Let the coil be denoted by ABCD. Let the center be O. So, A, B, C and D are 4 corners of the coil. The distance between the coil and the center is 2a/2 = a. This is the perpendicular distance between any side and the center O.
The magnetic field strength/intensity B due to a finite conductor carrying a current of I, and of length L at a point at a distance d is :
Here α and β are the angles OAB and OBA.
This expression can be derived from the Biot-Savart's Law and integrating over the length of the conductor.
where r' is the relative vector from the small element dl
carrying current I to the point P at which dB
So now, we have α = π/4 and β = π/4 and d = a.B = (μ₀ I / 4 π a) [1/√2 + 1/√2 ] = μ I / (2√2 π a)
The expression (1) can be derived as:
Let us take an element dy (or dl) at P at a distance y from the center of a conductor AB of length L.
We want to find magnetic field induction strength B at O, whose perpendicular distance from AB is d. Let the angle dy (or AB) makes with PO = θ. Let PO make angle Φ with the perpendicular OC from O onto AB. θ = π/2 - Φ. r' = distance PO.
y = d tanΦ and so dy = d sec² Ф dФ
r' = d / Cos Ф = d SecФ
So we get: