A bar magnet having a magnetic moment of `2 xx 10 ^(4) JT ^(-1)` is free to rotate in a horizontal plane. A horizontal magnetic field `B = 6 xx 10 ^(-4)` T exists in the space. The work done in taking the magnet slowly from a direction parallel to the field to a direction `60^(@)` from the field is

Options

A. 12 J
B. 6 J
C. 2 J
D. `0.6 J`

Answer

To find the work done in taking the magnet from a direction parallel to the magnetic field to a direction 60 degrees from the field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Magnetic moment, \( M = 2 \times 10^4 \, \text{J T}^{-1} \)
- Magnetic field, \( B = 6 \times 10^{-4} \, \text{T} \)
- Initial angle, \( \theta_1 = 0^\circ \) (parallel to the field)
- Final angle, \( \theta_2 = 60^\circ \)

2. **Understand the Potential Energy of the Magnetic Moment:**
The potential energy \( U \) of a magnetic moment \( M \) in a magnetic field \( B \) is given by:
\[ U = -MB \cos(\theta) \]
where \( \theta \) is the angle between the magnetic moment and the magnetic field.

3. **Calculate Initial Potential Energy \( U_i \):**
For \( \theta_1 = 0^\circ \):
\[ U_i = -MB \cos(0^\circ) = -MB \cdot 1 = -MB \]

4. **Calculate Final Potential Energy \( U_f \):**
For \( \theta_2 = 60^\circ \):
\[ U_f = -MB \cos(60^\circ) = -MB \cdot \frac{1}{2} = -\frac{MB}{2} \]

5. **Calculate the Work Done \( W \):**
The work done in moving the magnet from the initial to the final position is given by the change in potential energy:
\[ W = U_f - U_i \]
Substituting the expressions we found:
\[ W = \left(-\frac{MB}{2}\right) - \left(-MB\right) = -\frac{MB}{2} + MB = MB - \frac{MB}{2} = \frac{MB}{2} \]

6. **Substitute the Values:**
\[ W = \frac{1}{2} \times (2 \times 10^4) \times (6 \times 10^{-4}) \]
\[ W = \frac{1}{2} \times 2 \times 6 \times 10^{4 - 4} = \frac{1}{2} \times 12 = 6 \, \text{J} \]

### Final Answer:
The work done in taking the magnet from a direction parallel to the field to a direction 60 degrees from the field is \( 6 \, \text{J} \).