= Solution
An <inductor> stores magnetic energy and opposes changes of current through <electromagnetic induction>. For a fixed linear device, the flux linkage is $\Psi=N\Phi=LI$, the self-induced <electromotive force> is $-L\,dI/dt$, and the passive terminal voltage is $V=L\,dI/dt$. Hence
$$
\boxed{U=\int_0^I LI'\,dI'=\tfrac12LI^2,\qquad |X_L|=\omega L.}
$$
The voltage leads a sinusoidal current by a quarter cycle. A long solenoid has $B\simeq\mu NI/l$ and \b[$L\simeq\mu N^2A/l$]; its energy agrees with the volume integral of $B^2/(2\mu)$.
For two coupled coils, $\Psi_1=L_1I_1+\mathcal M I_2$, $\Psi_2=\mathcal M I_1+L_2I_2$, with signs set by coil orientations. Reciprocity gives the same <mutual inductance> in both equations. For the <energy stored in a nonlinear inductor>, $U=\int I\,d\Psi$ replaces the simple quadratic formula, and <hysteresis> can dissipate energy. Real coils also have resistance, core loss and parasitic capacitance, so an ideal <inductance> model has a finite useful frequency range.<Electromagnetism by David Tong>[Section 4.1.2: inductance and magnetostatic energy]{ref}
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