Magnetic energy

 ( Forms Of Energy ) 

The potential magnetic energy of a magnet or magnetic moment $\backslash mathbf\{m\}$ in a magnetic field $\backslash mathbf\{B\}$ is defined as the 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 work is done by 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.

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 paramagnetism or diamagnetism 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.

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.

• Magnetic Energy, Richard Fitzpatrick Professor of Physics The University of Texas at Austin.