The physics
Every small piece of a wire adds its own small field at a point. This is the Biot–Savart law.
dB = (μ0/4π) I dl sin θ / r²
The field of a piece points along dl × r. Out of the page is ⊙, into it is ⊗. Add the pieces, and keep their signs.
straight piece: B = (μ0I/4πd)(sin α + sin β)
arc at its centre: B = μ0Iθ/4πR
A straight piece gives nothing at a point on its own line. A long lead that ends beside the point gives μ0I/4πd.
Here μ0/4π = 10⁻⁷ T m/A.