Magnetism and Matter Set-2

Q1. A long solenoid has turns per unit length $n=400\ \mathrm{m^{-1}}$ and carries a steady current $I=2\ \mathrm{A}$. Its core is filled with a linear material of susceptibility $\chi=0.50$. Neglect edge effects. The magnetic field inside the solenoid is

Q2. A ferromagnetic specimen has an approximately rectangular hysteresis loop. Its coercive field is $H_c=400\ \mathrm{A/m}$ and its remanent flux density is $B_r=0.60\ \mathrm{T}$. Using the $B$–$H$ plot approximation, estimate the energy loss per cycle per unit volume (in $\mathrm{J/m^3}$).

Q3. Assertion (A): A magnetic dipole of moment $\vec{\mu}$ placed in a uniform magnetic field $\vec{B}$ and oriented perpendicular to $\vec{B}$ experiences a net force but no torque.
Reason (R): In general $\vec{\tau}=\vec{\mu}\times\vec{B}$ and $\vec{F}=\nabla(\vec{\mu}\cdot\vec{B})$, so for a uniform field $\vec{F}=0$.

Q4. A magnetic dipole $\vec{\mu}=0.50\ \mathrm{A\cdot m^2}\,\hat{x}$ is placed in a region where $\vec{B}(x)=\big(0.02+0.005x\big)\,\hat{x}\ \mathrm{T}$ (with $x$ in metres). Find the force on the dipole at $x=2.0\ \mathrm{m}$.

Q5. A long solenoid (ideal, length $\gg$ radius) is connected to an ideal battery that maintains a constant current $I$. A linear paramagnetic rod ($\chi>0$) is inserted slowly into the solenoid. Which of the following statements are correct?
I. There is a mechanical force pulling the rod into the solenoid.
II. The magnetic energy stored in the solenoid increases.
III. The battery supplies additional energy during insertion.

Q6. A long solenoid has $n=800\ \mathrm{turns/m}$ and carries current $I=1.50\ \mathrm{A}$. A linear magnetic sample placed entirely inside the solenoid yields a measured magnetic field $B=1.60\times10^{-3}\ \mathrm{T}$. Using $B=\mu_0(H+M)$ and $H=nI$, estimate the magnetization $M$ of the sample (in $\mathrm{A/m}$).

Q7. Three materials show the following $B$ versus $H$ behaviour near the origin (qualitatively):
Material X — straight line through origin with slope slightly less than $\mu_0$;
Material Y — straight line through origin with slope slightly greater than $\mu_0$;
Material Z — shows a wide hysteresis loop with remanence and coercivity.
Which classification is correct?

Q8. Assertion (A): For a linear, isotropic magnetic medium the magnetic energy density is $u=\tfrac{1}{2}\mathbf{B}\cdot\mathbf{H}$.
Reason (R): For such a medium $\mathbf{H}$ is proportional to $\mathbf{B}$, and $u=\int_0^{\mathbf{B}}\mathbf{H}\cdot d\mathbf{B}$ evaluates to $\tfrac{1}{2}\mathbf{B}\cdot\mathbf{H}$.

Q9. A rectangular loop of dimensions $a=0.10\ \mathrm{m}$ and $b=0.20\ \mathrm{m}$ carries current $I=5.0\ \mathrm{A}$. It is placed in a uniform magnetic field $B=0.60\ \mathrm{T}$ so that the plane of the loop makes an angle $30^\circ$ with $\vec{B}$. What is the magnitude of the torque on the loop?

Q10. A ferromagnetic core in a solenoid is driven from low to very high applied field $H$ so the material passes from the initial linear region into saturation. Which statements are correct?
I. Magnetization $M$ approaches a constant saturation value at very large $H$.
II. Relative permeability $\mu_r$ approaches unity at very large $H$.
III. Inductance $L$ of the coil increases without bound as $I$ increases into the saturation region.