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Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential 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 (actually magnetic torque) on the re-alignment of the vector of the magnetic dipole moment and is equal to:

while the energy stored in an inductor (of inductance ${\displaystyle L}$) when a current ${\displaystyle I}$ flows through it is given by: This second expression forms the basis for superconducting magnetic energy storage.

Energy is also stored in a magnetic field. The energy per unit volume in a region of space of permeability ${\displaystyle \mu _{0}}$ containing magnetic field ${\displaystyle \mathbf {B} }$ is:

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 ${\displaystyle \mathbf {H} }$, then it can be shown that the magnetic field stores an energy of

where the integral is evaluated over the entire region where the magnetic field exists. ^[1]

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: ^[1]

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. ^[2]

References

External links

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