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 of calculating the work done when a magnetic dipole with magnetic moment \( M \) is rotated by \( 180^\circ \) in a magnetic field \( B \), we can follow these steps:

### Step 1: Understand the Initial Configuration
The magnetic dipole is initially placed at right angles (90 degrees) to the magnetic field \( B \). This means that the angle \( \theta_0 \) between the magnetic moment \( M \) and the magnetic field \( B \) is \( 90^\circ \).

### Step 2: Determine the Final Configuration
When the magnetic dipole is rotated by \( 180^\circ \), the new angle \( \theta_1 \) between the magnetic moment \( M \) and the magnetic field \( B \) becomes \( 90^\circ + 180^\circ = 270^\circ \).

### Step 3: Use the Work Done Formula
The work done \( W \) in rotating the magnetic dipole in a magnetic field is given by the formula:
\[ W = -M B (\cos \theta_1 - \cos \theta_0) \]
where:
- \( M \) is the magnetic moment,
- \( B \) is the magnetic field strength,
- \( \theta_0 \) is the initial angle,
- \( \theta_1 \) is the final angle.

### Step 4: Substitute the Values
Substituting the angles into the formula:
- \( \theta_0 = 90^\circ \) → \( \cos 90^\circ = 0 \)
- \( \theta_1 = 270^\circ \) → \( \cos 270^\circ = 0 \)

Thus, the work done becomes:
\[ W = -M B (0 - 0) = -M B (0) = 0 \]

### Step 5: Conclusion
The total work done in rotating the magnetic dipole by \( 180^\circ \) in the magnetic field \( B \) is \( 0 \) joules.

### Final Answer
The correct answer is \( 0 \) joules.

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