The dipole moment of a circular loop carrying a current \(I\), is \(m\) and the magnetic field at the centre of the loop is \(B_1 \). When the dipole moment is doubled by keeping the current constant, the magnetic field at the centre of the loop is \(B_2\). The ratio \(\frac{B_1}{B_2}\) is:
1. \(2\)
2. \(\sqrt{3}\)
3. \(\sqrt{2}\)
4. \(\frac{1}{\sqrt{2}}\)

A square loop carries a steady current \(I\) and has a magnetic dipole moment of magnitude \(m.\) The square loop is then replaced by a circular loop made from the same wire, carrying the same current. What will be the magnitude of the magnetic dipole moment of the circular loop?
 1. \( \dfrac{4 m}{\pi} \)

2. \(\dfrac{3 m}{\pi} \)

3. \(\dfrac{2 m}{\pi} \)

4. \(\dfrac{m}{\pi}\)

A wire carrying current \(I\) is bent in the shape \(ABCDEFA\) as shown, where rectangle \(ABCDA\) and \(ADEFA\) are perpendicular to each other. If the sides of the rectangles are of lengths \(a\) and \(b\), then the magnitude and direction of magnetic moment of the loop \(ABCDEFA\) is:

                     
1. \(\sqrt{2}abI\) along \(\left [ \dfrac{\hat{j}}{\sqrt{2}}+\dfrac{\hat{k}}{\sqrt{2}} \right ]\)
2. \(\sqrt{2}abI\) along \(\left [ \dfrac{\hat{j}}{\sqrt{5}}+\dfrac{2 \hat{k}}{\sqrt{5}} \right ]\)
3. \(abI\) along \(\left [\dfrac{\hat{j}}{\sqrt{2}}+\dfrac{\hat{k}}{\sqrt{2}} \right ]\)
4. \(abI\) along \(\left [ \dfrac{\hat{j}}{\sqrt{5}}+\dfrac{2 \hat{k}}{\sqrt{5}} \right]\)

A circular coil has moment of inertia \(0.8~\text{kgm}^2\) around any diameter and is carrying current to produce a magnetic moment of \(20~\text{Am}^2\). The coil is kept initially in a vertical position and it can rotate freely around a horizontal diameter. When a uniform magnetic field of \(4~\text{T}\) is applied along the vertical, it starts rotating around its horizontal diameter. The angular speed the coil acquires after rotating by \(60^\circ\) will be:
1. \( 20~ \text{rad}\text{s}^{-1} \)
2. \( 20 \pi ~\text{rad} \text{s}^{-1} \)
3. \( 10 \pi ~\text{rad} \text{s}^{-1} \)
4. \( 10 ~\text{rad} \text{s}^{-1} \)
 

An iron rod of volume \(10^{-3}~\text{m}^3\) and relative permeability \(1000\) is placed as core in a solenoid with \(10~\text{turns/cm}\). If a current of \(0.5~\text{A}\) is passed through the solenoid, then the magnetic moment of the rod will be:
1. \( 0.5 \times 10^2~\text{Am}^2 \)
2. \( 50 \times 10^2~\text{Am}^2 \)
3. \(500 \times 10^2 ~\text{Am}^2 \)
4. \(5 \times 10^2 ~\text{Am}^2 \)

A charged particle going around in a circle can be considered to be a current loop. A particle of mass \(m\) carrying charge \(q\) is moving in a plane with speed \(v\) under the influence of a magnetic field \(\vec{B}.\) The magnetic moment of this moving particle:
1. \( \frac{m v^2 \vec{B}}{B^2} \)
2. \( -\frac{m v^2 \vec{B}}{2 \pi B^2} \)
3. \( \frac{m v^2 \vec{B}}{2 B^2} \)
4. \( -\frac{m v^2 \vec{B}}{2 B^2}\)