Magnetism and Matter Set-4

Q1. A long solenoid (length ≫ radius) has $500$ turns per metre and carries a current $3.0\ \mathrm{A}$. Its core is filled with a linear magnetic material of volume susceptibility $\chi=0.4$. Assuming negligible demagnetizing effects, the magnetization $M$ of the material is:

Q2. A short cylindrical sample (demagnetizing factor along the field $N=0.02$) of a magnetic material with volume susceptibility $\chi=500$ is placed in a uniform external magnetic field $H_0=80\ \mathrm{A/m}$. For linear response the equilibrium magnetization satisfies $M=\chi\,(H_0 - N\,M)$. The value of $M$ (in A/m) is approximately:

Q3. A spherical sample of a linear magnetic material with susceptibility $\chi=2.0$ is placed in a uniform external magnetic field $H_0=600\ \mathrm{A/m}$. For a sphere the demagnetizing factor is $N=\frac{1}{3}$. Using $M=\chi(H_0-NM)$ and $B=\mu_0(H+M)$, the magnetic induction $B$ inside the sphere (in tesla) is closest to:

Q4. Assertion (A): A uniformly magnetized sphere with magnetization $\mathbf{M}$ produces, at every point outside the sphere, the same magnetic field as that of a point dipole of moment $\mathbf{m}=\mathbf{M}V$ placed at the centre.
Reason (R): The bound surface current on the sphere is equivalent to a single circular current loop at the equator; that loop produces the identical external field of a point dipole.

Q5. A paramagnetic material at temperature $T=300\ \mathrm{K}$ has volume susceptibility $\chi=2.0\times 10^{-3}$. The number density of magnetic ions is $N=2.5\times 10^{28}\ \mathrm{m^{-3}}$. Using Curie's law in SI units $\displaystyle \chi=\frac{\mu_0 N \mu^2}{3k_B T}$, determine the magnetic moment $\mu$ per ion in units of the Bohr magneton $\mu_B$ (take $\mu_B=9.274\times 10^{-24}\ \mathrm{A\cdot m^2}$):

Q6. A very long solenoid has $n=1000\ \text{turns/m}$ and carries a steady current $I=2.0\ \text{A}$. A long soft‑iron rod (linear magnetic susceptibility $\chi_m=2000$) is fully inserted along its axis. Neglect end effects and demagnetizing fields. The magnetic induction $B$ inside the rod is approximately

Q7. A small paramagnetic sphere of susceptibility $\chi=0.01$ is placed in a uniform external magnetic field $H_0=1.0\times10^4\ \mathrm{A/m}$. Taking the demagnetizing factor for a sphere as $N=\tfrac{1}{3}$ and using linear response, the magnetization $M$ inside the sphere is

Q8. A small paramagnetic sphere (volume $V=1.0\ \text{cm}^3$) with susceptibility $\chi=0.01$ is placed at a point where the magnetic field has magnitude $H=2.0\times10^4\ \mathrm{A/m}$ and an axial gradient $dH/dx=5.0\times10^3\ \mathrm{A/m}^2$. Using the approximation $U=-\tfrac{1}{2}\mu_0\chi V H^2$ for the magnetic potential energy of a linear sample, the magnitude of the force $F_x\simeq -\partial U/\partial x$ on the sphere is closest to

Q9. Statement–Reason type. Read the two statements and choose the correct option.
Statement 1 (S1): When a ferromagnetic specimen completes one full hysteresis cycle in the $B$–$H$ plane, the net thermal energy dissipated per unit volume equals the area enclosed by the hysteresis loop.
Statement 2 (S2): The infinitesimal work done per unit volume by the applied magnetic field when $B$ changes by $dB$ is $H\,dB$, so the total work done over a closed cycle is $\displaystyle\oint H\,dB$, which equals the area of the loop and appears as heat.

Q10. A very long solid cylinder (radius $R$) is uniformly magnetized along its axis with magnetization $\mathbf{M}=M\,\hat{z}$. Which of the following gives the bound surface current density and the magnetic induction deep inside the cylinder (far from ends)?

Q11. A long solenoid has $n=2000\ \text{turns/m}$ and carries current $I=0.50\ \text{A}$. Its core is a linear magnetic material with volume susceptibility $\chi_m=0.02$ and completely fills the solenoid. Using $H=nI$ and $M=\chi_m H$, the magnetization $M$ (in A/m) inside the core is:

Q12. A small spherical sample of a linear magnetic material with susceptibility $\chi=1.00$ is placed in a uniform applied magnetic field $H_0=600\ \mathrm{A/m}$. Taking the demagnetizing factor for a sphere as $N=\tfrac{1}{3}$ and using the self-consistent relation $M=\chi\big(H_0-NM\big)$, the magnetization $M$ inside the sphere (in A/m) is:

Q13. A long straight wire carries current $I_1$. A rectangular loop of width $a$ (distance between its two sides parallel to the wire) and length $b$ lies in the same plane with its near side at distance $d$ from the wire and its far side at $d+a$. The loop carries current $I_2$ in the same direction as the long wire. Using $B(r)=\dfrac{\mu_0 I_1}{2\pi r}$ for the wire and $F=I_2 L\times B$, the magnitude and direction of the net force on the loop is:

Q14. A solenoid has inductance $L(x)$ that increases when a soft-iron rod is pushed further in by an infinitesimal $dx$, so $dL>0$. The solenoid is connected to an ideal current source that maintains constant current $I$. For this small insertion, which statement is correct regarding energy and mechanical work?

Q15. A very long straight solid conductor of radius $a$ carries total steady current $I$ uniformly distributed. The coaxial region $a<r<b$ is filled with a linear magnetic material of relative permeability $\mu_r$, while the region $r>b$ is vacuum. Using Ampère's law in terms of $H$ and $B=\mu_0\mu_r H$ inside the magnetic shell, the magnetic induction $B(r)$ for $a<r<b$ is: