Physics:Magnetic energy
Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or magnetic moment [math]\displaystyle{ \mathbf{m} }[/math] in a magnetic field [math]\displaystyle{ \mathbf{B} }[/math] 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: [math]\displaystyle{ E_\text{p,m} = -\mathbf{m} \cdot \mathbf{B} }[/math] while the energy stored in an inductor (of inductance [math]\displaystyle{ L }[/math]) when a current [math]\displaystyle{ I }[/math] flows through it is given by: [math]\displaystyle{ E_\text{p,m} = \frac{1}{2} LI^2. }[/math] 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 [math]\displaystyle{ \mu _0 }[/math] containing magnetic field [math]\displaystyle{ \mathbf{B} }[/math] is: [math]\displaystyle{ u = \frac{1}{2} \frac{B^2}{\mu_0} }[/math]
More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates [math]\displaystyle{ \mathbf{B} }[/math] and the magnetization [math]\displaystyle{ \mathbf{H} }[/math], then it can be shown that the magnetic field stores an energy of [math]\displaystyle{ E = \frac{1}{2} \int \mathbf{H} \cdot \mathbf{B} \, \mathrm{d}V }[/math] 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] [math]\displaystyle{ E = \frac{1}{2} \int \mathbf{J} \cdot \mathbf{A}\, \mathrm{d}V }[/math] where [math]\displaystyle{ \mathbf{J} }[/math] is the current density field and [math]\displaystyle{ \mathbf{A} }[/math] is the magnetic vector potential. This is analogous to the electrostatic energy expression [math]\displaystyle{ \frac{1}{2}\int \rho \phi \, \mathrm{d}V }[/math]; note that neither of these static expressions apply in the case of time-varying charge or current distributions.[2]
References
- ↑ 1.0 1.1 Jackson, John David (1998). Classical Electrodynamics (3 ed.). New York: Wiley. pp. 212–onwards.
- ↑ "The Feynman Lectures on Physics, Volume II, Chapter 15: The vector potential". https://feynmanlectures.caltech.edu/II_15.html.
External links
- Magnetic Energy, Richard Fitzpatrick Professor of Physics The University of Texas at Austin.
 |  |
