Magnetic field

In physics, a magnetic field (commonly denoted by H) is proportional to a magnetic force (a vector) defined 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 physical source of a magnetic field can be the presence of

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

(or combinations of the three).

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 dimension of the magnetic field is ampere⋅turn/meter (SI units) or oersted (Gaussian units); one oersted is equivalent to 1000/4π A⋅turn/m.

In modern texts on electricity and magnetism, the vector H is seen as the magnetic analogue of the electric displacement D. In older texts, in which one introduces Coulomb's law for magnetic poles, one finds the emphasis on the analogy of H and the electric field E. Since magnetic poles do not occur in nature this analogy is not stressed very often anymore.

The magnetic field H is closely related to the magnetic induction B (also a vector field). The relation in SI units is

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

where 1 is the 3×3 unit matrix, χ the magnetic susceptibility tensor, and μ₀ the magnetic permeability of the vacuum (magnetic constant). Most non-ferromagnetic materials are linear and isotropic; in that case the latter tensor is equal to χ_m1, and H can easily be solved

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

with the relative magnetic permeability μ_r = 1 + χ_m.

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.