Each turn contributes its own ring of field. Stack enough of them and the contributions cancel everywhere except along the axis, where they add into something almost perfectly uniform. Every line here is a Biot–Savart sum over the whole winding — no idealisation.
whatA helical winding with the field summed over every discretised current element, rather than assumed uniform.
lawBiot–Savart, integrated along the wire. In the long-coil limit the interior tends to μ₀nI and the exterior nearly vanishes.
tryShorten the coil until it is wider than it is long.
noticeThe bore stops being uniform from the ends inward, and field starts leaking out the sides. Length-to-radius, not turn count alone, buys uniformity.
in lifeMRI scanners are enormous solenoids, built long and narrow for exactly the uniformity shown here.
found byAmpère coined the word and built the first coils in the 1820s.