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Concept
Mathematical descriptions of the electromagnetic field
Summary
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.
Classical electromagnetism
The most common description of the electromagnetic field uses two three-dimensional vector fields called the electric field and the magnetic field. 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 E(x, y, z, t) (electric field) and B(x, y, z, t) (magnetic field).
If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field (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.
Maxwell's equations
The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed by Maxwell-Heaviside's equations:
{| class="toccolours collapsible" width="400px" style="background-color:#ECFCF4; padding:6; cellpadding=6;text-align:left;border:2px solid #50C878"
|-
|text-align="center" colspan="2"|Maxwell's equations (vector fields)
|-
| || Gauss's law
|-
| || Gauss's law for magnetism
|-
| || Faraday's law
|-
| || Ampère–Maxwell law
|}
where ρ is the charge density, which can (and often does) depend on time and position, ε0 is the electric constant, μ0 is the magnetic constant, and J is the current per unit area, also a function of time and position. The equations take this form with the International System of Quantities.
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