= Long-solenoid approximation
{title2=$\mathbf B_{\rm in}=\mu_0nI\,\widehat{\mathbf z},\quad\mathbf B_{\rm out}=0$}
In <magnetostatics>, an ideal infinitely long <solenoid> is a cylindrical sheet of circumferential <surface current density> $K=nI$. It has negligible winding advance, no magnetic core, and no imposed external <magnetic field>. Choose positive <electric current> to produce <magnetic field> along $+z$. In this ideal model the interior <magnetic field> is uniform and axial, $B=\mu_0nI$, and the exterior <magnetic field> is zero; $\mu_0$ is the <vacuum permeability>. For a finite, closely wound air-core coil, the same result is an approximation well inside the coil, several radii from either end. Air is treated as having the <magnetic permeability> of vacuum. https://www.feynmanlectures.caltech.edu/II_13.html#Ch13-S5[Feynman lectures, II.13–5: long-solenoid field and the return field near the ends]{ref}.
For a short proof, the translational and rotational <symmetries> of the ideal cylindrical <current sheet>, together with the <Biot-Savart law>, give an axial <magnetic field>. <Ampère's law> on rectangles wholly outside makes that axial exterior value independent of radius; the condition of no imposed <magnetic field> makes it zero at infinity and hence everywhere outside. Rectangles wholly inside similarly show that the interior value is independent of radius. Now choose an <Amperian loop> with axial length $s$, one long side inside and the other outside. The transverse sides are perpendicular to the <magnetic field>, so their contributions to the <line integral> vanish. The loop crosses $ns$ turns of the winding, giving
$$
\oint_\Gamma\mathbf B\cdot d\mathbf l
=Bs=\mu_0(ns)I,
\qquad B=\mu_0nI.
$$
Here $s$ is the integration-path length, not the whole coil <length> $\ell$. An <Amperian loop> surrounding both sides of the winding encloses equal and opposite <electric currents>; zero net circulation alone would not prove zero <magnetic field> at every exterior point.
\Image[/solenoid-ampere-loop.png]
{title=<Ampère's law> for the ideal <long-solenoid approximation>}
{description=Axial cross-section of an infinite ideal winding: paired dots and crosses show <electric current> out of and into the page. The indicated <right-hand rule> produces interior <magnetic field> along $+z$. The counterclockwise <Amperian loop> crosses only the upper row, with $ns=6$ positive crossings; its interior side runs along $+z$, and its exterior return lies where the ideal <magnetic field> is zero.}
For the $1000$-turn, $0.50\,\mathrm m$ coil with <electric current> $I=0.50\,\mathrm A$, the <long-solenoid approximation> gives
$$
B\approx(1.256637\times10^{-6})(2000)(0.50)\,\mathrm T
=1.256637\times10^{-3}\,\mathrm T
\approx1.257\,\mathrm{mT}.
$$
This illustrative calculation uses <vacuum permeability> $\mu_0\approx1.256637\times10^{-6}\,\mathrm{H/m}$, a rounded value, not an exact constant. Since the 2019 redefinition of <SI>, <vacuum permeability> is determined experimentally.<The International System of Units (SI), ninth edition, SI Brochure>[§2.3.1: the ampere and vacuum permeability]{ref} Doubling $I$ doubles the ideal <magnetic field>; doubling $N$ at fixed $\ell$ does the same. Increasing $\ell$ while keeping $N$ fixed decreases $n$ and hence the ideal <magnetic field>.
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