The wire of length L is bent in the form of square and circle.The ratio of their magnetic moment at their centre is

Options

A. `pi//4`
B. `pi//6`
C. `pi//2`
D. `pi`

Explanation

To find the ratio of the magnetic moments of a wire bent into the shape of a square and a circle, we can follow these steps:

### Step 1: Determine the side length of the square
The total length of the wire is given as \( L \). For a square, the perimeter \( P \) is equal to the length of the wire:
\[ P = 4 \times \text{side length} = L \]
Thus, the side length \( S \) of the square can be calculated as:
\[ S = \frac{L}{4} \]

### Step 2: Calculate the area of the square
The area \( A \) of the square can be calculated using the formula:
\[ A = S^2 = \left(\frac{L}{4}\right)^2 = \frac{L^2}{16} \]

### Step 3: Calculate the magnetic moment of the square
The magnetic moment \( M_1 \) for a single loop (N=1) carrying current \( I \) is given by:
\[ M_1 = N \cdot I \cdot A = 1 \cdot I \cdot \frac{L^2}{16} = \frac{I L^2}{16} \]

### Step 4: Determine the radius of the circle
For a circle, the circumference \( C \) is equal to the length of the wire:
\[ C = 2\pi r = L \]
Thus, the radius \( r \) of the circle can be calculated as:
\[ r = \frac{L}{2\pi} \]

### Step 5: Calculate the area of the circle
The area \( A \) of the circle can be calculated using the formula:
\[ A = \pi r^2 = \pi \left(\frac{L}{2\pi}\right)^2 = \pi \cdot \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi} \]

### Step 6: Calculate the magnetic moment of the circle
The magnetic moment \( M_2 \) for the circular loop is given by:
\[ M_2 = N \cdot I \cdot A = 1 \cdot I \cdot \frac{L^2}{4\pi} = \frac{I L^2}{4\pi} \]

### Step 7: Find the ratio of the magnetic moments
Now, we can find the ratio of the magnetic moments \( M_1 \) and \( M_2 \):
\[ \text{Ratio} = \frac{M_1}{M_2} = \frac{\frac{I L^2}{16}}{\frac{I L^2}{4\pi}} = \frac{4\pi}{16} = \frac{\pi}{4} \]

### Conclusion
Thus, the ratio of the magnetic moments at their center is:
\[ \frac{M_1}{M_2} = \frac{\pi}{4} \]