Magnetism and Matter Set-3

Q1. A long solenoid (length ≫ radius) has $n=800\ \text{turns m}^{-1}$ and carries current $I=0.50\ \mathrm{A}$. The core is filled with a linear magnetic material of volume susceptibility $\chi=2.5$. Assuming the demagnetizing factor is negligible, calculate the magnetic induction $B$ inside the core. (Use $\mu_0=4\pi\times10^{-7}\ \mathrm{N/A^2}$ and $B=\mu_0(1+\chi)nI$.)

Q2. A cylindrical specimen with demagnetizing factor $N=0.10$ (axis along applied field) is placed in an external uniform field $H_0=1000\ \mathrm{A/m}$. The material is linear with susceptibility $\chi=50$. Using $H_{\rm int}=\dfrac{H_0}{1+N\chi}$ and $B=\mu_0(1+\chi)H_{\rm int}$, find the magnetic induction $B$ inside the specimen. (Take $\mu_0=4\pi\times10^{-7}\ \mathrm{N/A^2}$.)

Q3. Two materials X and Y are subjected to the same cyclic magnetising field and yield the following qualitative hysteresis loops: Loop X is very narrow with a small coercive field and small area; Loop Y is wide with large coercivity and large loop area. Which option best follows from these observations?

Q4. Assertion: For a very long slender ferromagnetic rod magnetised along its axis, the demagnetising field inside is negligible compared to the applied field, so the internal magnetising field is approximately the applied field.
Reason: The demagnetising factor $N$ for an axially magnetised long rod tends to zero because the surface pole distribution produces little opposing internal field.

Q5. A small spherical paramagnetic particle (volume susceptibility $\chi=2.0\times10^{-2}$) is placed in vacuum in a uniform external magnetic induction $B_0=0.100\ \mathrm{T}$. For a sphere the demagnetising factor is $N=1/3$. Assuming linear response, compute the magnetisation $M$ inside the sphere. (Use $\mu_0=4\pi\times10^{-7}\ \mathrm{N/A^2}$ and $H_0=B_0/\mu_0$.)

Q6. A linear, non-hysteretic magnetic material has relative permeability $\mu_r=100$ and is subjected to an applied field $H=800\ \mathrm{A/m}$. Using the relation for magnetic energy density in a linear medium $u=\tfrac{1}{2}\,B\cdot H=\tfrac{1}{2}\mu_0\mu_r H^2$, calculate the magnetic energy stored per unit volume.

Q7. A long ferromagnetic rod of length $L$ is magnetised along its axis by an external field $H_0$, producing magnetisation $M_0$. The rod is then cut into two identical rods of length $L/2$ and the halves are separated so each is isolated in the same external field $H_0$. Neglecting interactions between pieces beyond the shape-dependent demagnetising factor, which statement best describes the magnetisation $M$ of each half compared to $M_0$?

Q8. For a ferromagnetic sample measured at two temperatures T1 and T2, the following qualitative M–H behaviours are observed: Curve P shows a hysteresis loop with remanence and coercivity; Curve Q is a straight line through the origin with a small positive slope. Which assignment is correct regarding the Curie temperature $T_C$?

Q9. A small paramagnetic bead of volume $V=1.0\times10^{-6}\ \mathrm{m^3}$ and susceptibility $\chi=1.0\times10^{-2}$ is placed near $x=0$ in a magnetic field $H(x)=H_0(1+\alpha x)$ with $H_0=1.0\times10^{4}\ \mathrm{A/m}$ and $\alpha=0.10\ \mathrm{m^{-1}}$. For a small linear sample, the magnetic force may be approximated as $\vec{F}=\dfrac{1}{2}\mu_0\chi V\nabla(H^2)$. Calculate the magnitude of the force at $x=0$ (use $\mu_0=4\pi\times10^{-7}\ \mathrm{N/A^2}$).

Q10. Assertion: In a linear isotropic magnetic material the vectors $\vec{B}$ and $\vec{H}$ are always parallel.
Reason: Magnetisation $\vec{M}$ is always perpendicular to the applied field $\vec{H}$ in magnetic materials.