Magnetism and Matter Set-6

Q1. A rectangular coil with $N$ turns and sides $a$ and $b$ carries current $I$. It is placed in a uniform magnetic field $\mathbf{B}$ such that the plane of the coil makes an angle $30^\circ$ with $\mathbf{B}$. The magnitude of the torque on the coil is

Q2. A long solenoid has $n$ turns per unit length and cross-sectional area $A$, and it carries a steady current $I$. A linear magnetic rod (same cross-section) of susceptibility $\chi_m$ is inserted so that a length $x$ of the rod lies inside the solenoid. Neglecting end effects, the magnitude of the force tending to pull the rod further into the solenoid is

Q3. The axial magnetic field of a circular coil on its axis is given by $B(z)=\dfrac{B_0}{1+(z/a)^2}$, where the coil centre is at $z=0$. A small magnetic dipole of moment $\boldsymbol{\mu}= \mu\,\hat{z}$ is placed at $z=a$ on the axis. What is the direction of the net force on the dipole?

Q4. A small spherical sample of a linear magnetic material with magnetic susceptibility $\chi_m=2$ is placed in an initially uniform external magnetic field $\mathbf{H}_0$ (directed along $+z$). Using the demagnetizing factor for a sphere ($N=\tfrac{1}{3}$), the magnetic induction $\mathbf{B}$ at the centre of the sphere equals

Q5. A small spherical paramagnetic particle (radius $r=0.50\ \mathrm{mm}$, density $\rho=2000\ \mathrm{kg\,m^{-3}}$, susceptibility $\chi=2.0\times10^{-3}$) is placed at $z=0$ in a vertical magnetic field $B(z)=B_0 e^{\alpha z}$ with $\alpha=1000\ \mathrm{m^{-1}}$. For a small particle the magnetic force may be approximated by $F_z=\dfrac{\chi V}{\mu_0}\,B\,\dfrac{\mathrm{d}B}{\mathrm{d}z}$. The minimum $B_0$ required to levitate the particle (i.e. $mg=F_z$ at $z=0$) is closest to:

Q6. A long solenoid of length $0.50\ \mathrm{m}$ has $1000$ turns and carries a steady current $2.0\ \mathrm{A}$. It is completely filled with a linear magnetic material of susceptibility $\chi_m=0.20$. Using $H=\dfrac{NI}{L}$ and $M=\chi H$, the magnitude of magnetization $M$ of the material is:

Q7. A small spherical sample of linear paramagnetic material with susceptibility $\chi=0.60$ is placed in a uniform magnetic induction $B_0=0.20\ \mathrm{T}$ produced in vacuum. For a sphere the demagnetizing factor is $N=\tfrac{1}{3}$. Taking the demagnetizing field into account, the magnetic induction inside the sphere $B_{\mathrm{inside}}$ is closest to:

Q8. A ferromagnetic core of mean length $l_{\rm core}=0.50\ \mathrm{m}$ and cross-sectional area $A=1.0\times10^{-4}\ \mathrm{m^2}$ has relative permeability $\mu_r=1000$. An air gap of length $g=1.0\times10^{-3}\ \mathrm{m}$ is introduced. A coil of $N=200$ turns carrying current $I=0.50\ \mathrm{A}$ is wound on the core. Neglect fringing. The magnetic flux $\Phi$ in the core is approximately:

Q9. A long solenoid of length $L$ and $N$ turns carries current $I$. Half its length is filled with a linear magnetic material of relative permeability $\mu_r$, the other half remains vacuum. Using the magnetic-circuit approximation (neglect fringing), the ratio $R=\dfrac{B_{\text{half}}}{B_{\text{vac}}}$ equals:

Q10. A small spherical paramagnetic particle (radius $r=0.50\ \mathrm{mm}$, density $\rho=2000\ \mathrm{kg\,m^{-3}}$, susceptibility $\chi=2.0\times10^{-3}$) is placed at $z=0$ in a vertical magnetic field $B(z)=B_0 e^{\alpha z}$ with $\alpha=1000\ \mathrm{m^{-1}}$. For a small particle the magnetic force may be approximated by $F_z=\dfrac{\chi V}{\mu_0}\,B\,\dfrac{\mathrm{d}B}{\mathrm{d}z}$. The minimum $B_0$ required to levitate the particle (i.e. $mg=F_z$ at $z=0$) is closest to:

Q11. A long solenoid (length ≫ radius) has $n=500\ \mathrm{turns/m}$ and carries current $I=0.8\ \mathrm{A}$. A linear magnetic material with susceptibility $\chi=4$ completely fills the solenoid. The magnetic induction $B$ inside the filled solenoid is approximately:

Q12. A very long solenoid has $n=1000\ \mathrm{turns/m}$ and carries current $I=2\ \mathrm{A}$. A paramagnetic rod of cross-sectional area $A=1.0\times10^{-4}\ \mathrm{m^2}$ and susceptibility $\chi=0.01$ is partially inserted so that a length $x$ is inside. Assuming the solenoid field is $H=nI$ and unaffected by insertion, the magnitude of the force pulling the rod further into the solenoid is $F=\dfrac{1}{2}\mu_0\chi (nI)^2 A$. The numerical value of $F$ is closest to:

Q13. A small sphere made of a linear magnetic material with susceptibility $\chi=3$ is placed in a uniform external magnetic field $H_0=800\ \mathrm{A/m}$. For a sphere the demagnetizing factor is $N=\tfrac{1}{3}$. Using the relation $M=\chi\bigl(H_0-NM\bigr)$, the magnetization $M$ inside the sphere is:

Q14. An infinitely long cylindrical rod of radius $R$ is uniformly magnetized along its axis: $\mathbf{M}=M_0\hat{\mathbf{z}}$. Neglect end effects. Which of the following gives the magnetic induction $\mathbf{B}$ (in vacuum) inside ($r<R$) and outside ($r>R$) the cylinder?

Q15. A circular coil of radius $R$ carries current $I$. The magnetic field on its axis a distance $x$ from the centre is $B(x)=\dfrac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}$. A small magnetic dipole of moment $m$ (aligned along the axis) is placed at the centre and is free to move along the axis. For small axial displacement $x$ the axial force is $F\approx m\,B''(0)\,x$. Which statement about the stability of the centre is correct?