Result

Energy in an inductor

(E10.3.1) E10.3 · Energy in a Magnetic Field →

U = \frac{1}{2} L I^{2}, u_{B} = \frac{B ^{2}}{2 μ _{0}}

Statement

Energy stored in an inductor carrying current I, and the corresponding energy density of a magnetic field.

Symbols and units

$U$	energy stored	J
$L$	inductance	H
$I$	current	A
$u_{B}$	magnetic energy density	J/m³
$μ_{0}$	permeability of free space	4π×10⁻⁷ T·m/A
$B$	magnetic field	T

When it applies

Any inductor with current I, or any region containing a magnetic field.

Don't use when

The ½LI² form is for a linear inductor with constant L — a core that saturates magnetically makes L current-dependent, and the energy must be found by integration instead.

Derivation

Integrate the work done against the back-emf while current builds from 0 to I:

U = \int_{0}^{I} L I^{'} d I^{'} = \frac{1}{2} L I^{2}

Watch out for

Exactly parallel to a capacitor's ½CV² — an inductor stores energy in its magnetic field the way a capacitor stores it in its electric field.

Connects to

Stored energy Energy density