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An inductor stores energy $U = \frac{1}{2}LI^2$ in its magnetic field, analogous to a capacitor's $U = \frac{1}{2}CV^2$. The energy density (per unit volume) is $u = B^2/(2\mu_0)$, paralleling the electric energy density $u_E = \varepsilon_0 E^2/2$. For a solenoid: total energy = energy density $\times$ volume = $\frac{B^2}{2\mu_0} \times Al$.

Energy considerations explain several phenomena: (1) Sparks when inductive circuits are opened — the stored energy $\frac{1}{2}LI^2$ must dissipate somewhere, producing high voltage across the switch gap. (2) The inductor's "inertia" — it resists current changes because energy must be supplied to (or extracted from) the magnetic field. (3) Electromagnetic wave energy — equal contributions from electric and magnetic fields.

In coupled coils: $U = \frac{1}{2}L_1I_1^2 + \frac{1}{2}L_2I_2^2 \pm MI_1I_2$. The condition $U \geq 0$ requires $M \leq \sqrt{L_1 L_2}$ (i.e., $k \leq 1$), providing a physical bound on mutual inductance.

The energy density formula $u = B^2/(2\mu_0)$ is universal — it applies to any magnetic field, not just solenoids. A 1 T field stores about $4 \times 10^5$ J/m$^3$, explaining the enormous energy in MRI magnets and the danger of sudden quenches.