The physics
Each small piece of a current-carrying wire makes a small field. This is the Biot–Savart law. Add the pieces of a round coil.
centre: B = μ0NI / 2R axis: B = μ0NIR² / 2(R² + x²)3/2
At the centre, every piece adds the same way. On the axis, the sideways parts cancel round the coil. Only the parts along the axis add.
Curl the fingers of your right hand with the current. Your thumb points along B.
Fields from several coils or wires add as arrows. Fields at right angles add by Pythagoras. Opposite fields subtract.
μ0 = 4π × 10⁻⁷ T m/A