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`

Explanation

To solve the problem, we need to find the ratio of the magnetic moments of two wires shaped into a square and a circle, given that they are of the same length and carry the same current.

### Step-by-Step Solution:

1. **Define the Length of the Wire**:
Let the total length of each wire be \( L \).

2. **Calculate the Side of the Square**:
The perimeter of a square is given by \( 4a \), where \( a \) is the side length. Since the perimeter equals the length of the wire, we have:
\[ 4a = L \implies a = \frac{L}{4} \]

3. **Calculate the Area of the Square**:
The area \( A \) of the square can be calculated as:
\[ A_{\text{square}} = a^2 = \left(\frac{L}{4}\right)^2 = \frac{L^2}{16} \]

4. **Calculate the Magnetic Moment of the Square**:
The magnetic moment \( \mu \) is given by the product of current \( I \) and area \( A \):
\[ \mu_{\text{square}} = I \cdot A_{\text{square}} = I \cdot \frac{L^2}{16} = \frac{IL^2}{16} \]

5. **Calculate the Radius of the Circle**:
The circumference of a circle is given by \( 2\pi r \). Setting this equal to the length of the wire:
\[ 2\pi r = L \implies r = \frac{L}{2\pi} \]

6. **Calculate the Area of the Circle**:
The area \( A \) of the circle is given by:
\[ A_{\text{circle}} = \pi r^2 = \pi \left(\frac{L}{2\pi}\right)^2 = \pi \cdot \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi} \]

7. **Calculate the Magnetic Moment of the Circle**:
The magnetic moment for the circle is:
\[ \mu_{\text{circle}} = I \cdot A_{\text{circle}} = I \cdot \frac{L^2}{4\pi} = \frac{IL^2}{4\pi} \]

8. **Calculate the Ratio of Magnetic Moments**:
Now, we find the ratio of the magnetic moments of the square to the circle:
\[ \text{Ratio} = \frac{\mu_{\text{square}}}{\mu_{\text{circle}}} = \frac{\frac{IL^2}{16}}{\frac{IL^2}{4\pi}} = \frac{IL^2}{16} \cdot \frac{4\pi}{IL^2} = \frac{4\pi}{16} = \frac{\pi}{4} \]

### Final Answer:
The ratio of the magnetic moments is \( \frac{\pi}{4} \).

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