Magnetic energy

Magnetic energy of an inductor

In an electronic circuit the magnetic energy, ${\displaystyle U_{\text{m}}}$, stored in an inductor (of inductance ${\displaystyle L}$) when a current ${\displaystyle I}$ flows through it is given by:$${\displaystyle U_{\text{m}}=\frac{1}{2}L{I}^2.}$$ This expression forms the basis for superconducting magnetic energy storage. It can be derived from a time average of the product of current and voltage across an inductor.

Magnetic energy stored in magnetic fields

Energy is also stored in a magnetic field itself. The energy per unit volume ${\displaystyle u}$ in a region of free space with vacuum permeability ${\displaystyle {\mu }_0}$ containing magnetic field ${\displaystyle \mathbf{B}}$ is: $${\displaystyle u=\frac{1}{2}\frac{B^2}{{\mu }_0}}$$More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates ${\displaystyle \mathbf{B}}$ and the magnetization ${\displaystyle \mathbf{H}}$ (for example ${\displaystyle \mathbf{H}=\mathbf{B}/\mu }$ where ${\displaystyle \mu }$ is the magnetic permeability of the material), then it can be shown that the magnetic field stores an energy of $${\displaystyle U_{\text{m}}=\frac{1}{2}\int \mathbf{H}\cdot \mathbf{B}\mathrm{d}V}$$ where the integral is evaluated over the entire region where the magnetic field exists.

For a magnetostatic system of currents in free space, the stored energy can be found by noting that ${\displaystyle \mathrm{\nabla }\times \overrightarrow{H}=\overrightarrow{J}}$ and ${\displaystyle \overrightarrow{B}=\mathrm{\nabla }\times \overrightarrow{A}}$, or by imagining the process of linearly turning on the currents and their generated magnetic field, arriving at a total energy of: $${\displaystyle U_{\text{m}}=\frac{1}{2}\int \mathbf{J}\cdot \mathbf{A}\mathrm{d}V}$$ where ${\displaystyle \mathbf{J}}$ is the current density field and ${\displaystyle \mathbf{A}}$ is the magnetic vector potential. This is analogous to the electrostatic energy expression $\frac{1}{2}\int \rho \varphi \mathrm{d}V$; note that neither of these static expressions apply in the case of time-varying charge or current distributions.