2.1i.2 Stored Energy of Inductors

Stored Energy of an Inductor

The stored energy $E_L$ of an inductor can be written as
$E_L=\frac{1}{2}L{I}_L(t{)}^2.$

Proof

Since we already know that $P=IV$ and that power is the time-integral of energy (work), we can substitute $V_L=L\frac{d{I}_L(t)}{dt}$: $P_L=L{I}_L(t)\frac{d{I}_L(t)}{dt}.$We then manipulate and integrate:
$\int P_{L}dt=L\int I_L(t)d{I}_L(t).$
Completing the integration we get:
$E=\frac{1}{2}L{I}_L(t{)}^2.$