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: Determine the Magnetic Moment (M)
The magnetic moment \( \vec{M} \) of a circular loop is given by the formula:
\[ \vec{M} = N \cdot I \cdot A \cdot \hat{j} \]
where:
- \( N \) is the number of turns,
- \( I \) is the current,
- \( A \) is the area of the loop,
- \( \hat{j} \) indicates the direction of the magnetic moment (upward for anti-clockwise current).

The area \( A \) of a circular loop is:
\[ A = \pi r^2 \]
Thus, we can write:
\[ \vec{M} = N \cdot I \cdot \pi r^2 \cdot \hat{j} \]

### Step 2: Identify the Magnetic Field (B)
The magnetic field in the region is given as:
\[ \vec{B} = B_0 \hat{i} \]

### Step 3: Calculate the Torque (τ)
The torque \( \vec{\tau} \) acting on the loop is given by the cross product of the magnetic moment and the magnetic field:
\[ \vec{\tau} = \vec{M} \times \vec{B} \]

Substituting the expressions for \( \vec{M} \) and \( \vec{B} \):
\[ \vec{\tau} = (N \cdot I \cdot \pi r^2 \cdot \hat{j}) \times (B_0 \hat{i}) \]

### Step 4: Evaluate the Cross Product
Using the right-hand rule and the properties of cross products:
\[ \hat{j} \times \hat{i} = -\hat{k} \]
Thus,
\[ \vec{\tau} = N \cdot I \cdot \pi r^2 \cdot B_0 \cdot (-\hat{k}) \]
This simplifies to:
\[ \vec{\tau} = -N \cdot I \cdot \pi r^2 \cdot B_0 \hat{k} \]

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