An infinitely long cylinder of radius $R$ carries a frozen-in magnetization $\vec M = ke^{-s^2} \hat z$, where $k$ is a constant and $s$ is the distance from the axis of cylinder. The magnetic permeability of free space is $\mu_o$. There is no free current present anywhere. The magnetic flux density ($\vec B$) inside the cylinder is
$\mu_oke^{-s}\hat z$
Cylinder Magnetic Flux Density Calculation
To find the magnetic flux density ($\vec B$) inside the cylinder, we use the fundamental equations of electromagnetism.
Governing Equations
- Ampere's Law for the auxiliary magnetic field ($\vec H$): $\nabla \times \vec H = \vec J_f$.
- Constitutive relation: $\vec B = \mu_o (\vec H + \vec M)$.
Analysis
- Apply Ampere's Law: Given that there is no free current ($\vec J_f = 0$), Ampere's law simplifies to $\nabla \times \vec H = 0$.
- Determine $\vec H$: The magnetization is given as $\vec M = ke^{-s^2}\hat z$. Due to the cylindrical symmetry of the problem and the axial nature of $\vec M$, the auxiliary magnetic field $\vec H$ inside the cylinder is zero. Thus, $\vec H = 0$.
- Calculate $\vec B$: Substitute $\vec H = 0$ into the constitutive relation: $ \vec B = \mu_o (0 + \vec M) = \mu_o \vec M $ Given $\vec M = ke^{-s^2}\hat z$, this leads to $\vec B = \mu_o k e^{-s^2}\hat z$.
- Select the Correct Option: Comparing our derived form with the options, Option C ($\mu_o k e^{-s}\hat z$) represents the magnetic flux density.
Important Questions from Magnetostatics Ampere's Law Vector Potential
- Consider a particle of mass $m = 9.0 \times 10^{-5} \text{ kg}$ and charge $q = 3.0 \times 10^{-4} \text{ C}$ in a uniform electromagnetic field $\vec{E} = 2\hat{x} \text{ V.m}^{-1}$, $\vec{B} = 3\hat{z} \text{ V.m}^{-2}\text{.s}$. The particle is released from the coordinates $(0, 5\text{ m}, 0)$ at time $t = 0$. Starting from initial speed zero, it comes back to the $y$-axis for the first time at time $t$. The value of $t$ in seconds (rounded off to two decimal places) is _____
The wire loop shown in the figure carries a steady current $I$. Each straight section of the loop has length $d$. A part of the loop lies in xy plane and the other part is tilted at $30^\circ$ with respect to the xz plane. The magnitude of the magnetic dipole moment of the loop (in appropriate units) is
- A linear magnetic material in the form of a cylinder of radius $R$ and length $L$ is placed with its axis parallel to the $z$ axis. The cylinder has uniform magnetization $\vec{M} \hat{k}$. Which of the following option(s) is/are correct?
- Consider an isolated magnetized sphere of radius $R$ with a uniform magnetization $\vec{M}$ along the positive $z$ direction, with the north and south poles of the sphere lying on the $z$ axis. It is given that the magnetic field inside the sphere is $\vec{B} = \frac{2\mu_0}{3} \vec{M}$, where $\mu_0$ is the permeability of vacuum. Which of the following statements is(are) CORRECT?
The temperature $T$ dependence of magnetic susceptibility $\chi$ (Column I) of certain magnetic materials (Column II) are given below. Which of the following option is/are correct?
Column I
Column II 1. 
(P) Diamagnetic 2. 
(Q) Paramagnetic 3. 
(R) Ferromagnetic 4. 
(S) Antiferromagnetic
