ΜThe point P is on the axis of the circular coil, I suppose. Let the center of the coil be O. Let OP be equal to x. Radius is a. Current is I. Let dL be a small element of the coil. So r = distance between dL and P = √(a²+x²).
Let the angle made by the line joining dL and P with the axis be θ. Here the angle θ is same for all elements on the coil. Also r is also same. The angle between the vectors dL and r is 90 deg. always. Let us say that the axis of the coil is x axis.
Cos θ = a / √(a² + x²) = a/r
Using Biot Savarts law:
The y and z components of the magnetic field induction By and Bz are zero, as the components along these axes will be cancelled by diametrically opposite elements.
B₀ at the center of the coil when x = 0, is: μ₀ I / (2 a)
Bx at a distance x on the axis will be B₀ / 8 when Cos θ = 1/2
ie., Cos² θ = 1/4 = a² / (a² + x²)
=> x² = 3 a²
=> x = √3 a