Energy Stored in an Inductor

Why establishing a current costs energy. As a current $I$ in an inductor is being built up from zero, the self-induced back emf $\mathcal{E}=-L\,dI/dt$ (Section 6.7) continuously opposes the increase, so whatever source is driving the current (a battery, say) must do work against this opposition at every instant, exactly as a person pushing a heavy trolley must do work against its mechanical inertia to speed it up. The instantaneous rate at which this work is done is

since the source must supply an emf that exactly matches the back emf's magnitude at every instant to keep pushing the current higher.

Total energy stored. Integrating this rate of work as the current builds up smoothly from $0$ to its final steady value $I$ gives the total energy delivered to the inductor:

This energy is not dissipated as heat (that would need resistance, a separate effect); it is instead STORED in the magnetic field the current has set up around the inductor, in exactly the same sense that the energy delivered while charging a capacitor is stored in its electric field. If the current is later reduced back to zero, this stored energy $\frac{1}{2}LI^2$ is released back into the circuit (this is what allows an inductor to momentarily keep a current flowing, and can even produce a large voltage spark, at the instant a current-carrying inductive circuit is suddenly switched off). …