Magnetic energy

This article may need to be rewritten to comply with Wikipedia's quality standards. You can help. The talk page may contain suggestions. (March 2023)

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]

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

This article about energy, its collection, its distribution, or its uses is a stub. You can help Wikipedia by expanding it.

• v
• t
• e

This electromagnetism-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e