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Summary

Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or magnetic moment $\mathbf {m}$ in a magnetic field $\mathbf {B}$ 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:

$E_{\rm {p,m}}=-\mathbf {m} \cdot \mathbf {B}$ while the energy stored in an inductor (of inductance $L$ ) when a current $I$ flows through it is given by:

$E_{\rm {p,m}}={\frac {1}{2}}LI^{2}.$ 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 $\mu _{0}$ containing magnetic field $\mathbf {B}$ is:

$u={\frac {1}{2}}{\frac {B^{2}}{\mu _{0}}}$ More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates $\mathbf {B}$ and $\mathbf {H}$ , then it can be shown that the magnetic field stores an energy of

$E={\frac {1}{2}}\int \mathbf {H} \cdot \mathbf {B} \ \mathrm {d} V$ where the integral is evaluated over the entire region where the magnetic field exists.

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:

$E={\frac {1}{2}}\int \mathbf {J} \cdot \mathbf {A} \ \mathrm {d} V$ where $\mathbf {J}$ is the current density field and $\mathbf {A}$ 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.

References

1. ^ a b Jackson, John David (1998). Classical Electrodynamics (3 ed.). New York: Wiley. pp. 212–onwards.
2. ^ "The Feynman Lectures on Physics, Volume II, Chapter 15: The vector potential".