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: $E_{\text{p,m}}=-\mathbf {m} \cdot \mathbf {B}$ The mechanical work takes the form of a torque ${\boldsymbol {N}}$ : $\mathbf {N} =\mathbf {m} \times \mathbf {B} =-\mathbf {r} \times \mathbf {\nabla } E_{\text{p,m}}$ 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 $L$ ) when a current $I$ flows through it is given by: $E_{\text{p,m}}={\frac {1}{2}}LI^{2}.$ 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 $u$ in a region of free space with vacuum 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 $\mathbf {H}$ (for example $\mathbf {H} =\mathbf {B} /\mu$ where $\mu$ is the magnetic permeability of the material), 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.^[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] $E={\frac {1}{2}}\int \mathbf {J} \cdot \mathbf {A} \,\mathrm {d} V$ 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.^[3]

References

^ Griffiths, David J. (2023). Introduction to electrodynamics (Fifth ed.). New York: Cambridge University Press. ISBN 978-1-009-39773-5.
^ ^a ^b 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".

External links

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

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[1] Griffiths, David J. (2023). Introduction to electrodynamics (Fifth ed.). New York: Cambridge University Press. ISBN 978-1-009-39773-5.

[Jackson-2] Jackson, John David (1998). Classical Electrodynamics (3 ed.). New York: Wiley. pp. 212–onwards.

[3] "The Feynman Lectures on Physics, Volume II, Chapter 15: The vector potential".

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