= Solenoid inductance
{title2=$L\approx\mu_0N^2A/\ell$}
= Self-inductance of a solenoid
{synonym}
For a long vacuum-core or air-core <solenoid> with fixed geometry and linear magnetic response, neglecting end effects gives <self-inductance>
$$
L\approx\frac{\mu_0N^2A}{\ell}
=\mu_0n^2A\ell,
\qquad A=\pi a^2.
$$
Here $N$ is the total number of turns, $n=N/\ell$ is the <solenoid turn density>, and $A$ is the interior cross-sectional <area>. The reason for the square of $N$ is physical: more turns produce more <magnetic field>, and more turns link the resulting <magnetic flux>. Approximately equal <magnetic flux> $\Phi\approx BA$ through each turn gives <magnetic flux linkage> $\Lambda\approx N\Phi\approx\mu_0N^2AI/\ell$; division by the <electric current> yields $L$. https://s3.amazonaws.com/edx-textbooks/mit_802_textbook/Chapter11.pdf[MIT 8.02 course notes, Example 11.2: self-inductance of a solenoid]{ref}.
For the same $N=1000$, $\ell=0.50\,\mathrm m$, and $a=0.025\,\mathrm m$, the <area> is $A=\pi(0.025)^2\,\mathrm{m^2}\approx1.9635\times10^{-3}\,\mathrm{m^2}$. The long-coil estimate is
$$
L\approx\frac{(1.256637\times10^{-6})(1000)^2\pi(0.025)^2}{0.50}\,\mathrm H
\approx4.9348\,\mathrm{mH}.
$$
At $I=0.50\,\mathrm A$, each turn links approximately $2.4674\times10^{-6}\,\mathrm{Wb}$ and the total <magnetic flux linkage> is approximately $2.4674\times10^{-3}\,\mathrm{Wb}$. Their ratio to $I$ is the stated <self-inductance>. A real finite coil need not give precisely these values: the axial centre correction above is not by itself a correction formula for <inductance>, which depends on <magnetic flux> across all the turns.
At fixed $A$ and $\ell$, doubling $N$ quadruples $L$. At fixed <solenoid turn density> $n$ and <area> $A$, doubling $\ell$ doubles both $N$ and $L$. These are different changes to the geometry, so there is no contradiction between $L\propto1/\ell$ at fixed $N$ and $L\propto\ell$ at fixed $n$. A magnetic core requires its own <magnetic permeability> model; nonlinear <magnetization> can make a constant-$L$ description inadequate.
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