A 250-turns recantagular coil of length 2.1 cm and width 1.25 cm carries a current of `85 muA` and subjected to magnetic field of strength `0.85 T`. Work done for rotating the coil by `180^(@)` against the torque is

Options

A. `9.1 muJ`
B. `4.55 muJ`
C. `2.3 muJ`
D. `1.5 muJ`

Explanation

To solve the problem, we need to calculate the work done in rotating the coil by 180 degrees against the torque in a magnetic field. Here’s a step-by-step solution:

### Step 1: Calculate the Area of the Coil
The area \( A \) of the rectangular coil can be calculated using the formula:
\[ A = \text{length} \times \text{width} \]
Given:
- Length = 2.1 cm = \( 2.1 \times 10^{-2} \) m
- Width = 1.25 cm = \( 1.25 \times 10^{-2} \) m

Calculating the area:
\[ A = (2.1 \times 10^{-2}) \times (1.25 \times 10^{-2}) = 2.625 \times 10^{-4} \, \text{m}^2 \]

### Step 2: Calculate the Magnetic Moment \( M \)
The magnetic moment \( M \) of the coil is given by:
\[ M = n \cdot I \cdot A \]
Where:
- \( n = 250 \) (number of turns)
- \( I = 85 \, \mu A = 85 \times 10^{-6} \, A \)

Substituting the values:
\[ M = 250 \cdot (85 \times 10^{-6}) \cdot (2.625 \times 10^{-4}) \]
Calculating \( M \):
\[ M = 250 \cdot 85 \cdot 2.625 \times 10^{-10} = 5.4684375 \times 10^{-5} \, \text{A m}^2 \]

### Step 3: Calculate the Work Done
The work done \( W \) in rotating the coil by \( 180^\circ \) is given by:
\[ W = M \cdot B \cdot (\cos \theta_1 - \cos \theta_2) \]
Where:
- \( B = 0.85 \, T \)
- \( \theta_1 = 0^\circ \) and \( \theta_2 = 180^\circ \)

Using the cosine values:
\[ \cos 0^\circ = 1 \quad \text{and} \quad \cos 180^\circ = -1 \]
Thus:
\[ W = M \cdot B \cdot (1 - (-1)) = M \cdot B \cdot 2 \]

Substituting the values:
\[ W = 2 \cdot (5.4684375 \times 10^{-5}) \cdot (0.85) \]
Calculating \( W \):
\[ W = 2 \cdot 5.4684375 \times 10^{-5} \cdot 0.85 = 9.3002 \times 10^{-5} \, J \]

### Step 4: Convert Work Done to Microjoules
To convert Joules to microjoules:
\[ W = 9.3002 \times 10^{-5} \, J = 9.3002 \times 10^{1} \, \mu J = 9.3 \, \mu J \]

### Conclusion
The work done for rotating the coil by \( 180^\circ \) against the torque is approximately \( 9.3 \, \mu J \).

### Final Answer
The closest option is \( 9.1 \, \mu J \).