Current `i` is carried in a wire of length `L`. If the wire is turned into a circular coil, the maximum magnitude of torque in a given magnetic field `B` will be

Options

A. `(L^(2)B^(2))/(2)`
B. `(L^(2)B)/(2)`
C. `(L^(2)iB)/(4pi)`
D. `(L^(2)B)/(4pi)`

Answer

To find the maximum magnitude of torque (\( \tau \)) produced by a circular coil made from a wire of length \( L \) carrying current \( i \) in a magnetic field \( B \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a wire of length \( L \) carrying a current \( i \). When this wire is bent into a circular coil, we need to determine the maximum torque it experiences in a magnetic field \( B \).

2. **Determine the Radius of the Coil**: The length of the wire remains constant when it is bent into a circle. The circumference of the circle is given by:
\[ C = 2\pi R \]
Setting this equal to the length of the wire, we have:
\[ 2\pi R = L \implies R = \frac{L}{2\pi} \]

3. **Calculate the Area of the Coil**: The area \( A \) of the circular coil is given by:
\[ A = \pi R^2 \]
Substituting the expression for \( R \):
\[ A = \pi \left(\frac{L}{2\pi}\right)^2 = \pi \cdot \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi} \]

4. **Determine the Magnetic Moment**: The magnetic moment \( m \) of the coil is defined as:
\[ m = i \cdot A \]
Substituting the area we found:
\[ m = i \cdot \frac{L^2}{4\pi} \]

5. **Calculate the Torque**: The torque \( \tau \) experienced by the coil in a magnetic field is given by:
\[ \tau = m \cdot B \cdot \sin \theta \]
For maximum torque, \( \sin \theta = 1 \):
\[ \tau_{\text{max}} = m \cdot B \]
Substituting the expression for \( m \):
\[ \tau_{\text{max}} = \left(i \cdot \frac{L^2}{4\pi}\right) \cdot B = \frac{i L^2 B}{4\pi} \]

6. **Final Result**: The maximum magnitude of torque in the given magnetic field \( B \) is:
\[ \tau_{\text{max}} = \frac{i L^2 B}{4\pi} \]