= Inductor
{wiki}
An <inductor> is an <electrical circuit> component designed to provide <self-inductance>, often a coil such as a <solenoid>. An ideal linear <inductor> has constant $L>0$ and zero <electrical resistance>. Its stored <magnetic energy>, with zero chosen at zero <electric current>, is
$$
U=\frac12LI^2.
$$
For <self-inductance> $0.5\,\mathrm H$ and <electric current> $3\,\mathrm A$, this gives $U=2.25\,\mathrm J$. Unlike <Joule heating>, ideal storage is recoverable: the <inductor> can return <energy> to an <electrical circuit> when its <electric current> decreases. https://openstax.org/books/university-physics-volume-2/pages/14-3-energy-in-a-magnetic-field[OpenStax University Physics 2, §14.3: energy stored by an inductor]{ref}.
The short derivation uses the terminal convention in <self-inductance>. The <power> absorbed by an ideal <inductor> is
$$
P_L=v_L I=LI\frac{dI}{dt}=\frac{d}{dt}\left(\frac12LI^2\right).
$$
Starting from zero <electric current>, the <integral> of this <power> gives
$$
U=\int_0^t P_L(t')\,dt'=\int_0^I Li\,di=\frac12LI^2.
$$
Positive $P_L$ increases stored <magnetic energy>; negative $P_L$ returns <energy>. At constant <electric current> $3\,\mathrm A$ in the example, $P_L=0$ and induced <electromotive force> is zero, but $U=2.25\,\mathrm J$ still resides in the <magnetic field>. The <battery> in the steady <RL circuit> supplies <Joule heating> in the <electrical resistance>, rather than further increasing the <inductor>'s <magnetic energy>.
\Image[/inductor-energy-storage-and-release.png]
{title=Stored <magnetic energy> and its release as <Joule heating> in an <RL circuit>}
{description=Left: the <inductor>'s <magnetic energy> during buildup and a separate decay experiment starting at $3\,\mathrm A$. Right: the <power> of <Joule heating> during decay; the shaded <integral> over all time is $2.25\,\mathrm J$. The decay path stays closed through <electrical resistance> $2\,\Omega$ and <self-inductance> $0.5\,\mathrm H$.}
During the closed-loop release in <RL circuit>, stored <magnetic energy> plus accumulated <Joule heating> stays $2.25\,\mathrm J$. A real coil also has <electrical resistance>, so its <Joule heating> must be included separately from ideal <inductor> storage. The constant-$L$ formulas assume a fixed linear magnetic response.
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