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 find the angle through which a magnet is to be rotated from its rest position in a magnetic field so that it experiences half of the maximum couple (torque), we can follow these steps:

### Step 1: Understand the Maximum Torque
The maximum torque (\( \tau_{max} \)) experienced by a bar magnet in a magnetic field is given by the formula:
\[ \tau_{max} = M \cdot B \]
where \( M \) is the magnetic moment of the magnet and \( B \) is the magnetic field strength. This maximum torque occurs when the angle (\( \theta \)) between the magnetic moment and the magnetic field is \( 90^\circ \) (i.e., when the magnet is perpendicular to the field).

### Step 2: Determine Half of the Maximum Torque
Half of the maximum torque can be expressed as:
\[ \tau = \frac{1}{2} \tau_{max} = \frac{1}{2} M \cdot B \]

### Step 3: Write the Torque Equation
The torque (\( \tau \)) experienced by the magnet at an angle \( \theta \) is given by:
\[ \tau = M \cdot B \cdot \sin(\theta) \]
To find the angle where the torque is half of the maximum torque, we set the two torque equations equal:
\[ M \cdot B \cdot \sin(\theta) = \frac{1}{2} M \cdot B \]

### Step 4: Simplify the Equation
Since \( M \) and \( B \) are common on both sides, we can cancel them out:
\[ \sin(\theta) = \frac{1}{2} \]

### Step 5: Solve for the Angle \( \theta \)
The angle \( \theta \) that satisfies \( \sin(\theta) = \frac{1}{2} \) is:
\[ \theta = 30^\circ \]

### Step 6: Determine the Rotation from the Rest Position
The rest position of the magnet is when it is aligned with the magnetic field (0 degrees). To find the angle through which the magnet must be rotated from the rest position to achieve this angle, we calculate:
\[ \text{Angle of rotation} = 90^\circ - \theta = 90^\circ - 30^\circ = 60^\circ \]

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