A magnetic dipole is placed in a uniform magnetic field of intensity B, oriented along the direction of the field. If the magnetic dipole moment is M, then the maximum work an external agent can perform in rotating the dipole will be

Options

A. `1/2 MB`
B. `4 MB`
C. `2MB`
D. `MB`

Answer

To solve the problem, we need to determine the maximum work done by an external agent in rotating a magnetic dipole in a uniform magnetic field. Here's a step-by-step solution:

### Step 1: Understand the Initial Configuration
- A magnetic dipole with magnetic moment \( M \) is placed in a uniform magnetic field \( B \).
- The dipole moment is aligned with the magnetic field, meaning the angle \( \theta \) between the dipole moment and the magnetic field is \( 0^\circ \).

### Step 2: Calculate the Initial Potential Energy
- The potential energy \( U \) of a magnetic dipole in a magnetic field is given by the formula:
\[ U = -M \cdot B \cdot \cos(\theta) \]
- Substituting \( \theta = 0^\circ \):
\[ U_{\text{initial}} = -M \cdot B \cdot \cos(0) = -M \cdot B \cdot 1 = -MB \]

### Step 3: Determine the Final Configuration
- To find the maximum work done, we need to consider the dipole being rotated to the position where it is anti-aligned with the magnetic field, which corresponds to \( \theta = 180^\circ \).

### Step 4: Calculate the Final Potential Energy
- Using the same formula for potential energy:
\[ U_{\text{final}} = -M \cdot B \cdot \cos(180^\circ) = -M \cdot B \cdot (-1) = MB \]

### Step 5: Calculate the Work Done
- The work done \( W \) by the external agent is equal to the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} \]
- Substituting the values we found:
\[ W = MB - (-MB) = MB + MB = 2MB \]

### Conclusion
- Therefore, the maximum work an external agent can perform in rotating the dipole is:
\[ \boxed{2MB} \]