Magnetic field

In physics, a magnetic field 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 vector field may be pictured as a set of arrows pointing from all points in space. The arrows represent magnetic force vectors. As for any vector, the magnetic force is defined by its length (the strength of the magnetic field at the point where the arrow is located) and by its direction.

The magnetic field is homogeneous if all vectors are parallel and of the same length.

The physical cause (source) of the magnetic force is the presence of one or more permanents magnets and/or the presence of one or more electric currents (see Biot-Savart's law). In general the strength of the magnetic field decreases with a power of the inverse of the distance to the source.

Mathematical description

The magnetic field, usually designated by 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.