Magnetic energy

The potentialmagnetic energyof amagnetormagnetic moment$\mathbf {m}$in amagnetic field$\mathbf {B}$is defined as themechanical workof the magnetic force on the re-alignment of the vector of themagnetic dipole momentand 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.

In anelectronic circuitthe energy stored in aninductor(ofinductance$L$) 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 withvacuum 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 isparamagneticordiamagneticso that a linear constitutive equation exists that relates$\mathbf {B}$and themagnetization$\mathbf {H}$(for example$\mathbf {H} =\mathbf {B} /\mu$where$\mu$is themagnetic permeabilityof 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.

For amagnetostaticsystem 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: $$E={\frac {1}{2}}\int \mathbf {J} \cdot \mathbf {A} \,\mathrm {d} V$$ where$\mathbf {J}$is the current density field and$\mathbf {A}$is themagnetic vector potential.This is analogous to theelectrostatic energyexpression${\textstyle {\frac {1}{2}}\int \rho \phi \,\mathrm {d} V}$;note that neither of these static expressions apply in the case of time-varying charge or current distributions.

External links

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