Magnetic field

In physics, a magnetic field (commonly denoted by H) yields a magnetic force (a vector) for 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 (Newtonian) world that we live in. The magnetic force can act on a permanent magnet (which is a magnetic dipole or—approximately—two magnetic monopoles) or on moving electric charges (through the Lorentz force).

The physical source of a magnetic field can be

• one or more permanent magnets (see Coulomb's law)
• one or more electric currents (see Biot-Savart's law),
• time-dependent electric fields (displacement currents),

or combinations of the three.

The SI unit of magnetic field strength is ampere⋅turn/meter; see solenoid for the origin of this unit. In the Gaussian system of units it is the oersted, with one oersted being equivalent to 1000/4π A⋅turn/m.

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

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). 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, 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., 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).

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.