Find the angle through which a magnet is to be rotated from rest position when it is suspended in a magnetic field so that it experiences half of the maximum couple

Options

A. `60^@`
B. `30^@`
C. `45^@`
D. `90^@`

Answer

To solve the problem of finding the angle through which a magnet must be rotated from its rest position to experience half of the maximum couple in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Torque in a Magnetic Field**:
The torque (\(\tau\)) experienced by a magnetic dipole in a magnetic field is given by the formula:
\[ \tau = \mathbf{M} \times \mathbf{B} \]
where \(\mathbf{M}\) is the magnetic dipole moment and \(\mathbf{B}\) is the magnetic field.

2. **Expressing Torque in Terms of Angle**:
The magnitude of torque can also be expressed as:
\[ \tau = MB \sin \theta \]
where \(\theta\) is the angle between the magnetic moment and the magnetic field.

3. **Finding Maximum Torque**:
The maximum torque (\(\tau_{\text{max}}\)) occurs when \(\sin \theta = 1\) (i.e., \(\theta = 90^\circ\)):
\[ \tau_{\text{max}} = MB \]

4. **Setting Up the Equation for Half Maximum Torque**:
We need to find the angle \(\theta\) when the torque is half of the maximum torque:
\[ \tau = \frac{\tau_{\text{max}}}{2} = \frac{MB}{2} \]

5. **Substituting into the Torque Equation**:
Substitute \(\tau\) into the torque equation:
\[ \frac{MB}{2} = MB \sin \theta \]

6. **Simplifying the Equation**:
Cancel \(MB\) from both sides (assuming \(M\) and \(B\) are not zero):
\[ \frac{1}{2} = \sin \theta \]

7. **Finding the Angle \(\theta\)**:
To find \(\theta\), take the inverse sine:
\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) \]
This gives:
\[ \theta = 30^\circ \]

8. **Calculating the Rotation from the Rest Position**:
The rest position of the magnet is at \(0^\circ\). To find the angle through which the magnet must be rotated:
\[ \text{Rotation} = 90^\circ - 30^\circ = 60^\circ \]

### Final Answer:
The angle through which the magnet is to be rotated from the rest position to experience half of the maximum couple is:
\[ \text{Rotation} = 60^\circ \]