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The potentialmagnetic energy of amagnet ormagnetic moment${\displaystyle \mathbf{m}}$ in amagnetic field${\displaystyle \mathbf{B}}$ is defined as thework of the magnetic force on the re-alignment of the vector of themagnetic dipole moment and is equal to:$${\displaystyle E_{\text{p,m}}=-\mathbf{m}\cdot \mathbf{B}}$$The work is done by a torque${\displaystyle \mathit{N}}$:$${\displaystyle \mathbf{N}=\mathbf{m}\times \mathbf{B}=-\mathbf{r}\times \mathrm{\nabla }{E}_{\text{p,m}}}$$which will act to "realign" the magnetic dipole with the magnetic field.^[1]

In anelectronic circuit the energy stored in aninductor (ofinductance${\displaystyle L}$) when a current${\displaystyle I}$ flows through it is given by:$${\displaystyle E_{\text{p,m}}=\frac{1}{2}L{I}^2.}$$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.

Energy is also stored in a magnetic field itself. The energy per unit volume${\displaystyle u}$ in a region of free space withvacuum 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 isparamagnetic ordiamagnetic so that a linear constitutive equation exists that relates${\displaystyle \mathbf{B}}$ and themagnetization${\displaystyle \mathbf{H}}$ (for example${\displaystyle \mathbf{H}=\mathbf{B}/\mu }$ where${\displaystyle \mu }$ is themagnetic permeability of the material), then it can be shown that the magnetic field stores an energy of$${\displaystyle E=\frac{1}{2}\int \mathbf{H}\cdot \mathbf{B}\mathrm{d}V}$$where the integral is evaluated over the entire region where the magnetic field exists.^[2]

For amagnetostatic 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:^[2]$${\displaystyle E=\frac{1}{2}\int \mathbf{J}\cdot \mathbf{A}\mathrm{d}V}$$where${\displaystyle \mathbf{J}}$ is the current density field and${\displaystyle \mathbf{A}}$ is themagnetic vector potential. This is analogous to theelectrostatic 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.^[3]

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

• Forms of energy
• Magnetism
• Electromagnetic quantities

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