Magnetic field

In physics, a magnetic field (commonly denoted by H) is a magnetic force (a vector) defined for every point in space.

In non-relativistic physics, space is the three-dimensional Euclidean space $\scriptstyle \mathbb {E} ^{3}$—the infinite (Newtonian) world that we live in. The vector field H may be pictured as a set of arrows, one arrow for each point of space. In this picture an arrow represent a magnetic 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.

The physical source of the magnetic force is 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). 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 Amperes/meter (SI units) or oersted (Gaussian units); one oersted equals 1000/4π A/m.

Mathematical description

The magnetic field H is in general a function of position. When we choose a Cartesian coordinate system for $\scriptstyle \mathbb {E} ^{3}$, a point P has coordinates x, y and z, and H is a vector function H(x,y,z), i.e.,

$\mathbf {H} (x,y,z)={\Big (}H_{x}(x,y,z),H_{y}(x,y,z),H_{z}(x,y,z){\Big )}\;{\hbox{and}}\;|\mathbf {H} (x,y,z)|\equiv {\sqrt {H_{x}^{2}+H_{y}^{2}+H_{z}^{2}}},$

where |H(x,y,z)| is the strength (also known as intensity) of the field at (x,y,z). If the vector H does not depend on position, the field is homogeneous.

Indicating unit vectors along the Cartesian coordinate axes by e_x, e_y, e_z, and the origin of coordinate system by O, we may equivalently write

${\overrightarrow {OP}}\equiv \mathbf {r} =\mathbf {e} _{x}x+\mathbf {e} _{y}y+\mathbf {e} _{z}z\quad \mathrm {and} \quad \mathbf {H} =\mathbf {e} _{x}H_{x}(x,y,z)+\mathbf {e} _{y}H_{y}(x,y,z)+\mathbf {e} _{z}H_{z}(x,y,z).$

This notation makes clear how rotation of the coordinate system affects r and H, and in particular it shows that both vectors obey the same rotation rule.