susceptibility essentially independent of temperature?
26.A domain in solid iron is in the form of a cube of side $1 \mu\text{m}$. Estimate the magnetic dipole moment of the domain. (Given: Density of iron = $7.9 \text{ g/cm}^3$, atomic mass = $55\text{ u}$, dipole moment of each iron atom = $9.27 \times 10^{-24} \text{ A m}^2$).
27.An iron rod of volume $10^{-4} \text{ m}^3$ and relative permeability $1000$ is placed inside a long solenoid wound with $500 \text{ turns/m}$. If a current of $0.5 \text{ A}$ is passed through the solenoid, find the magnetic moment of the iron rod.
28.Why does a paramagnetic material display much weaker magnetism compared to a ferromagnetic material, even though both have unpaired electrons?
29.The magnetic susceptibility of magnesium at $300\text{ K}$ is $1.2 \times 10^{-5}$. At what absolute temperature will its susceptibility become $1.8 \times 10^{-5}$?
30.Draw the hysteresis loop ($B-H$ curve) for a ferromagnetic material. Clearly mark the points indicating retentivity and coercivity on the graph.
31.Based on the hysteresis loops, explain why soft iron is preferred over steel for making the cores of transformers and AC electromagnets.
32.What is the physical significance of the area enclosed by a $B-H$ hysteresis loop?
33.A transformer core has a volume of $0.01 \text{ m}^3$ and the area of its $B-H$ loop is $100 \text{ J/m}^3$ per cycle. If it operates at a frequency of $50 \text{ Hz}$, calculate the power loss due to hysteresis.
34.A ferromagnetic material has a coercivity of $4 \times 10^3 \text{ A/m}$. A short iron rod of length $10 \text{ cm}$ made of this material is placed in a solenoid of 500 turns. What current must be passed through the solenoid to completely demagnetize the rod?
35.Distinguish between the required magnetic properties of a permanent magnet and an electromagnet.
36.Why do permanent magnets tend to lose their magnetism when dropped repeatedly or heated strongly? Explain using domain theory.
37.Can a material with extremely high retentivity but very low coercivity be used to manufacture a reliable permanent magnet? Justify your reasoning.
38.Mention the composition of Alnico. Why is it highly suited for making permanent magnets?
