The ratio of the magnetic field at the centre of a current carrying circular wire and the magnetic field at the centre of a square coil made from the same length of wire will be

Options

A. `(pi^(2))/(4sqrt(2))`
B. `(pi^(2))/(8sqrt(2))`
C. `(pi)/(2sqrt(2))`
D. `(pi)/(4sqrt(2))`

Explanation

To find the ratio of the magnetic field at the center of a current-carrying circular wire and the magnetic field at the center of a square coil made from the same length of wire, we can follow these steps:

### Step 1: Determine the magnetic field at the center of the circular wire
The formula for the magnetic field \( B \) at the center of a circular loop of radius \( R \) carrying a current \( I \) is given by:

\[ B_{\text{circle}} = \frac{\mu_0 I}{2R} \]

### Step 2: Relate the radius of the circular wire to the length of the wire
The length of the wire used to form the circular loop is given by the circumference:

\[ L = 2\pi R \implies R = \frac{L}{2\pi} \]

### Step 3: Substitute \( R \) in the magnetic field formula
Substituting \( R \) in the magnetic field formula:

\[ B_{\text{circle}} = \frac{\mu_0 I}{2 \left( \frac{L}{2\pi} \right)} = \frac{\mu_0 I \cdot 2\pi}{2L} = \frac{\mu_0 I \pi}{L} \]

### Step 4: Determine the magnetic field at the center of the square coil
For a square coil, the magnetic field at the center can be calculated using the Biot-Savart law. The formula for the magnetic field \( B \) at the center of a square coil of side \( a \) carrying a current \( I \) is:

\[ B_{\text{square}} = \frac{2\mu_0 I}{\pi a} \]

### Step 5: Relate the side of the square to the length of the wire
The length of the wire used to form the square coil is given by:

\[ L = 4a \implies a = \frac{L}{4} \]

### Step 6: Substitute \( a \) in the magnetic field formula for the square coil
Substituting \( a \) in the magnetic field formula:

\[ B_{\text{square}} = \frac{2\mu_0 I}{\pi \left( \frac{L}{4} \right)} = \frac{2\mu_0 I \cdot 4}{\pi L} = \frac{8\mu_0 I}{\pi L} \]

### Step 7: Calculate the ratio of the magnetic fields
Now, we can find the ratio of the magnetic fields at the center of the circular wire and the square coil:

\[ \text{Ratio} = \frac{B_{\text{circle}}}{B_{\text{square}}} = \frac{\frac{\mu_0 I \pi}{L}}{\frac{8\mu_0 I}{\pi L}} = \frac{\pi^2}{8} \]

### Conclusion
Thus, the ratio of the magnetic field at the center of the current-carrying circular wire to the magnetic field at the center of the square coil is:

\[ \frac{\pi^2}{8} \]