# Magnetic field

A vector-valued function of space-time variables describing a force acting on a moving electrical charge and proportional to the charge velocity in the following sense. The total force $\overset{\rightharpoonup} { F }$ which acts on a charge $q$ moving with velocity $\overset{\rightharpoonup }{ v }$ is given by the Lorentz force law

\begin{equation*} \vec { F } = q ( \vec { E } + \vec { v } \times \vec { B } ), \end{equation*}

where $\overset{\rightharpoonup }{ E }$ is the electric intensity field. The vector $\overset{\rightharpoonup} { B }$ is called the magnetic field (see, e.g., [a1], Sect. 1). However, quite often $\overset{\rightharpoonup} { B }$ is considered as a vector of magnetic induction (see, e.g., [a2], Sect. 29), while the vector $\overset{\rightharpoonup }{ H }$ related to $\overset{\rightharpoonup} { B }$ by the material equation $\overset{\rightharpoonup} { B } = \mu \overset{\rightharpoonup}{ H }$ is named magnetic intensity field or simply magnetic field. Here, $\mu$ denotes the material permeability. Together with the electric intensity field $\overset{\rightharpoonup }{ E }$, the vector $\overset{\rightharpoonup} { B }$ satisfies the Maxwell equations.

#### References

[a1] | R. Feynman, R. Leighton, M. Sands, "The Feynman lectures on physics" , 2 , Addison-Wesley (1964) |

[a2] | L.D. Landau, Ye.M. Lifshits, "Course of theoretical physics: Electrodynamics of continuous media" , VIII , Nauka (1992) (In Russian) (English transl.: Pergamon) |

**How to Cite This Entry:**

Magnetic field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Magnetic_field&oldid=50424