Magnetic energy

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The potential magnetic energy of a magnet or magnetic moment ${\displaystyle \mathbf {m} }$ in a magnetic field ${\displaystyle \mathbf {B} }$ 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: $${\displaystyle E_{\text{p,m}}=-\mathbf {m} \cdot \mathbf {B} }$$The mechanical work takes the form of a torque ${\displaystyle {\boldsymbol {N}}}$:$${\displaystyle \mathbf {N} =\mathbf {m} \times \mathbf {B} =-\mathbf {r} \times \mathbf {\nabla } E_{\text{p,m}}}$$ which will act to "realign" the magnetic dipole with the magnetic field.^[1]

In an electronic circuit the energy stored in an inductor (of inductance ${\displaystyle L}$) when a current ${\displaystyle I}$ flows through it is given by:$${\displaystyle E_{\text{p,m}}={\frac {1}{2}}LI^{2}.}$$ 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 ${\displaystyle u}$ 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 $${\displaystyle E={\frac {1}{2}}\int \mathbf {H} \cdot \mathbf {B} \,\mathrm {d} V}$$ where the integral is evaluated over the entire region where the magnetic field exists.^[2]

For 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:^[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 the magnetic vector potential. This is analogous to the electrostatic energy expression ${\textstyle {\frac {1}{2}}\int \rho \phi \,\mathrm {d} V}$; note that neither of these static expressions apply in the case of time-varying charge or current distributions.^[3]