A circular coil of 100turns radius 10cm, carries a current of 5A. It is suspended vertically in a uniform horizontal magnetic field of 0.5T and the field lines make an angle of `60^(@)` with the plane of the coil. The magnitude of the torque that must be applied on it to prevent it from turning is

Options

A. 2.93Nm
B. 3.41Nm
C. 3.93Nm
D. 4.93Nm

Explanation

To find the magnitude of the torque that must be applied on the circular coil to prevent it from turning, we can follow these steps:

### Step 1: Identify the given values
- Number of turns (N) = 100
- Radius of the coil (R) = 10 cm = 0.1 m
- Current (I) = 5 A
- Magnetic field strength (B) = 0.5 T
- Angle (θ) = 60°

### Step 2: Calculate the area of the coil
The area (A) of a circular coil is given by the formula:
\[ A = \pi R^2 \]
Substituting the value of R:
\[ A = \pi (0.1)^2 \]
\[ A = \pi (0.01) \]
\[ A \approx 3.14 \times 10^{-2} \, \text{m}^2 \]

### Step 3: Determine the angle with respect to the normal
The angle that the magnetic field makes with the normal to the plane of the coil is:
\[ \theta' = 90° - θ = 90° - 60° = 30° \]

### Step 4: Use the torque formula
The torque (τ) on the coil due to the magnetic field is given by the formula:
\[ \tau = N I B A \sin(\theta') \]
Substituting the values:
\[ \tau = 100 \times 5 \times 0.5 \times (3.14 \times 10^{-2}) \times \sin(30°) \]

### Step 5: Calculate sin(30°)
We know that:
\[ \sin(30°) = 0.5 \]

### Step 6: Substitute and calculate the torque
Now substituting the values into the torque formula:
\[ \tau = 100 \times 5 \times 0.5 \times (3.14 \times 10^{-2}) \times 0.5 \]
\[ \tau = 100 \times 5 \times 0.5 \times 3.14 \times 10^{-2} \times 0.5 \]
\[ \tau = 100 \times 5 \times 0.5 \times 0.5 \times 3.14 \times 10^{-2} \]
\[ \tau = 100 \times 5 \times 0.25 \times 3.14 \times 10^{-2} \]
\[ \tau = 100 \times 1.25 \times 3.14 \times 10^{-2} \]
\[ \tau = 125 \times 3.14 \times 10^{-2} \]
\[ \tau \approx 3.925 \, \text{N m} \]

### Conclusion
The magnitude of the torque that must be applied on the coil to prevent it from turning is approximately:
\[ \tau \approx 3.925 \, \text{N m} \]

---