Two wires of same length are shaped into a square and a circle. If they carry same current, ratio of the magnetic moment is

Options

A. `2:pi`
B. `pi:2`
C. `pi:4`
D. `4:pi`

Answer

To find the ratio of the magnetic moments of two wires shaped into a square and a circle, we can follow these steps:

### Step 1: Determine the length of the wire
Let the length of the wire be \( L \).

### Step 2: Calculate the side length of the square
For a square, the perimeter is given by:
\[ \text{Perimeter} = 4A \]
where \( A \) is the side length of the square. Since the length of the wire is \( L \), we have:
\[ L = 4A \implies A = \frac{L}{4} \]

### Step 3: Calculate the area of the square
The area \( A_s \) of the square is:
\[ A_s = A^2 = \left(\frac{L}{4}\right)^2 = \frac{L^2}{16} \]

### Step 4: Calculate the magnetic moment of the square
The magnetic moment \( M_s \) for a current \( I \) flowing in the square loop is given by:
\[ M_s = I \cdot A_s = I \cdot \frac{L^2}{16} = \frac{I L^2}{16} \]

### Step 5: Calculate the radius of the circle
For a circle, the circumference is given by:
\[ \text{Circumference} = 2\pi R \]
where \( R \) is the radius of the circle. Since the length of the wire is \( L \), we have:
\[ L = 2\pi R \implies R = \frac{L}{2\pi} \]

### Step 6: Calculate the area of the circle
The area \( A_c \) of the circle is:
\[ A_c = \pi R^2 = \pi \left(\frac{L}{2\pi}\right)^2 = \pi \cdot \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi} \]

### Step 7: Calculate the magnetic moment of the circle
The magnetic moment \( M_c \) for a current \( I \) flowing in the circular loop is given by:
\[ M_c = I \cdot A_c = I \cdot \frac{L^2}{4\pi} = \frac{I L^2}{4\pi} \]

### Step 8: Find the ratio of the magnetic moments
Now, we can find the ratio of the magnetic moment of the square to that of the circle:
\[ \text{Ratio} = \frac{M_s}{M_c} = \frac{\frac{I L^2}{16}}{\frac{I L^2}{4\pi}} = \frac{4\pi}{16} = \frac{\pi}{4} \]

### Conclusion
Thus, the ratio of the magnetic moment of the square to that of the circle is:
\[ \frac{\pi}{4} \]