Magnetic field at the centre of regular polygon of 'n' sides which is formed by wire, which carries current I and side of polygon is ‘a’

The magnetic field at the center of a regular polygon formed by a wire carrying current I and with side length 'a' can be calculated using Ampere's Circuital Law and the Biot-Savart Law. Here are the steps:

1. First, we need to understand that the magnetic field at the center due to each side of the polygon will be the same because of the symmetry of the polygon.

2. Let's consider one side of the polygon. The magnetic field at the center due to this side can be calculated using the Biot-Savart Law, which states that dB = (μ0/4π) * (Idl x r/r^3), where dB is the magnetic field, μ0 is the permeability of free space, I is the current, dl is the length element, and r is the distance from the current element to the point where we are calculating the magnetic field.

3. In this case, dl is the length of the side of the polygon 'a', and r is the distance from the center to the midpoint of the side, which can be calculated using the properties of regular polygons.

4. The angle between dl and r is 90 degrees, so the cross product simplifies to dlr. Therefore, the magnetic field due to one side is dB = (μ0/4π) * (Ia/r^2).

5. Since there are 'n' sides, the total magnetic field at the center is B = n*dB.

6. Substitute the values of dB and r (calculated using the properties of regular polygons) into this equation to get the final expression for the magnetic field at the center of the polygon.

Please note that this is a simplified explanation and the actual calculation may involve more complex geometry depending on the specific properties of the polygon.