Magnetism and Matter Set-5

Q1. A long solenoid of length $L=0.50\ \mathrm{m}$ has $N=2000$ turns and carries current $I=1.50\ \mathrm{A}$. It is completely filled with a linear magnetic rod of susceptibility $\chi=0.02$. The rod's cross-sectional area is $1.00\ \mathrm{cm}^2$. Using $H=\dfrac{N}{L}I$, $M=\chi H$ and magnetic moment $m=M\cdot V$, the induced magnetic moment of the rod is approximately:

Q2. A paramagnetic solid has molar mass $200\ \mathrm{g\,mol^{-1}}$ and density $2.0\ \mathrm{g\,cm^{-3}}$. Each ion carries magnetic moment $2.0\,\mu_B$ ($\mu_B=9.274\times10^{-24}\ \mathrm{J\,T^{-1}}$). At temperature $T=300\ \mathrm{K}$ it is placed in a uniform magnetic field $H=1.0\times10^{5}\ \mathrm{A\,m^{-1}}$. Using the Curie (Langevin) approximation $\chi=\dfrac{\mu_0 N \mu^2}{3k_B T}$ and $M=\chi H$, the magnetization $M$ (in A/m) is closest to:

Q3. A ferromagnetic core of volume $V=2.0\times10^{-4}\ \mathrm{m^3}$ has a hysteresis loop approximated by a rectangle with coercive field $H_c=80\ \mathrm{A\,m^{-1}}$ and remanent flux density $B_r=0.60\ \mathrm{T}$ (corners at $(\pm H_c,\pm B_r)$). The energy loss per cycle per unit volume equals the area $\oint H\,dB$ of the $B$–$H$ loop. Using this rectangular approximation, the power loss due to hysteresis at frequency $f=50\ \mathrm{Hz}$ is approximately:

Q4. An infinitely long hollow cylindrical shell (neglect end effects) with inner radius $a$ and outer radius $b$ is uniformly magnetized along its axis, $\mathbf{M}=M\hat{z}$. Determine the magnetic induction $\mathbf{B}$ in the three regions: (i) $r<a$ (inside the hollow), (ii) $a<r<b$ (within the shell), (iii) $r>b$ (outside the shell).

Q5. A small sphere made of a linear magnetic material (susceptibility $\chi$) is placed in a uniform external field $\mathbf{H}_0=H_0\hat{z}$. For a sphere the demagnetizing factor is $N=\tfrac{1}{3}$ so $\mathbf{H}_{\mathrm{int}}=\mathbf{H}_0 - N\mathbf{M}$ and $\mathbf{M}=\chi\mathbf{H}_{\mathrm{int}}$. If $\chi=2.0$, the magnetization inside the sphere $\mathbf{M}$ equals:

Q6. A long solenoid of length $L$ has $N$ turns and carries a steady current $I$. A linear magnetic material with relative permeability $\mu_r$ completely fills the solenoid. Neglecting edge effects, the magnetic field inside the solenoid is:

Q7. A small spherical sample of a linear magnetic material (magnetic susceptibility $\chi$) is placed in a uniform external magnetic field $\mathbf{H}_0$. Taking the demagnetizing factor of a sphere as $N=\tfrac{1}{3}$, the resulting magnetization $\mathbf{M}$ inside the sphere (in terms of $\mathbf{H}_0$) is:

Q8. A long solid non-conducting cylinder of radius $R$ has magnetization $\mathbf{M}=k\,\rho\,\hat{\boldsymbol{\phi}}$ (cylindrical coordinates), where $k$ is a constant and $\rho$ is the radial coordinate. The bound volume current density $\mathbf{J}_b=\nabla\times\mathbf{M}$ inside the cylinder and the bound surface current density $\mathbf{K}_b=\mathbf{M}\times\hat{\mathbf{n}}$ at $\rho=R$ are, respectively:

Q9. A linear magnetic material is tested in two sample shapes in the same uniform external magnetic field. For a spherical sample the apparent susceptibility measured is $\chi_{\rm sph}=0.75$, and for a very thin flat plate with its normal along the field the apparent susceptibility is $\chi_{\rm plate}=0.50$. Taking demagnetizing factors $N_{\rm sph}=\tfrac{1}{3}$ and $N_{\rm plate}=1$, the true susceptibility $\chi$ of the material is:

Q10. Two identical small non-conducting spheres (radius $a$, centre-to-centre separation $r\gg a$) made of a linear magnetic material with susceptibility $\chi$ are placed along the axis of a uniform external magnetic field $\mathbf{H}_0$ (field directed along the line joining the centres). In the induced-dipole approximation, how do the spheres interact for $\chi>0$ (paramagnetic) and for $\chi<0$ (diamagnetic), respectively?

Q11. A long solenoid has $n=2000\ \mathrm{turns/m}$ and carries a current $I=0.5\ \mathrm{A}$. A soft-iron rod of relative permeability $\mu_r=300$ is inserted completely along its axis. Assuming the solenoid is long and end effects are negligible, the magnetic flux density inside the rod is $B=\mu_0\mu_r n I$. Calculate $B$.

Q12. A small paramagnetic sphere of volume $V=2.0\times10^{-6}\ \mathrm{m^3}$ and susceptibility $\chi=0.01$ is placed on the axis of a solenoid. At that point the magnetic field magnitude is $B=0.02\ \mathrm{T}$ and the axial gradient is $\dfrac{\partial B}{\partial z}=0.5\ \mathrm{T\,m^{-1}}$. Using the energy/force approximation for a linear magnetic material,
$F_z=\dfrac{\chi V}{2\mu_0}\dfrac{d(B^2)}{dz}$,
calculate the magnitude and direction of the force on the sphere.

Q13. A magnetic core has mean magnetic path length $l_{\rm core}=0.50\ \mathrm{m}$ and cross-sectional area $A=2.0\times10^{-4}\ \mathrm{m^2}$. The core material has relative permeability $\mu_r=2000$ and there is an air gap of length $l_g=1.0\times10^{-3}\ \mathrm{m}$. A coil of $N=200$ turns carrying current $I=0.50\ \mathrm{A}$ is wound on the core. Neglect fringing and assume flux confined to core+gap. Calculate the magnetic flux density $B$ in the gap (and in the core).

Q14. A uniform magnetic field in medium 1 (relative permeability $\mu_1=1$) makes an angle $\theta_1=60^\circ$ with the normal to a plane boundary with medium 2 (relative permeability $\mu_2=4$). Using boundary conditions $B_{n1}=B_{n2}$ and $H_{t1}=H_{t2}$, determine the angle $\theta_2$ that the magnetic field in medium 2 makes with the normal and state whether the field bends toward or away from the normal.

Q15. A spherical sample of a paramagnetic material has Curie constant $C=150\ \mathrm{K}$. For a sphere the demagnetizing factor is $N=\dfrac{1}{3}$. The susceptibility (without demagnetizing effects) follows Curie's law $\chi=\dfrac{C}{T}$. Accounting for the demagnetizing field, the magnetization under an external field $H_0$ is $M=\dfrac{C\,H_0}{T+N C}$. If the same sample is at temperatures $T_1=100\ \mathrm{K}$ and $T_2=200\ \mathrm{K}$, what is the ratio $M_1/M_2$?