Two wires of the 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 solve the problem of finding the ratio of the magnetic moments of two wires shaped into a square and a circle, we can follow these steps:

### Step 1: Define the Problem
We have two wires of the same length, one shaped into a square and the other into a circle, both carrying the same current. We need to find the ratio of their magnetic moments.

### Step 2: Determine the Length of the Wires
Let the length of each wire be \( L \).

For the square:
- The perimeter of the square is \( 4a \), where \( a \) is the side length of the square.
- Therefore, \( 4a = L \) or \( a = \frac{L}{4} \).

For the circle:
- The circumference of the circle is \( 2\pi r \), where \( r \) is the radius of the circle.
- Thus, \( 2\pi r = L \) or \( r = \frac{L}{2\pi} \).

### Step 3: Calculate the Areas
The area \( A \) is needed to calculate the magnetic moment.

For the square:
- The area \( A_s \) of the square is given by:
\[ A_s = a^2 = \left(\frac{L}{4}\right)^2 = \frac{L^2}{16} \]

For the circle:
- The area \( A_c \) of the circle is given by:
\[ A_c = \pi r^2 = \pi \left(\frac{L}{2\pi}\right)^2 = \pi \cdot \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi} \]

### Step 4: Calculate the Magnetic Moments
The magnetic moment \( M \) is given by the formula:
\[ M = NIA \]
where \( N \) is the number of turns (which is 1 for both shapes), and \( I \) is the current.

For the square:
- The magnetic moment \( M_s \) is:
\[ M_s = 1 \cdot I \cdot A_s = I \cdot \frac{L^2}{16} \]

For the circle:
- The magnetic moment \( M_c \) is:
\[ M_c = 1 \cdot I \cdot A_c = I \cdot \frac{L^2}{4\pi} \]

### Step 5: Find the Ratio of the Magnetic Moments
Now we can find the ratio of the magnetic moments:
\[ \frac{M_s}{M_c} = \frac{I \cdot \frac{L^2}{16}}{I \cdot \frac{L^2}{4\pi}} = \frac{\frac{L^2}{16}}{\frac{L^2}{4\pi}} = \frac{4\pi}{16} = \frac{\pi}{4} \]

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

### Final Answer
The ratio of the magnetic moment of the square to the magnetic moment of the circle is \( \frac{\pi}{4} \).