Magnetic energy

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 $${\displaystyle U_{\text{m}}={\frac {1}{2}}\int \mathbf {H} \cdot \mathbf {B} \,\mathrm {d} V}$$ 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 noting that ${\displaystyle \nabla \times {\vec {H}}={\vec {J}}}$ and ${\displaystyle {\vec {B}}=\nabla \times {\vec {A}}}$, or by imagining the process of linearly turning on the currents and their generated magnetic field, arriving at a total energy of:^[2] $${\displaystyle U_{\text{m}}={\frac {1}{2}}\int \mathbf {J} \cdot \mathbf {A} \,\mathrm {d} V}$$ 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]

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