A wire of length l is formed into a circular loop of one turn only and is suspended in a magnetic field B. When a current i is passed through the loop, the maximum torque experienced by it is

Options

A. `((1)/(4pi))Bil`
B. `((1)/(4pi))l^(2)iB`
C. `((1)/(4pi))B^(2)il`
D. `((1)/(4pi))Bi^(2)l`

Answer

To find the maximum torque experienced by a circular loop of wire in a magnetic field when a current is passed through it, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Radius of the Loop:**
The length of the wire is given as \( L \). When the wire is formed into a circular loop, the circumference of the loop is equal to the length of the wire.
\[ 2\pi r = L \]
From this, we can solve for the radius \( r \):
\[ r = \frac{L}{2\pi} \]

2. **Calculate the Area of the Loop:**
The area \( A \) of the circular loop can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Substituting the expression for \( r \):
\[ A = \pi \left(\frac{L}{2\pi}\right)^2 = \pi \cdot \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi} \]

3. **Calculate the Magnetic Moment:**
The magnetic moment \( m \) of the loop is given by the formula:
\[ m = n \cdot i \cdot A \]
Since there is only one turn (one loop), \( n = 1 \):
\[ m = i \cdot A = i \cdot \frac{L^2}{4\pi} \]

4. **Determine the Maximum Torque:**
The torque \( \tau \) experienced by the loop in a magnetic field \( B \) 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 \( m \):
\[ \tau_{\text{max}} = \left(i \cdot \frac{L^2}{4\pi}\right) \cdot B \]
Thus, we can express the maximum torque as:
\[ \tau_{\text{max}} = \frac{i \cdot L^2 \cdot B}{4\pi} \]

5. **Final Result:**
Therefore, the maximum torque experienced by the loop is:
\[ \tau_{\text{max}} = \frac{i \cdot L^2 \cdot B}{4\pi} \]