1.A bar magnet of magnetic moment $M$ is bent into the shape of a semi-circle. Calculate its new magnetic dipole moment.
2.A straight steel wire of length $L$ has a magnetic moment $M$. When it is bent into an arc of a circle subtending an angle of $60^\circ$ at its center, what will be its new magnetic moment?
3.Two short magnetic dipoles of moments $m_1$ and $m_2$ are placed coaxially (along the same line) with their centers separated by a distance $r$. Derive an expression for the force of interaction between them.
4.A thin rectangular magnet suspended freely has a period of oscillation $T$ in a uniform magnetic field. If it is cut into four equal parts (two cuts, one parallel to its length and one parallel to its width) and one piece is allowed to oscillate in the same field, what will be its new time period?
5.A magnetic dipole $\vec{m} = m_0 \hat{i}$ is placed at the origin. Determine the magnitude and direction of the magnetic field at a point $P(a, a)$ in the x-y plane.
6.A short bar magnet produces a neutral point on its equatorial line at a distance $d$ from its center. If the magnet is rotated by $90^\circ$ in the horizontal plane, what will be the magnitude of the net magnetic field at the same point $d$? (Let Earth's horizontal magnetic field be $B_H$).
7.A vibration magnetometer placed in magnetic meridian has a small bar magnet. The magnet executes oscillations with a time period of $2 \text{ s}$ in Earth's horizontal magnetic field of $24 \mu\text{T}$. When a horizontal field of $18 \mu\text{T}$ is produced opposite to the Earth's field, what will be the new time period of the magnet?
8.A small coil of $10$ turns, area $1 \text{ cm}^2$, and carrying a current of $2 \text{ A}$ is placed at a distance of $1 \text{ m}$ on the axial line of a short bar magnet of magnetic moment $10 \text{ A m}^2$. The coil's axis is perpendicular to the magnet's axis. Calculate the torque acting on the coil.
9.A magnetic needle has a magnetic moment of $5 \times 10^{-2} \text{ A m}^2$ and moment of inertia of $8.5 \times 10^{-7} \text{ kg m}^2$. It performs 10 complete oscillations in 6.70 s. Calculate the magnitude of the magnetic field.
10.Two identical short bar magnets are placed with their axes perpendicular to each other, their centers separated by distance $d$. If the magnetic moment of each is $M$, find the net magnetic field at the mid-point of the line joining their centers.
11.A dip circle is mounted such that its vertical plane makes an angle $\theta$ with the magnetic meridian. If the apparent dip observed is $\delta_1$, and when the plane is rotated by $90^\circ$ the apparent dip is $\delta_2$, prove that the true dip $\delta$ is given by: $\cot^2\delta = \cot^2\delta_1 + \cot^2\delta_2$.
12.If the true angle of dip at a place is $30^\circ$, find the apparent angle of dip observed in a vertical plane making an angle of $45^\circ$ with the magnetic meridian.
13.Assuming the Earth's magnetic field is produced by a perfectly centrally placed short magnetic dipole, derive the expression for the total magnetic field $B$ at a geomagnetic latitude $\lambda$.
14.If the horizontal component of Earth's magnetic field at the magnetic equator is $0.4 \text{ G}$, use the dipole model to calculate the total magnetic field at a geomagnetic latitude of $30^\circ$.
15.At a certain location, the Earth's magnetic field has a magnitude of $0.6 \text{ G}$ and a dip angle of $45^\circ$. If an anomaly causes the total field strength to increase by $10\%$ without changing the horizontal component, what is the new angle of dip?
16.A ship sails directly from the geographic equator to the geographic North Pole. Assuming the magnetic poles perfectly coincide with the geographic poles, write down the function describing the rate of change of the angle of dip with respect to latitude.
17.In a tangent galvanometer experiment, the magnetic needle shows a deflection of $60^\circ$ when a current $I$ is passed. If the current is reduced to $I/3$, what will be the new deflection of the needle?
18.A short bar magnet is placed in the magnetic meridian with its north pole pointing Geographic South. A neutral point is found at a distance of $30 \text{ cm}$ from the center of the magnet. If $B_H = 0.34 \text{ G}$, calculate the magnetic moment of the magnet.
19.If the bar magnet in the previous question is rotated by $180^\circ$ so its North pole points Geographic North, where will the new neutral points be located?
20.An electron beam passes through a region of crossed uniform electric and magnetic fields. If the electric field is turned off, the electrons travel in a circle of radius $R$. If the magnetic field is produced by the Earth (vertical component $B_V$), find an expression for the required accelerating voltage of the electron gun.
21.An iron ring of mean radius $15 \text{ cm}$ and cross-sectional area $20 \text{ cm}^2$ is uniformly wound with $500$ turns of wire carrying a current of $2 \text{ A}$. If the relative permeability of iron under these conditions is $1500$, calculate the Magnetic Intensity ($H$), Magnetization ($M$), and Total Magnetic Field ($B$).
22.The magnetic susceptibility of a paramagnetic material is $1.2 \times 10^{-5}$ at $300 \text{ K}$. Calculate its susceptibility at $200 \text{ K}$ and at $600 \text{ K}$.
23.A domain in ferromagnetic iron is in the shape of a cube of side length $1 \mu\text{m}$. Estimate the maximum possible dipole moment of this domain and its magnetization. (Density of iron $= 7.9 \text{ g/cm}^3$, Atomic mass $= 55 \text{ g/mol}$, Dipole moment per atom $= 9.27 \times 10^{-24} \text{ A m}^2$).
24.A superconducting material is placed in a uniform magnetic field $\vec{B}_{ext}$. What is the internal magnetic field, relative permeability, and magnetic susceptibility of the superconductor? Relate this to magnetic levitation.
25.A rod of a ferromagnetic material with dimensions $10 \text{ cm} \times 0.5 \text{ cm} \times 0.2 \text{ cm}$ is placed in a magnetizing field of $2 \times 10^3 \text{ A/m}$, which produces a magnetic field of $3.14 \text{ T}$ inside it. Calculate the pole strength of the rod.
26.Using the relation $\vec{B} = \mu_0(\vec{H} + \vec{M})$, show that the magnetic susceptibility $\chi$ and relative permeability $\mu_r$ are related by $\mu_r = 1 + \chi$.
27.A paramagnetic liquid is taken in a U-tube and one arm of the U-tube is placed between the pole pieces of a strong electromagnet. What happens to the level of the liquid in that arm when the field is switched on? Derive an expression for the rise in height $h$.
28.At what temperature will the magnetization of a paramagnetic sample equal $50\%$ of its saturation magnetization, assuming an applied field $B = 2.0 \text{ T}$ and atomic dipole moment $\mu \approx 10^{-23} \text{ J/T}$? (Hint: Analyze the high-temperature / weak-field limit approximation vs standard).
29.Explain the Curie-Weiss law. How does it modify Curie's law for ferromagnetic materials above their Curie temperature?
30.A specimen of iron of permeability $8 \times 10^{-3} \text{ Wb/Am}$ is placed in a magnetic field of strength $160 \text{ A/m}$. Find the magnetic induction and the magnetization of the iron.
31.The coercivity of a small bar magnet is $3 \times 10^3 \text{ A/m}$. It is placed inside a solenoid of $500$ turns and length $10 \text{ cm}$. What current must be passed through the solenoid to completely demagnetize the magnet?
32.A transformer core of volume $10^{-3} \text{ m}^3$ is subjected to an alternating magnetic field of frequency $50 \text{ Hz}$. If the area of the B-H hysteresis loop for the core material is $0.1 \text{ J/m}^3$, calculate the power dissipated as heat in the core.
33.Show from fundamental principles that the work done per unit volume by the magnetizing field in taking a ferromagnetic material through one complete hysteresis cycle is given by $\oint H \, dB$.
34.Material A has high retentivity and high coercivity. Material B has low retentivity and low coercivity. Material C has high retentivity and low coercivity. Which material is best suited for (i) Permanent Magnets, (ii) Electromagnets, and (iii) Magnetic recording tapes?
35.A magnetic material has a hysteresis loss of $300 \text{ J/m}^3$ per cycle. If the density of the material is $7500 \text{ kg/m}^3$ and its specific heat capacity is $400 \text{ J/(kg K)}$, find the rate of rise of temperature of the material if it operates at $50 \text{ Hz}$ in an AC field, assuming no heat is lost to the surroundings.
36.Explain why soft iron cores are laminated in transformers. How does lamination differ in purpose from selecting a material with a narrow hysteresis loop?
37.The B-H curve for a ferromagnet has a retentivity of $1.2 \text{ T}$ and a coercivity of $5000 \text{ A/m}$. If a toroidal coil of radius $10 \text{ cm}$ and $1000$ turns is wound on a core of this material, what minimum reverse current is needed to bring the magnetic flux density to zero?
38.Discuss the transition of magnetic properties of a ferromagnetic domain when heated from Absolute Zero ($0 \text{ K}$) up to and beyond its Curie Temperature ($T_C$).
39.A permanent magnet made of steel and an electromagnet made of soft iron are designed to have the same maximum field strength. Discuss the difference in their mass and volume required to achieve this.
40.Is the magnetic field inside a permanently magnetized object uniform? Relate the internal "demagnetizing field" to the physical shape of the permanent magnet (e.g., a short cylinder vs. a long needle).
