A rectangular coil of length `0.12 m` and width `0.1 m` having 50 turns of wire is suspended vertically in uniform magnetic field of strength 0.2 `Weber//m^(2)`. The coil carries a current of 2 A. If the plane of the coil is inclined at an angle,e of `30^(@)` with the direction of the field the torque required to keep the coil in stable equilibrium will be

Options

A. `0.12Nm`
B. `0.15Nm`
C. `0.20Nm`
D. `0.24Nm`

Answer

To find the torque required to keep the rectangular coil in stable equilibrium, we can follow these steps:

### Step 1: Identify the parameters
- Length of the coil (l) = 0.12 m
- Width of the coil (w) = 0.1 m
- Number of turns (N) = 50
- Current (I) = 2 A
- Magnetic field strength (B) = 0.2 Weber/m²
- Angle (θ) = 30°

### Step 2: Calculate the area of the coil
The area (A) of the rectangular coil can be calculated using the formula:
\[ A = l \times w \]
Substituting the values:
\[ A = 0.12 \, \text{m} \times 0.1 \, \text{m} = 0.012 \, \text{m}^2 \]

### Step 3: Determine the angle with respect to the magnetic field
Since the coil is inclined at an angle of 30° with the magnetic field, the angle (φ) between the magnetic moment and the magnetic field is:
\[ φ = 90° - θ = 90° - 30° = 60° \]

### Step 4: Calculate the magnetic moment (m)
The magnetic moment (m) of the coil is given by:
\[ m = N \times I \times A \]
Substituting the values:
\[ m = 50 \times 2 \, \text{A} \times 0.012 \, \text{m}^2 = 1.2 \, \text{A m}^2 \]

### Step 5: Calculate the torque (τ)
The torque (τ) acting on the coil in a magnetic field is given by:
\[ τ = m \times B \times \sin(φ) \]
Substituting the values:
\[ τ = 1.2 \, \text{A m}^2 \times 0.2 \, \text{Wb/m}^2 \times \sin(60°) \]
Using the value of \(\sin(60°) = \frac{\sqrt{3}}{2}\):
\[ τ = 1.2 \times 0.2 \times \frac{\sqrt{3}}{2} \]
\[ τ = 0.24 \times \frac{\sqrt{3}}{2} \]
\[ τ = 0.12 \sqrt{3} \, \text{N m} \]

### Step 6: Calculate the numerical value
Using \(\sqrt{3} \approx 1.732\):
\[ τ \approx 0.12 \times 1.732 \approx 0.20784 \, \text{N m} \]

### Final Answer
The torque required to keep the coil in stable equilibrium is approximately:
\[ τ \approx 0.21 \, \text{N m} \]