A magnet of moment `4Am^(2)` is kept suspended in a magnetic field of induction `5xx10^(-5)T`. The workdone in rotating it through `180^(@)` is

Options

A. `4xx10^(-4) J`
B. `5xx10^(-4) J`
C. `2xx10^(-4) J`
D. `10^(-4) J`

Answer

To solve the problem of calculating the work done in rotating a magnet of moment \( M = 4 \, \text{Am}^2 \) through an angle of \( 180^\circ \) in a magnetic field of induction \( B = 5 \times 10^{-5} \, \text{T} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Torque**:
The torque \( \tau \) experienced by a magnetic dipole in a magnetic field is given by:
\[ \tau = M \cdot B \cdot \sin(\theta) \]
where \( M \) is the magnetic moment, \( B \) is the magnetic field induction, and \( \theta \) is the angle between the magnetic moment and the magnetic field.

2. **Setting Up the Work Done**:
The work done \( W \) in rotating the magnet from an angle \( \theta_1 \) to \( \theta_2 \) is given by:
\[ W = \int_{\theta_1}^{\theta_2} \tau \, d\theta \]
For our case, we are rotating from \( \theta = 0^\circ \) to \( \theta = 180^\circ \) (which is \( \pi \) radians).

3. **Substituting the Torque Expression**:
Substitute the expression for torque into the work done formula:
\[ W = \int_{0}^{\pi} M \cdot B \cdot \sin(\theta) \, d\theta \]

4. **Calculating the Integral**:
The integral of \( \sin(\theta) \) is:
\[ \int \sin(\theta) \, d\theta = -\cos(\theta) \]
Therefore, we have:
\[ W = M \cdot B \left[-\cos(\theta)\right]_{0}^{\pi} \]
Evaluating this from \( 0 \) to \( \pi \):
\[ W = M \cdot B \left[-\cos(\pi) + \cos(0)\right] \]
Since \( \cos(\pi) = -1 \) and \( \cos(0) = 1 \):
\[ W = M \cdot B \left[-(-1) + 1\right] = M \cdot B \left[1 + 1\right] = 2MB \]

5. **Substituting the Values**:
Now substitute the values of \( M \) and \( B \):
\[ W = 2 \cdot 4 \, \text{Am}^2 \cdot 5 \times 10^{-5} \, \text{T} \]
\[ W = 40 \times 10^{-5} \, \text{J} \]
\[ W = 4 \times 10^{-4} \, \text{J} \]

### Final Answer:
The work done in rotating the magnet through \( 180^\circ \) is:
\[ W = 4 \times 10^{-4} \, \text{J} \]