A magnetic dipole is placed at right angles to the direction of lines of force of magnetic induction B. If it is rotated through an angle of `180^(@)` then the work done is

Options

A. `2MB`
B. `MB`
C. `-2MB`
D. Zero

Answer

To solve the problem of finding the work done when a magnetic dipole is rotated through an angle of 180 degrees in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Position**:
- The magnetic dipole is initially placed at right angles to the magnetic field lines. This means that the angle \( \theta \) between the magnetic moment \( \vec{m} \) and the magnetic field \( \vec{B} \) is \( 90^\circ \).

2. **Calculating Initial Potential Energy**:
- The potential energy \( U \) of a magnetic dipole in a magnetic field is given by the formula:
\[ U = -\vec{m} \cdot \vec{B} = -mB \cos(\theta) \]
- Substituting \( \theta = 90^\circ \):
\[ U_{\text{initial}} = -mB \cos(90^\circ) = -mB \cdot 0 = 0 \]

3. **Understanding the Final Position**:
- After rotating the dipole through \( 180^\circ \), the new angle \( \theta \) becomes \( 270^\circ \) (or equivalently \( -90^\circ \)).

4. **Calculating Final Potential Energy**:
- Now, we calculate the potential energy at this new position:
\[ U_{\text{final}} = -mB \cos(270^\circ) = -mB \cdot 0 = 0 \]

5. **Calculating Work Done**:
- The work done \( W \) by an external agent in rotating the dipole is equal to the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} = 0 - 0 = 0 \]

6. **Conclusion**:
- Therefore, the work done in rotating the magnetic dipole through an angle of \( 180^\circ \) is \( 0 \).

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

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