A wire of length 2 m is bent into a circular loop. When a current of 1 A is passed through the loop, then the magnetic moment of the loop is

Options

A. `2pi A m^(2)`
B. `pi //2 Am^(2)`
C. `pi //4 A m^(2)`
D. `1//pi Am^(2)`

Explanation

To find the magnetic moment of a circular loop formed by bending a wire of length 2 m through which a current of 1 A is passed, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a wire of length \( L = 2 \, \text{m} \) bent into a circular loop. The current flowing through the loop is \( I = 1 \, \text{A} \). We need to calculate the magnetic moment \( M \) of the loop.

2. **Identify the Formula for Magnetic Moment**: The magnetic moment \( M \) of a circular loop is given by the formula:
\[ M = I \cdot A \]
where \( I \) is the current and \( A \) is the area of the loop.

3. **Calculate the Radius of the Loop**: The circumference of the circular loop is equal to the length of the wire:
\[ C = 2\pi r = L \]
Substituting \( L = 2 \, \text{m} \):
\[ 2\pi r = 2 \]
Solving for \( r \):
\[ r = \frac{2}{2\pi} = \frac{1}{\pi} \, \text{m} \]

4. **Calculate the Area of the Loop**: The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi \left(\frac{1}{\pi}\right)^2 = \pi \cdot \frac{1}{\pi^2} = \frac{1}{\pi} \, \text{m}^2 \]

5. **Calculate the Magnetic Moment**: Now substituting the values of \( I \) and \( A \) into the magnetic moment formula:
\[ M = I \cdot A = 1 \cdot \frac{1}{\pi} = \frac{1}{\pi} \, \text{A m}^2 \]

6. **Final Answer**: The magnetic moment of the loop is:
\[ M = \frac{1}{\pi} \, \text{A m}^2 \]