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
To solve this problem, we need to calculate the magnetic dipole moment of the loop carrying a steady current \(I\). The loop consists of straight sections, each of length \(d\). The key to solving this problem is understanding the geometry of the loop and its orientation in the 3D plane.
The magnetic dipole moment \(\mathbf{m}\) of a current loop is given by the formula:
m = I \cdot A
where \(I\) is the current and \(A\) is the area of the loop.
In this problem, the loop consists of two distinct areas: one in the xy-plane and one tilted at \(30^\circ\) in the xz-plane.
- **Calculate the area in the xy-plane:**
The loop forms a complete rectangle in the xy-plane with each side having a length \(d\). Therefore, the area \(A_1\) in the xy-plane is: A_1 = d \times d = d^2 - **Calculate the area tilted at \(30^\circ\):**
The tilted section is also a rectangle but oriented at an angle. The effective area for the magnetic dipole moment is influenced by the cosine of the angle of tilt (since only the component perpendicular to the area contributes to the dipole moment). Thus, the area \(A_2\) becomes: A_2 = d \times d \times \cos(30^\circ) = d^2 \times \frac{\sqrt{3}}{2} - **Combine the areas to find the effective area for the dipole moment:**
Since both these areas contribute to the magnetic dipole moment, we need to sum these contributions effectively considering direction and orientation: A_{\text{effective}} = d^2 + \left(d^2 \times \frac{\sqrt{3}}{2}\right) = d^2 + \frac{\sqrt{3}}{2}d^2 - **Calculate the magnetic dipole moment:**
By summing the contributions, we find: m = I \times A_{\text{effective}} = I \times d^2 This effectively leads us to the provided correct choice: \(Id^2\).
Based on the calculations and conceptual understanding, the correct answer is \(Id^2\), which matches the given correct option.
**Conclusion:** The magnitude of the magnetic dipole moment of the loop is \(Id^2\), based on the combination of areas taking into account their contributions through orientation and tilt.
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 _____
- 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?
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
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
