A current i is flowing in a square coil of side a. The value of magnetic field at the centre of the coil is

Question

Accepted Answer

To determine the magnetic field at the center of a square coil carrying a current, we need to apply the principles of electromagnetism, specifically the Biot-Savart Law. The Biot-Savart Law helps us calculate the magnetic field produced by a current-carrying conductor.

### Explanation for Option 1:

1. **Biot-Savart Law Application**: For a straight segment of wire carrying a current $i$, the magnetic field $dB$ at a point located at a perpendicular distance $r$ from the wire is given by:
   $$
   dB = \frac{\mu_0 i}{4\pi} \frac{dl \sin \theta}{r^2}
   $$
   where $dl$ is the length element of the wire, $\theta$ is the angle between the wire and the line joining the point to the wire, and $\mu_0$ is the permeability of free space.

2. **Square Coil Configuration**: For a square coil with side $a$, the magnetic field at the center due to one side can be calculated and then multiplied by 4 (since the coil has four sides).

3. **Calculation for One Side**: The magnetic field at the center due to one side of the square is:
   $$
   B_{\text{one side}} = \frac{\mu_0 i}{4\pi} \frac{2}{a/\sqrt{2}}
   $$
   Simplifying, we find:
   $$
   B_{\text{one side}} = \frac{\mu_0 i \sqrt{2}}{\pi a}
   $$

4. **Total Magnetic Field**: Since the square has four sides, the total magnetic field at the center is:
   $$
   B_{\text{total}} = 4 \times \frac{\mu_0 i \sqrt{2}}{\pi a} = \frac{2\sqrt{2}\mu_0 i}{\pi a}
   $$

Thus, the correct expression for the magnetic field at the center of the square coil is given by Option 1: $\frac{2\sqrt{2}\mu_0 i}{\pi a}$.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $\frac{2\sqrt{2}\mu_0 i}{a}$**: This option does not include the $\pi$ factor in the denominator, which is essential as derived from the Biot-Savart Law and the geometry of the square coil.

- **Option 3: $\frac{\sqrt{2}\mu_0 i}{\pi a}$**: This option only accounts for the magnetic field contribution from one side of the square coil, not the total contribution from all four sides.

- **Option 4: $\frac{\sqrt{2}\mu_0 i}{a}$**: Similar to Option 3, this option lacks the $\pi$ factor and only considers the contribution from one side, not the entire coil.

In conclusion, Option 1 is the correct answer as it accurately represents the total magnetic field at the center of a square coil with side $a$ carrying a current $i$, considering the contributions from all four sides and the necessary geometric and physical constants.