Magnetic field

In physics, a magnetic field (commonly denoted by H) describes a magnetic field (a vector) at every point in space; it is a vector field. In non-relativistic physics, the space in question is the three-dimensional Euclidean space $\scriptstyle \mathbb {E} ^{3}$—the infinite world that we live in.

In general H is seen as an auxiliary field useful when a magnetizable medium is present. The magnetic flux density B is usually seen as the fundamental magnetic field, see the article about B for more details about magnetism.

The SI unit of magnetic field strength is ampere⋅turn/meter; a unit that is based on the magnetic field of a solenoid. In the Gaussian system of units |H| has the unit oersted, with one oersted being equivalent to (1000/4π)⋅A⋅turn/m.

Relation between H and B

The magnetic field H is closely related to the magnetic induction B (also a vector field). It is the vector B that enters the expression for magnetic force on moving charges (Lorentz force). Historically, the theory of magnetism developed from Coulomb's law, where H played a pivotal role and B was an auxiliary field, which explains its historic name "magnetic induction". At present the roles have swapped and some authors give B the name magnetic field (and do not give a name to H other than "auxiliary field").

In the general case, H is introduced in terms of B as:

${\begin{aligned}\mathbf {H} &={\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} \qquad &{\hbox{in SI units}}\\\mathbf {H} &=\mathbf {B} -4\pi \mathbf {M} \qquad \;\;&{\hbox{in Gaussian units}},\\\end{aligned}}$

with M(r, t) the magnetization of the medium.

For the most common case of linear materials, M is linear in H, and in SI units,

$\mathbf {B} =\mu _{0}(\mathbf {1} +{\boldsymbol {\chi }})\mathbf {H} ,$

where 1 is the 3×3 unit matrix, χ the magnetic susceptibility tensor of the magnetizable medium, and μ₀ the magnetic permeability of the vacuum (also known as magnetic constant). In Gaussian units the relation is

$\mathbf {B} =(\mathbf {1} +4\pi {\boldsymbol {\chi }})\mathbf {H} ,$

Many non-ferromagnetic materials are linear and isotropic; in the isotropic case the susceptibility tensor is equal to χ_m1, and H can easily be solved (in SI units)

$\mathbf {H} ={\frac {\mathbf {B} }{\mu _{0}(1+\chi _{m})}}\equiv {\frac {\mathbf {B} }{\mu _{0}\mu _{r}}},$

with the relative magnetic permeability μ_r = 1 + χ_m.

For example, at standard temperature and pressure (STP) air, a mixture of paramagnetic oxygen and diamagnetic nitrogen, is paramagnetic (i.e., has positive χ_m), the χ_m of air is 4⋅10⁻⁷. Argon at STP is diamagnetic with χ_m = −1⋅10⁻⁸. For most ferromagnetic materials χ_m depends on H, with a non-linear relation between H and B and is large (depending on the material) from, say, 50 to 10000 and strongly varying as a function of H.

The magnetic flux density B is a solenoidal (divergence-free, transverse) vector field because of one of Maxwell's equations

${\boldsymbol {\nabla }}\cdot \mathbf {B} =0\ ,$

This equation denies the existence of magnetic monopoles (magnetic charges) and hence also of magnetic currents.

The magnetic field H is not necessarily solenoidal, however, because it satisfies:

${\boldsymbol {\nabla }}\cdot \mathbf {H} =-{\boldsymbol {\nabla }}\cdot \mathbf {M} \ ,$

which need not be zero.