Editing Maxwell's equations (section)

{{short description|Equations describing classical electromagnetism}}
{{About||thermodynamic relations|Maxwell relations}}
{{Electromagnetism|cTopic=Electrodynamics}}
[[File:James Clerk Maxwell Statue Equations.jpg|thumb|Maxwell's equations on a plaque on his statue in Edinburgh]]
'''Maxwell's equations''', or '''Maxwell–Heaviside equations''', are a set of coupled [[partial differential equation]]s that, together with the [[Lorentz force]] law, form the foundation of [[classical electromagnetism]], classical [[optics]], [[Electrical network|electric]] and [[Magnetic circuit|magnetic]] circuits. 
The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, [[wireless]] communication, lenses, radar, etc. They describe how [[electric field|electric]] and [[magnetic field]]s are generated by [[electric charge|charges]], [[electric current|currents]], and changes of the fields.<ref group="note">''Electric'' and ''magnetic'' fields, according to the [[theory of relativity]], are the components of a single electromagnetic field.</ref> The equations are named after the physicist and mathematician [[James Clerk Maxwell]], who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to [[Oliver Heaviside]].<ref name="Hampshire">{{cite journal |title=A derivation of Maxwell's equations using the Heaviside notation |first1=Damian P. |last1=Hampshire |date=29 October 2018 |doi=10.1098/rsta.2017.0447 |volume=376 |issue=2134 |series= |issn=1364-503X |journal= Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|pmid=30373937 |pmc=6232579 |arxiv=1510.04309 |bibcode=2018RSPTA.37670447H }}</ref>

Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum, ''[[Speed of light|c]]'' ({{physconst|c|after=&nbsp;m/s|unit=no}}). Known as [[electromagnetic radiation]], these waves occur at various wavelengths to produce a [[Electromagnetic spectrum|spectrum]] of radiation from [[radio wave]]s to [[gamma ray]]s.

In [[partial differential equation]] form and a [[coherent system of units]], Maxwell's microscopic equations can be written as (top to bottom: Gauss's law, Gauss's law for magnetism, Faraday's law, Ampère-Maxwell law)
<math display="block">\begin{align}
\nabla \cdot \mathbf{E} \,\,\, &= \frac{\rho}{\varepsilon_0} \\
\nabla \cdot \mathbf{B} \,\,\, &= 0 \\
\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \times \mathbf{B} &= \mu_0 \left(\mathbf{J} + \varepsilon_0  \frac{\partial \mathbf{E}}{\partial t} \right)
\end{align}</math>
With <math>\mathbf{E}</math> the electric field, <math>\mathbf{B}</math> the magnetic field, <math>\rho</math> the [[electric charge density]] and <math>\mathbf{J}</math> the [[current density]]. <math>\varepsilon_0</math> is the [[vacuum permittivity]] and <math>\mu_0</math> the [[vacuum permeability]].

The equations have two major variants:
* The ''microscopic'' equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the [[atomic scale]].
* The ''macroscopic'' equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic-scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials.

The term "Maxwell's equations" is often also used for [[#Alternative formulations|equivalent alternative formulations]]. Versions of Maxwell's equations based on the [[electric potential|electric]] and [[magnetic scalar potential]]s are preferred for explicitly solving the equations as a [[boundary value problem]], [[Lorenz force#Lorentz force and analytical mechanics|analytical mechanics]], or for use in [[quantum mechanics]]. The [[Covariant formulation of classical electromagnetism|covariant formulation]] (on [[spacetime]] rather than space and time separately) makes the compatibility of Maxwell's equations with [[special relativity]] [[manifest covariance|manifest]]. [[Maxwell's equations in curved spacetime]], commonly used in [[Particle physics|high-energy]] and [[gravitational physics]], are compatible with [[general relativity]].<ref group="note">In general relativity, however, they must enter, through its [[stress–energy tensor]], into [[Einstein field equations]] that include the spacetime curvature.</ref> In fact, [[Albert Einstein]] developed special and general relativity to accommodate the invariant speed of light, a consequence of  Maxwell's equations, with the principle that only relative movement has physical consequences.

The publication of the equations marked the [[Unification (physics)|unification]] of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation.
Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a [[classical field theory|classical]] limit of the more precise theory of [[quantum electrodynamics]].

{{TOC limit|4}}