Magnetic energy

The potential magnetic 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 on the re-alignment of the vector of the magnetic dipole moment and is equal to:

$$E_{\text{p,m}}=-\mathbf {m} \cdot \mathbf {B}$$

while the energy stored in an inductor (of inductance $L$) when a current $I$ flows through it is given by:

$$E_{\text{p,m}}={\frac {1}{2}}LI^{2}.$$

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 the magnetization $\mathbf {H}$, 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.

External links

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