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

Explanation

To find the torque required to keep the rectangular coil in stable equilibrium, we can use the formula for torque (\( \tau \)) on a current-carrying coil in a magnetic field:

\[ \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \]

Where:
- \( n \) = number of turns of the coil
- \( I \) = current flowing through the coil
- \( A \) = area of the coil
- \( B \) = magnetic field strength
- \( \theta \) = angle between the normal to the coil and the magnetic field

### 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 = \( 0.12 \, m \)
- Width = \( 0.1 \, m \)

Calculating the area:

\[ A = 0.12 \, m \times 0.1 \, m = 0.012 \, m^2 \]

### Step 2: Identify the values
We have:
- Number of turns \( n = 50 \)
- Current \( I = 2 \, A \)
- Magnetic field strength \( B = 0.2 \, \text{Weber/m}^2 \)
- Angle \( \theta = 30^\circ \)

### Step 3: Calculate the angle for torque calculation
The angle \( \theta \) used in the torque formula is the angle between the magnetic field and the normal to the coil. Since the coil is inclined at \( 30^\circ \) to the magnetic field, the angle between the normal to the coil and the magnetic field is:

\[ \theta = 90^\circ - 30^\circ = 60^\circ \]

### Step 4: Calculate the torque
Now we can substitute the values into the torque formula:

\[ \tau = n \cdot I \cdot A \cdot B \cdot \sin(60^\circ) \]

First, calculate \( \sin(60^\circ) \):

\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

Now substituting the values:

\[ \tau = 50 \cdot 2 \cdot 0.012 \cdot 0.2 \cdot \frac{\sqrt{3}}{2} \]

Calculating step by step:

1. Calculate \( 50 \cdot 2 = 100 \)
2. Calculate \( 100 \cdot 0.012 = 1.2 \)
3. Calculate \( 1.2 \cdot 0.2 = 0.24 \)
4. Now multiply by \( \frac{\sqrt{3}}{2} \):

\[ \tau = 0.24 \cdot \frac{\sqrt{3}}{2} = 0.12\sqrt{3} \]

Using \( \sqrt{3} \approx 1.732 \):

\[ \tau \approx 0.12 \cdot 1.732 \approx 0.20784 \, \text{N m} \]

### Final Result
The torque required to keep the coil in stable equilibrium is approximately:

\[ \tau \approx 0.2 \, \text{N m} \]