Editing Electromagnetic field (section)

== Mathematical description ==
{{main|Mathematical descriptions of the electromagnetic field}}
There are different mathematical ways of representing the electromagnetic field. The first one views the electric and magnetic fields as three-dimensional [[vector field]]s. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as {{math|'''E'''(''x'', ''y'', ''z'', ''t'')}} ([[electric field]]) and {{math|'''B'''(''x'', ''y'', ''z'', ''t'')}} ([[magnetic field]]).

If only the electric field ({{math|'''E'''}}) is non-zero, and is constant in time, the field is said to be an [[electrostatic field]]. Similarly, if only the magnetic field ({{math|'''B'''}}) is non-zero and is constant in time, the field is said to be a [[magnetostatic field]]. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using [[Maxwell's equations]].{{sfnp|ps=|Wangsness|1986|loc=Intermediate-level textbook}}

With the advent of [[special relativity]], physical laws became amenable to the formalism of [[tensor]]s. Maxwell's equations can be written in tensor form, generally viewed by physicists as a more elegant means of expressing physical laws.

The behavior of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or [[electrodynamics]] (electromagnetic fields), is governed by Maxwell's equations. In the vector field formalism, these are:

; [[Gauss's law]] : <math>\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}</math>
; [[Gauss's law for magnetism]] : <math>\nabla \cdot \mathbf{B} = 0</math>
; [[Faraday's law of induction#Maxwell–Faraday equation|Faraday's law]] : <math>\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}</math>
; [[Ampère's circuital law#Extending the original law: the Ampère–Maxwell equation|Ampère–Maxwell law]] : <math>\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}</math>
where <math>\rho</math> is the charge density, which is a function of time and position, <math>\varepsilon_0</math> is the [[vacuum permittivity]], <math>\mu_0</math> is the [[vacuum permeability]], and {{math|'''J'''}} is the current density vector, also a function of time and position. Inside a linear material, Maxwell's equations change by switching the permeability and permittivity of free space with the permeability and permittivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these terms are often represented by complex numbers, or tensors.

The [[Lorentz force law]] governs the interaction of the electromagnetic field with charged matter.

When a field travels across to different media, the behavior of the field changes according to the properties of the media.{{sfnp|ps=|Edminister|1995|loc= Examples and practice problems}}