Magnetic energy

Energy from the work of a magnetic force

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

E p,m = m B {\displaystyle E_{\text{p,m}}=-\mathbf {m} \cdot \mathbf {B} }
while the energy stored in an inductor (of inductance L {\displaystyle L} ) when a current I {\displaystyle I} flows through it is given by:
E p,m = 1 2 L I 2 . {\displaystyle 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 μ 0 {\displaystyle \mu _{0}} containing magnetic field B {\displaystyle \mathbf {B} } is:

u = 1 2 B 2 μ 0 {\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 B {\displaystyle \mathbf {B} } and the magnetization H {\displaystyle \mathbf {H} } , then it can be shown that the magnetic field stores an energy of

E = 1 2 H B d V {\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.[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]

E = 1 2 J A d V {\displaystyle E={\frac {1}{2}}\int \mathbf {J} \cdot \mathbf {A} \,\mathrm {d} V}
where J {\displaystyle \mathbf {J} } is the current density field and A {\displaystyle \mathbf {A} } is the magnetic vector potential. This is analogous to the electrostatic energy expression 1 2 ρ ϕ d V {\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

  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".

External links

  • Magnetic Energy, Richard Fitzpatrick Professor of Physics The University of Texas at Austin.
  • v
  • t
  • e
Energy
Fundamental
conceptsTypesEnergy carriersPrimary energyEnergy system
componentsUse and
supplyMisc.
  • Category
  • Commons
  • Portal
  • WikiProject


Stub icon

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

  • v
  • t
  • e
Stub icon

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

  • v
  • t
  • e