Find the magnitude of torque that must be applied on a 100 turns circular coil having a radius of 10 cm, such that the coil is prevented from turning when it is suspended vertically in a uniform horizontal magnetic field of 0.5 T and also the field lines make an angle of 60° with the plane of the coil. Take the current flowing in the coil to be 5 A.

Options

A. 2.93 Nm
B. 3.43 Nm
C. 3.93 Nm
D. 4.93 Nm

Explanation

To find the magnitude of the torque that must be applied on the circular coil, we will use the formula for torque (\( \tau \)) in a magnetic field, which is given by:

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

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

### 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 \( r = 10 \) cm = 0.1 m, we can calculate the area:

\[ A = \pi (0.1)^2 = \pi (0.01) = 0.0314 \, \text{m}^2 \]

### Step 2: Identify the angle \( \theta \)
The problem states that the magnetic field lines make an angle of \( 60^\circ \) with the plane of the coil. Therefore, the angle \( \theta \) that we will use in our torque formula is:

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

### Step 3: Substitute the values into the torque formula
Now we can substitute the values into the torque formula:

- \( n = 100 \) turns
- \( B = 0.5 \, T \)
- \( I = 5 \, A \)
- \( A = 0.0314 \, m^2 \)
- \( \theta = 30^\circ \)

Thus, the torque \( \tau \) is:

\[ \tau = 100 \cdot 0.5 \cdot 5 \cdot 0.0314 \cdot \sin(30^\circ) \]

Since \( \sin(30^\circ) = 0.5 \):

\[ \tau = 100 \cdot 0.5 \cdot 5 \cdot 0.0314 \cdot 0.5 \]

Calculating this step-by-step:

1. \( 100 \cdot 0.5 = 50 \)
2. \( 50 \cdot 5 = 250 \)
3. \( 250 \cdot 0.0314 = 7.85 \)
4. \( 7.85 \cdot 0.5 = 3.925 \)

Thus, the magnitude of the torque that must be applied is:

\[ \tau \approx 3.925 \, \text{N m} \]

### Final Answer:
The magnitude of the torque that must be applied is approximately \( 3.93 \, \text{N m} \).