A circular coil of radius 4 cm having 50 turns carries a current of 2 A . It is placed in a uniform magnetic field of the intensity of `0.1` Weber `m^(-2)` . The work done to rotate the coil from the position by `180^@` from the equilibrium position

Options

A. 0.1 J
B. 0.2 J
C. 0.4 J
D. 0.8 J

Answer

To solve the problem, we need to calculate the work done in rotating a circular coil in a magnetic field. Here are the steps to arrive at the solution:

### Step 1: Calculate the Area of the Coil
The area \( A \) of a circular coil is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the coil. Given that the radius \( r = 4 \) cm, we first convert it to meters:
\[ r = 4 \, \text{cm} = 0.04 \, \text{m} \]
Now, substituting the value of \( r \):
\[ A = \pi (0.04)^2 = \pi (0.0016) = 0.0016\pi \, \text{m}^2 \]

### Step 2: Calculate the Magnetic Moment
The magnetic moment \( \mu \) of the coil is given by:
\[ \mu = n \cdot I \cdot A \]
where \( n \) is the number of turns, \( I \) is the current, and \( A \) is the area calculated in Step 1. Given \( n = 50 \) turns and \( I = 2 \) A:
\[ \mu = 50 \cdot 2 \cdot (0.0016\pi) = 100 \cdot 0.0016\pi = 0.16\pi \, \text{A m}^2 \]

### Step 3: Calculate the Work Done
The work done \( W \) in rotating the coil in a magnetic field is given by the change in potential energy:
\[ W = \mu B (\cos \theta_i - \cos \theta_f) \]
where:
- \( B \) is the magnetic field intensity (given as \( 0.1 \, \text{Wb/m}^2 \)),
- \( \theta_i \) is the initial angle (0 degrees),
- \( \theta_f \) is the final angle (180 degrees).

Substituting the values:
\[ \cos \theta_i = \cos(0) = 1 \]
\[ \cos \theta_f = \cos(180) = -1 \]
Thus,
\[ W = \mu B (1 - (-1)) = \mu B (1 + 1) = 2\mu B \]

### Step 4: Substitute the Values
Now substituting \( \mu = 0.16\pi \) and \( B = 0.1 \):
\[ W = 2 \cdot (0.16\pi) \cdot (0.1) = 0.032\pi \, \text{J} \]

### Step 5: Calculate the Numerical Value
Using \( \pi \approx 3.14 \):
\[ W \approx 0.032 \cdot 3.14 \approx 0.10048 \, \text{J} \]

### Final Answer
The work done to rotate the coil by \( 180^\circ \) from the equilibrium position is approximately:
\[ W \approx 0.1 \, \text{J} \]