Magnetism and Matter Set-1

Q1. A long solenoid of length $0.40\ \mathrm{m}$ has $500$ turns and carries a current $2.0\ \mathrm{A}$. A soft-iron core with relative permeability $\mu_r=200$ is fully inserted. Neglect end effects; calculate the magnetic induction $B$ inside the solenoid.

Q2. Assertion (A): A very long solid cylinder of radius $R$ uniformly magnetised along its axis with $\mathbf{M}=M\hat{z}$ produces a uniform magnetic induction inside $B_{\text{in}}=\mu_0 M$ and (to a good approximation) negligible magnetic field outside.
Reason (R): The uniform magnetisation is equivalent to an azimuthal bound surface current density $\mathbf{K}_b=\mathbf{M}\times\hat{n}=M\hat{\phi}$, which produces the same inside/outside field pattern as an ideal infinite solenoid.

Q3. A small magnetic dipole of magnetic moment magnitude $m=3.0\ \mathrm{A\cdot m^2}$ is oriented along the $+z$ axis and placed at $z=0$ in a magnetic field $\mathbf{B}(z)=\big(0,0,\,0.02+0.01z\big)\ \mathrm{T}$ (with $z$ in metres). Calculate the magnitude and direction of the force on the dipole.

Q4. Three ferromagnetic samples A, B and C are cycled by the same alternating magnetising field and produce the following qualitative hysteresis loops ($B$ vs $H$): loop A is very narrow with very small coercivity $H_c$ and small remanence $B_r$; loop B has moderate $H_c$ and $B_r$; loop C is wide with large $H_c$ and large $B_r$. Which assignment of practical applications is most appropriate?

Q5. Assertion (A): A freely movable diamagnetic rod placed in a non‑uniform external magnetic field moves toward the region of weaker magnetic field.
Reason (R): The induced magnetic moment in a diamagnetic material is opposite to the applied field, so the system’s magnetic potential energy is minimised when the rod moves to a region of smaller field.

Q6. A toroidal core has mean radius $r=0.10\ \mathrm{m}$ and cross‑sectional area $A=1.0\times10^{-4}\ \mathrm{m^2}$. It has $N=500$ turns and carries a current $I=2.0\ \mathrm{A}$. The core’s relative permeability is $\mu_r=1000$. Assuming the magnetic field is confined to the core, calculate the total magnetic energy stored in the toroid. (Use $u=\dfrac{B^2}{2\mu}$ and $B=\dfrac{\mu_0\mu_r NI}{2\pi r}$ for the mean path.)

Q7. A long solenoid produces an almost uniform internal magnetic field $\mathbf{B}_0$ directed along $+z$ and negligible field outside. A small bar magnet with magnetic moment $\mathbf{m}$ is placed just outside the solenoid on its axis where $B\approx 0$ and is free to move along $z$. If $\mathbf{m}$ is aligned along $+z$, which of the following describes the initial motion of the magnet?

Q8. Assertion (A): For a uniformly magnetised solid sphere with magnetisation $\mathbf{M}$, the internal magnetic field intensity is $\mathbf{H}_{\text{in}}=-\dfrac{1}{3}\mathbf{M}$ (the demagnetising field).
Reason (R): Bound surface currents on the sphere produce, inside the sphere, a uniform field that opposes $\mathbf{M}$ and has magnitude $M/3$.

Q9. A qualitative plot of magnetisation $M$ versus applied field $H$ at room temperature for three materials I, II and III is described as follows: I is a straight line with a small negative slope through the origin; II is a straight line with a small positive slope through the origin; III shows a very steep initial rise and then saturates at high $H$ (nonlinear). Identify I, II and III respectively.

Q10. A ferromagnetic sample's hysteresis loop in the $B$–$H$ plane is approximately rectangular and symmetric about the origin with coercive field $H_c=200\ \mathrm{A/m}$ and remanent induction $B_r=0.50\ \mathrm{T}$. Using this rectangular approximation, estimate the magnetic energy loss per cycle per unit volume in $\mathrm{J/m^3}$. (Use $W_{\text{loss}}=\oint H\,dB$.)