A "bar" magnet is at right angles to a uniform magnetic field. The couple acting on the magnet is to be one fourth by rotating it from the position. The angle of rotation is

Options

A. `60^(@)`
B. `45^(@)`
C. `30^(@)`
D. `75^(@)`

Answer

To solve the problem step by step, we will use the concepts of torque acting on a magnetic dipole in a magnetic field.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- A bar magnet is initially positioned at a right angle (90 degrees) to a uniform magnetic field. This means that the angle θ = 90°.

2. **Calculating the Initial Torque**:
- The torque (τ) acting on a magnetic dipole in a magnetic field is given by the formula:
\[ \tau = m \cdot B \cdot \sin(\theta) \]
- Here, \(m\) is the magnetic moment of the bar magnet, \(B\) is the magnetic field strength, and \(\theta\) is the angle between the magnetic moment and the magnetic field.
- Since θ = 90°, we have:
\[ \tau_1 = m \cdot B \cdot \sin(90°) = m \cdot B \cdot 1 = mB \]

3. **Condition After Rotation**:
- The problem states that after rotating the magnet, the torque acting on it becomes one-fourth of the initial torque:
\[ \tau_2 = \frac{1}{4} \tau_1 = \frac{1}{4} mB \]

4. **Setting Up the Equation for the New Torque**:
- After rotation, let the new angle be θ. Thus, we can express the new torque as:
\[ \tau_2 = m \cdot B \cdot \sin(\theta) \]
- Setting this equal to the expression for \(\tau_2\) we derived:
\[ m \cdot B \cdot \sin(\theta) = \frac{1}{4} mB \]

5. **Simplifying the Equation**:
- We can cancel \(mB\) from both sides (assuming \(mB \neq 0\)):
\[ \sin(\theta) = \frac{1}{4} \]

6. **Finding the Angle θ**:
- To find θ, we take the inverse sine:
\[ \theta = \sin^{-1}\left(\frac{1}{4}\right) \]

7. **Calculating the Angle of Rotation**:
- The angle of rotation from the original position (90°) is:
\[ \text{Angle of rotation} = 90° - \sin^{-1}\left(\frac{1}{4}\right) \]

8. **Final Calculation**:
- Using a calculator, we find:
\[ \sin^{-1}\left(\frac{1}{4}\right) \approx 15° \]
- Therefore, the angle of rotation is:
\[ 90° - 15° = 75° \]

### Final Answer:
The angle of rotation is **75 degrees**.