= Solenoid
{title2=$B=\mu_0nI$}
{wiki}
A <solenoid> is a coil of <electrical conductor>, usually wire wound into a <helix> around a <circular cylinder>. The same <electric current> $I$ flows through its successive turns. Each turn contributes a <magnetic field>; the <superposition principle> makes their axial contributions reinforce inside the coil. Curl the fingers of your right hand in the direction of the <electric current> around the turns: the thumb gives the interior <magnetic field> direction by the <right-hand rule>. Throughout this branch, $N$ is the total number of turns, $\ell$ is the coil <length>, and $n=N/\ell$ is the <solenoid turn density>. https://www.feynmanlectures.caltech.edu/II_13.html#Ch13-S5[Feynman, Leighton and Sands, The Feynman Lectures on Physics, II.13–5: the field of a long coil]{ref}.
The attraction of a long <solenoid> is a controllable, nearly uniform <magnetic field> over an interior working region: change the <electric current> to tune its strength, or reverse the <electric current> to reverse its direction. This makes it a useful setting for studying the <Lorentz force>, measuring <magnetic flux>, and constructing an <inductor>. Its geometry turns the many contributions from individual turns into the simple <long-solenoid approximation>. A real finite coil has end fields and leakage; its exterior <magnetic field> is not zero. The <on-axis magnetic field of a finite solenoid> below quantifies that difference.
Historically, the coil helped connect <magnetic fields> to <electric currents>. In his memoir on the mathematical theory of electrodynamic phenomena, Ampère modeled the interactions of magnets using the ends of solenoids in place of microscopic magnetic constituents. This was a route toward explaining magnetism by <electric currents>, rather than a modern microscopic account of <magnetization>. https://www.e-rara.ch/zut/download/pdf/1038813[Ampère, Mémoire sur la théorie mathématique des phénomènes électro-dynamiques, concluding discussion on p. 289]{ref}. The historical setting is documented by the https://ampere-archives.fr/histoire-electricite/parcours-historique/lois-courants/ampere-loi[Ampère archives, investigations of current interactions in 1820–1826]{ref}.
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