A magnetic dipole with magnetic moment M is placed at right angles to a magnetic field B. If it is rotated by an angle of `180^(@)` , the total work done is

Options

A. zero
B. 2MB
C. -2MB
D. MB

Answer

To solve the problem, we need to calculate the work done when a magnetic dipole with magnetic moment \( M \) is rotated from an initial position at right angles to a magnetic field \( B \) to a final position that is 180 degrees away.

### Step-by-Step Solution:

1. **Identify Initial and Final Angles**:
- The magnetic moment \( M \) is initially at right angles to the magnetic field \( B \). Therefore, the initial angle \( \theta_1 \) is \( 90^\circ \).
- When the dipole is rotated by \( 180^\circ \), the final angle \( \theta_2 \) becomes \( 90^\circ + 180^\circ = 270^\circ \).

2. **Use the Work Done Formula**:
- The work done \( W \) in rotating a magnetic dipole in a magnetic field is given by the formula:
\[ W = -M B (\cos \theta_2 - \cos \theta_1) \]
- Here, \( \theta_1 = 90^\circ \) and \( \theta_2 = 270^\circ \).

3. **Calculate Cosines**:
- Calculate \( \cos \theta_1 \):
\[ \cos 90^\circ = 0 \]
- Calculate \( \cos \theta_2 \):
\[ \cos 270^\circ = 0 \]

4. **Substitute Values into the Formula**:
- Substitute the values of \( \cos \theta_1 \) and \( \cos \theta_2 \) into the work done formula:
\[ W = -M B (0 - 0) = -M B \times 0 = 0 \]

5. **Conclusion**:
- The total work done in rotating the magnetic dipole by \( 180^\circ \) is \( 0 \).

### Final Answer:
The total work done is \( 0 \).

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