Magnetic energy

{{Short description|Energy from the work of a magnetic force}}

{{Cleanup rewrite|date=March 2023}}

The potential magnetic energy of a magnet or magnetic moment $\backslash mathbf\{m\}$ in a magnetic field $\backslash mathbf\{B\}$ is defined as the mechanical work of the magnetic force on the re-alignment of the vector of the magnetic dipole moment and is equal to: $$E\_\backslash text\{p,m\}\; =\; -\backslash mathbf\{m\}\; \backslash cdot\; \backslash mathbf\{B\}$$The mechanical work takes the form of a torque $\backslash boldsymbol\{N\}$:$$\backslash mathbf\{N\}=\backslash mathbf\{m\}\backslash times\backslash mathbf\{B\}=-\backslash mathbf\{r\}\backslash times\backslash mathbf\{\backslash nabla\}E\_\backslash text\{p,m\}$$

which will act to "realign" the magnetic dipole with the magnetic field.{{Cite book |last=Griffiths |first=David J. |title=Introduction to electrodynamics |date=2023 |publisher=Cambridge University Press |isbn=978-1-009-39773-5 |edition=Fifth |location=New York}}

In an electronic circuit the energy stored in an inductor (of inductance $L$) when a current $I$ flows through it is given by:$$E\_\backslash text\{p,m\}\; =\; \backslash frac\{1\}\{2\}\; LI^2.$$

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.

Energy is also stored in a magnetic field itself. The energy per unit volume $u$ in a region of free space with vacuum permeability $\backslash mu\; \_0$ containing magnetic field $\backslash mathbf\{B\}$ is:

$$u\; =\; \backslash frac\{1\}\{2\}\; \backslash frac\{B^2\}\{\backslash mu\_0\}$$More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates $\backslash mathbf\{B\}$ and the magnetization $\backslash mathbf\{H\}$ (for example $\backslash mathbf\{H\}=\backslash mathbf\{B\}/\backslash mu$ where $\backslash mu$ is the magnetic permeability of the material), then it can be shown that the magnetic field stores an energy of

$$E\; =\; \backslash frac\{1\}\{2\}\; \backslash int\; \backslash mathbf\{H\}\; \backslash cdot\; \backslash mathbf\{B\}\; \backslash ,\; \backslash mathrm\{d\}V$$

where the integral is evaluated over the entire region where the magnetic field exists.{{cite book |last1=Jackson|first1=John David |title=Classical Electrodynamics |date=1998 |publisher=Wiley|location=New York |edition=3 |pages=212–onwards}}

For a magnetostatic system of currents in free space, the stored energy can be found by imagining the process of linearly turning on the currents and their generated magnetic field, arriving at a total energy of:

$$E\; =\; \backslash frac\{1\}\{2\}\; \backslash int\; \backslash mathbf\{J\}\; \backslash cdot\; \backslash mathbf\{A\}\backslash ,\; \backslash mathrm\{d\}V$$

where $\backslash mathbf\{J\}$ is the current density field and $\backslash mathbf\{A\}$ is the magnetic vector potential. This is analogous to the electrostatic energy expression $\backslash frac\{1\}\{2\}\backslash int\; \backslash rho\; \backslash phi\; \backslash ,\; \backslash mathrm\{d\}V$; note that neither of these static expressions apply in the case of time-varying charge or current distributions.{{cite web| url=https://feynmanlectures.caltech.edu/II_15.html| title=The Feynman Lectures on Physics, Volume II, Chapter 15: The vector potential}}