A circular coil having N turns and radius r carries a current I. It is held in the XZ plane in a magnetic field `Bhati`. The torque on the coil due to the magnetic field is :

Options

A. `(Br^(2)I)/(piN)`
B. `Bpir^(2)IN`
C. `(Bpir^(2)(I))/(N)`
D. Zero

Answer

To find the torque on a circular coil placed in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Parameters:**
- Number of turns of the coil, \( N \)
- Radius of the coil, \( r \)
- Current flowing through the coil, \( I \)
- Magnetic field, \( \mathbf{B} = B \hat{i} \) (along the x-direction)

2. **Determine the Area of the Coil:**
The area \( A \) of a circular coil is given by the formula:
\[ A = \pi r^2 \]

3. **Calculate the Magnetic Moment \( \mathbf{m} \):**
The magnetic moment \( \mathbf{m} \) of the coil is given by:
\[ \mathbf{m} = N \cdot I \cdot A \]
Substituting the area \( A \):
\[ \mathbf{m} = N \cdot I \cdot (\pi r^2) \]

4. **Determine the Direction of the Magnetic Moment:**
Since the coil is in the xz-plane, the area vector \( \mathbf{A} \) (and hence the magnetic moment \( \mathbf{m} \)) will be perpendicular to the plane of the coil. Therefore, it points in the y-direction:
\[ \mathbf{m} = N \cdot I \cdot \pi r^2 \hat{j} \]

5. **Calculate the Torque \( \mathbf{\tau} \):**
The torque \( \mathbf{\tau} \) on the coil in a magnetic field is given by the cross product:
\[ \mathbf{\tau} = \mathbf{m} \times \mathbf{B} \]
Substituting the values of \( \mathbf{m} \) and \( \mathbf{B} \):
\[ \mathbf{\tau} = (N \cdot I \cdot \pi r^2 \hat{j}) \times (B \hat{i}) \]

6. **Evaluate the Cross Product:**
Using the right-hand rule and the properties of cross products:
\[ \mathbf{\tau} = N \cdot I \cdot \pi r^2 \cdot B (\hat{j} \times \hat{i}) \]
Since \( \hat{j} \times \hat{i} = -\hat{k} \), we have:
\[ \mathbf{\tau} = -N \cdot I \cdot \pi r^2 \cdot B \hat{k} \]

7. **Magnitude of the Torque:**
The magnitude of the torque is:
\[ |\mathbf{\tau}| = N \cdot I \cdot \pi r^2 \cdot B \]

### Final Result:
The torque on the coil due to the magnetic field is:
\[ |\mathbf{\tau}| = N \cdot I \cdot \pi r^2 \cdot B \]