Magnetic field

In physics, a magnetic field (commonly denoted by H) describes a magnetic force (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.

In general the strength of a magnetic field decreases as a low power of 1/R, the inverse of the distance R of the field point to the source.

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 gives the 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").

The relation between B and H is for the most common case of linear materials 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

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

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

Vector field

As any vector field, H may be pictured as a set of arrows, one arrow for each point of space. In this picture an arrow represents a magnetic force (or rather B, proportional to H, is the force). As for any vector, the magnetic force is defined by its length (the strength of the magnetic field) and by its direction.

A magnetic field is called homogeneous if all vectors are parallel and of the same length. If the vectors vary from point to point in length or direction, the field is called non-homogeneous.

The vectors may be time-dependent, i.e., the length and direction of the vectors may change as a function of time; in that case H is said to be a time-dependent field.

Both magnetic fields, H and B, are solenoidal (divergence-free, transverse) because of one of Maxwell's equations

${\boldsymbol {\nabla }}\cdot \mathbf {H} ={\boldsymbol {\nabla }}\cdot \mathbf {B} =0.$