A magnet of magnetic moment `M` is situated with its axis along the direction of a magnetic field of strength `B`. The work done in rotating it by an angle of `180^(@)` will be

Options

A. (a)`-MB`
B. (b)`+MB`
C. (c)0
D. (d)`+2MB`

Answer

To find the work done in rotating a magnet of magnetic moment \( M \) by an angle of \( 180^\circ \) in a magnetic field of strength \( B \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial and Final Positions**:
- Initially, the magnetic moment \( M \) is aligned with the magnetic field \( B \). This means the angle \( \theta_1 = 0^\circ \).
- After rotating the magnet by \( 180^\circ \), the magnetic moment will be opposite to the magnetic field direction, so the angle \( \theta_2 = 180^\circ \).

2. **Using the Formula for Work Done**:
- The work done \( W \) in rotating a magnetic moment in a magnetic field is given by the formula:
\[ W = -M B (\cos \theta_2 - \cos \theta_1) \]
- Here, \( \theta_1 \) is the initial angle and \( \theta_2 \) is the final angle.

3. **Substituting the Angles**:
- Substitute \( \theta_1 = 0^\circ \) and \( \theta_2 = 180^\circ \) into the formula:
\[ W = -M B (\cos 180^\circ - \cos 0^\circ) \]

4. **Calculating the Cosine Values**:
- We know that:
\[ \cos 0^\circ = 1 \quad \text{and} \quad \cos 180^\circ = -1 \]
- Therefore, substituting these values gives:
\[ W = -M B (-1 - 1) = -M B (-2) = 2 M B \]

5. **Final Result**:
- The work done in rotating the magnet by \( 180^\circ \) is:
\[ W = 2 M B \]

### Summary:
The work done in rotating the magnet by \( 180^\circ \) in a magnetic field of strength \( B \) is \( 2 M B \).