Energy Stored in a Magnetic Field

Building up a steady current in an inductor requires doing work AGAINST the self-induced back-emf, and this work does not vanish -- it is stored as energy in the magnetic field the current establishes, exactly as the work done charging a capacitor is stored as energy ($U_E=\frac{1}{2}\frac{q^2}{C}=\frac{1}{2}CV^2$) in the electric field between its plates.

To find this stored energy, note that the self-induced emf while the current is rising is $e=L\frac{di}{dt}$, so the work done in moving a small charge dq (= i,dt) against this emf, in time dt, is $dW = e\,dq = L\frac{di}{dt}\cdot i\,dt = Li\,di$. Integrating this from i=0 up to the final steady current $i_0$ gives the total work done: $W=\int_0^{i_0} Li\,di = \frac{1}{2}Li_0^2$. This work IS the energy $U_B$ stored in the magnetic field once the current has been established: $U_B=\frac{1}{2}Li_0^2$. …