A circular loop with N turns has radius r . It lies in the x-y plane carrying current I in the anti-clockwise direction . If the magnetic field in the region is `vecB = B_0hati` , then find the torque `(vecr)` acting on the loop .

Options

A. `B_0NI pir^2hatj`
B. `B_0Ipir^2hatj`
C. `B_0NIpir^2hatk`
D. `B_0pir^2atk`

Answer

To find the torque acting on a circular loop carrying current in a magnetic field, we can follow these steps:

### Step 1: Identify the Area Vector
The area vector \( \vec{A} \) for a circular loop lies perpendicular to the plane of the loop. Since the loop is in the x-y plane, the area vector points in the positive z-direction. The magnitude of the area \( A \) of the loop is given by:
\[ A = \pi r^2 \]
Thus, the area vector can be expressed as:
\[ \vec{A} = \pi r^2 \hat{k} \]

### Step 2: Determine the Magnetic Field
The magnetic field \( \vec{B} \) in the region is given as:
\[ \vec{B} = B_0 \hat{i} \]
This indicates that the magnetic field is directed along the x-axis.

### Step 3: Calculate the Torque
The torque \( \vec{\tau} \) acting on the loop can be calculated using the formula:
\[ \vec{\tau} = N \vec{I} \times \vec{A} \times \vec{B} \]
Where \( N \) is the number of turns, \( I \) is the current, and \( \vec{A} \) is the area vector.

Since the current is flowing in the anti-clockwise direction, the effective current for \( N \) turns is:
\[ \vec{I} = N I \]

Now substituting the values:
\[ \vec{\tau} = N I \vec{A} \times \vec{B} \]

### Step 4: Compute the Cross Product
We need to compute the cross product \( \vec{A} \times \vec{B} \):
\[ \vec{A} = \pi r^2 \hat{k}, \quad \vec{B} = B_0 \hat{i} \]
Using the right-hand rule for cross products:
\[ \hat{k} \times \hat{i} = \hat{j} \]
Thus,
\[ \vec{A} \times \vec{B} = \pi r^2 \hat{k} \times B_0 \hat{i} = \pi r^2 B_0 \hat{j} \]

### Step 5: Final Expression for Torque
Now substituting back into the torque equation:
\[ \vec{\tau} = N I (\pi r^2 B_0 \hat{j}) = N I \pi r^2 B_0 \hat{j} \]

### Final Answer
The torque acting on the loop is:
\[ \vec{\tau} = N I \pi r^2 B_0 \hat{j} \]

---