View source for Maxwell's equations

{{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]]'' ({{val|299792458|u=m/s}}).<ref name=NIST>{{cite web | url =https://physics.nist.gov/cgi-bin/cuu/Value?c | title =The NIST Reference on Constants, Units, and Uncertainty}}</ref> 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 [[SI unit]]s, Maxwell's microscopic equations can be written as 
<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}}

==History of the equations==
{{main|History of Maxwell's equations}}

==Conceptual descriptions==
===Gauss's law===
{{Main|Gauss's law}}
[[File:VFPt charges plus minus thumb.svg|thumb|upright=0.5|Electric field from positive to negative charges]]
[[Gauss's law]] describes the relationship between an [[electric field]] and [[electric charge]]s: an electric field points away from positive charges and towards negative charges, and the net [[electric flux|outflow]] of the electric field through a [[Gaussian surface|closed surface]] is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the [[vacuum permittivity|permittivity of free space]].

===Gauss's law for magnetism===
{{Main|Gauss's law for magnetism}}
[[Image:VFPt dipole magnetic1.svg|right|thumb|[[Gauss's law for magnetism]]: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.]]

[[Gauss's law for magnetism]] states that electric charges have no magnetic analogues, called [[magnetic monopole]]s; no north or south magnetic poles exist in isolation.<ref name=VideoGlossary>{{cite web | url =http://videoglossary.lbl.gov/#n45 | title =Maxwell's equations | last =Jackson | first =John | website =Science Video Glossary | publisher =Berkeley Lab | access-date =2016-06-04 | archive-date =2019-01-29 | archive-url =https://web.archive.org/web/20190129113142/https://videoglossary.lbl.gov/#n45 | url-status =dead }}</ref> Instead, the magnetic field of a material is attributed to a [[dipole]], and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total [[magnetic flux]] through a Gaussian surface is zero, and the magnetic field is a [[solenoidal vector field]].<ref group="note">The absence of sinks/sources of the field does not imply that the field lines must be closed or escape to infinity. They can also wrap around indefinitely, without self-intersections. Moreover, around points where the field is zero (that cannot be intersected by field lines, because their direction would not be defined), there can be the simultaneous begin of some lines and end of other lines. This happens, for instance, in the middle between two identical cylindrical magnets, whose north poles face each other. In the middle between those magnets, the field is zero and the axial field lines coming from the magnets end. At the same time, an infinite number of divergent lines emanate radially from this point. The simultaneous presence of lines which end and begin around the point preserves the divergence-free character of the field. For a detailed discussion of non-closed field lines, see L.&nbsp;Zilberti [https://zenodo.org/record/4518772#.YCJU_WhKjIU "The Misconception of Closed Magnetic Flux Lines"], IEEE Magnetics Letters, vol.&nbsp;8, art. 1306005, 2017.</ref>

===Faraday's law===
{{Main|Faraday's law of induction}}
[[File:Magnetosphere rendition.jpg|thumb|upright=1.45|left|In a [[geomagnetic storm]], solar wind plasma impacts [[Earth's magnetic field]] causing a time-dependent change in the field, thus inducing electric fields in Earth's atmosphere and conductive [[lithosphere]] which can destabilize [[power grid]]s. (Not to scale.)]]

The [[Faraday's law of induction#Maxwell–Faraday equation|Maxwell–Faraday]] version of [[Faraday's law of induction]] describes how a time-varying [[magnetic field]] corresponds to [[Curl (mathematics)|curl]] of an [[electric field]].<ref name="VideoGlossary" /> In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface.

The [[electromagnetic induction]] is the operating principle behind many [[electric generator]]s: for example, a rotating [[bar magnet]] creates a changing magnetic field and generates an electric field in a nearby wire.

===Ampère's law with Maxwell's addition===
{{Main|Ampère's circuital law}}
[[Image:Magnetic core.jpg|right|thumb|[[Magnetic-core memory]] (1954) is an application of [[Ampère's law]]. Each [[magnetic core|core]] stores one [[bit]] of data.]]

The original law of Ampère states that magnetic fields relate to [[electric current]]. [[Ampère's circuital law|Maxwell's addition]] states that magnetic fields also relate to changing electric fields, which Maxwell called [[displacement current]]. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.

Maxwell's addition to Ampère's law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields.<ref>J. D. Jackson, ''Classical Electrodynamics'', section 6.3</ref>{{clarify|date=May 2022}} As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field.<ref name="VideoGlossary" /><ref>[https://books.google.com/books?id=1DZz341Pp50C&pg=PA809 ''Principles of physics: a calculus-based text''], by R. A. Serway, J. W. Jewett, page 809.</ref> A further consequence is the existence of self-sustaining [[electromagnetic waves]] which [[electromagnetic wave equation|travel through empty space]].

The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents,<ref group="note">The quantity we would now call {{math|1/{{sqrt|''ε''{{sub|0}}''μ''{{sub|0}}}}}}, with units of velocity, was directly measured before Maxwell's equations, in an 1855 experiment by [[Wilhelm Eduard Weber]] and [[Rudolf Kohlrausch]]. They charged a [[leyden jar]] (a kind of [[capacitor]]), and measured the [[Coulomb's law|electrostatic force]] associated with the potential; then, they discharged it while measuring the [[Ampère's force law|magnetic force]] from the current in the discharge wire. Their result was {{val|3.107|e=8|ul=m/s}}, remarkably close to the speed of light. See Joseph F. Keithley, [https://books.google.com/books?id=uwgNAtqSHuQC&pg=PA115 ''The story of electrical and magnetic measurements: from 500 B.C. to the 1940s'', p.&nbsp;115].</ref> matches the [[speed of light]]; indeed, [[light]] ''is'' one form of [[electromagnetic radiation]] (as are [[X-ray]]s, [[radio wave]]s, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of [[electromagnetism]] and [[optics]].

==Formulation in terms of electric and magnetic fields (microscopic or in vacuum version)==
<!-- please do not change to Electromagnetic field: we want to (modestly) stress that in this formulation Electric and Magnetic fields play an intertwined but separate role -->
In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate [[Physical law|law of nature]], the [[Lorentz force]] law, describes how, conversely, the electric and magnetic fields act on charged particles and currents. A version of this law was included in the original equations by Maxwell but, by convention, is included no longer. The [[vector calculus]] formalism below, the work of [[Oliver Heaviside]],<ref>Bruce J. Hunt (1991) ''[[The Maxwellians]]'', chapter 5 and appendix, [[Cornell University Press]]</ref><ref>{{cite web|url=http://ethw.org/Maxwell's_Equations|title=Maxwell's Equations |date=29 October 2019 |publisher=Engineering and Technology History Wiki |access-date=2021-12-04}}</ref> has become standard. It is manifestly rotation invariant, and therefore mathematically much more transparent than Maxwell's original 20 equations in x,y,z components. The [[#Relativistic formulations|relativistic formulations]] are even more symmetric and manifestly Lorentz invariant. For the same equations expressed using tensor calculus or differential forms, see ''{{section link|#Alternative formulations}}''.

The differential and integral formulations are mathematically equivalent; both are useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely ''local'' and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using [[finite element analysis]].<ref>{{cite book |title=Partial differential equations and the finite element method |last=Šolín |first=Pavel |year=2006 |publisher=John Wiley and Sons |isbn=978-0-471-72070-6 |page=273 |url=https://books.google.com/books?id=-hIG3NZrnd8C&pg=PA273}}</ref>

===Key to the notation===
<!---please do not make this list much longer – we used to have a gigantic table of all the constants, variables, terminology, and units, which was converted into prose as it probably should for an encyclopedia (see the history in pre-2013). Editors (by all means in good faith) may add the units, alternative names and symbols, etc. to the list and make it longer and denser, then eventually there would be a good reason to resurrect the big table format again...--->

Symbols in '''bold''' represent [[Vector (geometric)|vector]] quantities, and symbols in ''italics'' represent [[scalar (physics)|scalar]] quantities, unless otherwise indicated.
The equations introduce the [[electric field]], {{math|'''E'''}}, a [[vector field]], and the [[magnetic field]], {{math|'''B'''}}, a [[pseudovector]] field, each generally having a time and location dependence.
The sources are
*the total electric [[charge density]] (total charge per unit volume), {{math|''ρ''}}, and
*the total electric [[current density]] (total current per unit area), {{math|'''J'''}}.

The [[universal constant]]s appearing in the equations (the first two ones explicitly only in the SI units formulation) are:
*the [[permittivity of free space]], {{math|''ε''<sub>0</sub>}}, and
*the [[permeability of free space]], {{math|''μ''<sub>0</sub>}}, and
*the [[speed of light]], <math>c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}</math>

====Differential equations====
In the differential equations, 
*the [[nabla symbol]], {{math|∇}}, denotes the three-dimensional [[gradient]] operator, [[del]],
*the {{math|∇⋅}} symbol (pronounced "del dot") denotes the [[divergence]] operator,
*the {{math|∇×}} symbol (pronounced "del cross") denotes the [[curl (mathematics)|curl]] operator.

====Integral equations====
In the integral equations, 
*{{math|Ω}} is any volume with closed [[boundary (topology)|boundary]] surface {{math|∂Ω}}, and
*{{math|Σ}} is any surface with closed boundary curve {{math|∂Σ}},
The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the [[differentiation under the integral sign]] in Faraday's law:
<math display="block"> \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} = \iint_{\Sigma}  \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{S}\,,</math>
Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss and Stokes formula appropriately. 
*{{oiint
   | intsubscpt=
   | integrand=}}<math>{\vphantom{\int}}_{\scriptstyle\partial \Omega}</math> is a [[surface integral]] over the boundary surface {{math|∂Ω}}, with the loop indicating the surface is closed
*<math>\iiint_\Omega</math> is a [[volume integral]] over the volume {{math|Ω}},
*<math>\oint_{\partial \Sigma}</math> is a [[line integral]] around the boundary curve {{math|∂Σ}}, with the loop indicating the curve is closed.
*<math>\iint_\Sigma</math> is a [[surface integral]] over the surface {{math|Σ}},
* The ''total'' [[electric charge]] {{math|''Q''}} enclosed in {{math|Ω}} is the [[volume integral]] over {{math|Ω}} of the [[charge density]] {{math|''ρ''}} (see the "macroscopic formulation" section below): <math display="block">Q = \iiint_\Omega \rho \ \mathrm{d}V,</math> where {{math|d''V''}} is the [[volume element]].
* The ''net'' [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} is the [[surface integral]] of the magnetic field {{math|'''B'''}} passing through a fixed surface, {{math|Σ}}: <math display="block">\Phi_B = \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d} \mathbf{S},</math>
* The ''net'' [[electric flux]] {{math|Φ<sub>''E''</sub>}} is the surface integral of the electric field {{math|'''E'''}} passing through {{math|Σ}}: <math display="block">\Phi_E = \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d} \mathbf{S},</math>
* The ''net'' [[electric current]] {{math|''I''}} is the surface integral of the [[electric current density]] {{math|'''J'''}} passing through {{math|Σ}}: <math display="block">I = \iint_{\Sigma} \mathbf{J} \cdot \mathrm{d} \mathbf{S},</math> where {{math|d'''S'''}} denotes the differential [[vector area|vector element]] of surface area {{math|''S''}}, [[Normal (geometry)|normal]] to surface {{math|Σ}}. (Vector area is sometimes denoted by {{math|'''A'''}} rather than {{math|'''S'''}}, but this conflicts with the notation for [[magnetic vector potential]]).

===Formulation in SI units convention===

{| class="wikitable"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | [[Integral]] equations 
! scope="col" | [[Partial differential equation|Differential]] equations
|-
|  [[Gauss's law]]
| {{oiint}}<math>\vphantom{\oint}_{\scriptstyle\partial \Omega}\mathbf{E}\cdot\mathrm{d}\mathbf{S} = \frac{1}{\varepsilon_0} \iiint_\Omega \rho \,\mathrm{d}V</math>
| <math>\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}</math>
|-
| [[Gauss's law for magnetism]]
| {{oiint}}<math>\vphantom{\oint}_{\scriptstyle \partial \Omega }\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0</math>
| <math>\nabla \cdot \mathbf{B} = 0</math>
|-
| Maxwell–Faraday equation ([[Faraday's law of induction]])
|<math>\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}  = - \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} </math>
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
|-
| [[Ampère's circuital law]] (with Maxwell's addition)
| <math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \left(\iint_{\Sigma} \mathbf{J} \cdot \mathrm{d}\mathbf{S} + \varepsilon_0 \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} \right) \\
\end{align}
</math>
| <math>\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) </math>
|}

===Formulation in Gaussian units convention===
{{main|Gaussian units}}

The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing [[dimensional analysis|dimensioned]] factors of {{math|''ε''<sub>0</sub>}} and {{math|''μ''<sub>0</sub>}} into the units of calculation, by convention. With a corresponding change in convention for the [[Lorentz force]] law this yields the same physics, i.e. trajectories of charged particles, or [[work (physics)|work]] done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the [[electromagnetic tensor]]: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension.<ref name=Jackson>{{cite book|author=J. D. Jackson|title=Classical Electrodynamics|edition=3rd|isbn=978-0-471-43132-9|date=1975-10-17|publisher=Wiley |url=https://archive.org/details/classicalelectro00jack_0}}</ref>{{rp|vii}}  Such modified definitions are conventionally used with the Gaussian ([[Centimetre gram second system of units#Alternate derivations of CGS units in electromagnetism|CGS]]) units. Using these definitions and conventions, colloquially "in Gaussian units",<ref name=Littlejohn>
{{cite web
 | url=http://bohr.physics.berkeley.edu/classes/221/0708/notes/emunits.pdf
 | title=Gaussian, SI and Other Systems of Units in Electromagnetic Theory
 | work=Physics 221A, University of California, Berkeley lecture notes
 | author=Littlejohn, Robert|author-link1=Robert Grayson Littlejohn
 | date=Fall 2007
 | access-date=2008-05-06
}}</ref>
the Maxwell equations become:<ref name=Griffiths>{{cite book
 | author=David J Griffiths
 | title=Introduction to electrodynamics
 | year=1999
 | edition=Third
 | pages=[https://archive.org/details/introductiontoel00grif_0/page/559 559–562]
 | publisher=Prentice Hall
 | isbn=978-0-13-805326-0
 | url=https://archive.org/details/introductiontoel00grif_0/page/559
 }}</ref>

{| class="wikitable"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | Integral equations
! scope="col" | Differential equations
|-
|[[Gauss's law]]
|{{oiint}}<math>\vphantom{\oint}_{\scriptstyle\partial \Omega }\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4\pi \iiint_\Omega \rho \,\mathrm{d}V</math>
| <math>\nabla \cdot \mathbf{E} = 4\pi\rho </math>
|-
|[[Gauss's law for magnetism]]
|{{oiint}}<math>\vphantom{\oint}_{\scriptstyle \partial \Omega }\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0</math>
| <math>\nabla \cdot \mathbf{B} = 0</math>
|-
| Maxwell–Faraday equation ([[Faraday's law of induction]])
| <math>\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = -\frac{1}{c}\frac{\mathrm{d}}{\mathrm{d}t}\iint_\Sigma \mathbf{B}\cdot\mathrm{d}\mathbf{S}</math>
| <math>\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}</math>
|-
|[[Ampère's circuital law]] (with Maxwell's addition)
| <math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B}\cdot\mathrm{d}\boldsymbol{\ell} = \frac{1}{c} \left( 4\pi \iint_\Sigma  \mathbf{J}\cdot\mathrm{d}\mathbf{S} + \frac{\mathrel{\mathrm{d}}}{\mathrm{d}t} \iint_\Sigma \mathbf{E}\cdot \mathrm{d}\mathbf{S}\right)
\end{align}
</math>
| <math>\nabla \times \mathbf{B} = \frac{1}{c}\left( 4\pi\mathbf{J} + \frac{\partial \mathbf{E}}{\partial t}\right)</math>
|}

The equations simplify slightly when a system of quantities is chosen in the speed of light, ''c'', is used for [[nondimensionalization]], so that, for example, seconds and lightseconds are interchangeable, and ''c'' = 1.

Further changes are possible by absorbing factors of {{math|4''π''}}. This process, called rationalization, affects whether [[Coulomb's law]] or [[Gauss's law]] includes such a factor (see ''[[Heaviside–Lorentz units]]'', used mainly in [[particle physics]]).

==Relationship between differential and integral formulations==
<!---PLEASE NOTE: This section on the "relation between int/diff forms" is independent of units and should not be made a subsection or merged with the above section on SI units&nbsp;— it should stay in its own section, yet as close as possible to the first mention of the equations. Thanks. --->
The equivalence of the differential and integral formulations are a consequence of the [[divergence theorem|Gauss divergence theorem]] and the [[Kelvin–Stokes theorem]].

===Flux and divergence===

[[File:Divergence theorem in EM.svg|thumb|Volume {{math|Ω}} and its closed boundary {{math|∂Ω}}, containing (respectively enclosing) a source {{math|(+)}} and sink {{math|(−)}} of a vector field {{math|'''F'''}}. Here, {{math|'''F'''}} could be the {{math|'''E'''}} field with source electric charges, but ''not'' the {{math|'''B'''}} field, which has no magnetic charges as shown. The outward [[unit normal]] is '''n'''.]]

According to the (purely mathematical) [[divergence theorem|Gauss divergence theorem]], the [[electric flux]] through the 
[[homology (mathematics)|boundary surface]] {{math|∂Ω}} can be rewritten as
:{{oiint}}<math>\vphantom{\oint}_{\scriptstyle\partial \Omega} \mathbf{E}\cdot\mathrm{d}\mathbf{S}=\iiint_{\Omega} \nabla\cdot\mathbf{E}\, \mathrm{d}V</math>
The integral version of Gauss's equation can thus be rewritten as 
<math display="block"> \iiint_{\Omega} \left(\nabla \cdot \mathbf{E} - \frac{\rho}{\varepsilon_0}\right) \, \mathrm{d}V = 0</math>
Since {{math|Ω}} is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied [[if and only if]] the integrand is zero everywhere. This is 
the differential equations formulation of Gauss equation up to a trivial rearrangement.

Similarly rewriting the [[magnetic flux]] in Gauss's law for magnetism in integral form gives  
:{{oiint}}<math>\vphantom{\oint}_{\scriptstyle\partial \Omega} \mathbf{B}\cdot\mathrm{d}\mathbf{S} = \iiint_{\Omega} \nabla \cdot \mathbf{B}\, \mathrm{d}V = 0.</math>
which is satisfied for all {{math|Ω}} if and only if <math> \nabla \cdot \mathbf{B} = 0</math> everywhere.

===Circulation and curl===
[[File:Curl theorem in EM.svg|thumb|Surface {{math|Σ}} with closed boundary {{math|∂Σ}}. {{math|'''F'''}} could be the {{math|'''E'''}} or {{math|'''B'''}} fields. Again, {{math|'''n'''}} is the [[unit normal]]. (The curl of a vector field does not literally look like the "circulations", this is a heuristic depiction.)]]

By the [[Stokes' theorem|Kelvin–Stokes theorem]] we can rewrite the [[line integral]]s of the fields around the closed boundary curve {{math|∂Σ}} to an integral of the "circulation of the fields" (i.e. their [[curl (mathematics)|curl]]s) over a surface it bounds, i.e.

<math display="block">\oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \iint_\Sigma (\nabla \times \mathbf{B}) \cdot \mathrm{d}\mathbf{S},</math>
Hence the modified Ampere law in integral form can be rewritten as 
<math display="block"> \iint_\Sigma  \left(\nabla \times \mathbf{B} - \mu_0 \left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)\right)\cdot \mathrm{d}\mathbf{S} = 0.</math>
Since {{math|Σ}} can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that the integrand is zero [[if and only if]] Ampere's modified law in differential equations form is satisfied.
The equivalence of Faraday's law in differential and integral form follows likewise.

The line integrals and curls are analogous to quantities in classical [[fluid dynamics]]: the [[circulation (fluid dynamics)|circulation]] of a fluid is the line integral of the fluid's [[flow velocity]] field around a closed loop, and the [[vorticity]] of the fluid is the curl of the velocity field.

==Charge conservation==
The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the modified Ampere's law has zero divergence by the [[Vector calculus identities#Divergence of curl is zero|div–curl identity]]. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives:

<math display="block">0 = \nabla\cdot (\nabla\times \mathbf{B}) = \nabla \cdot \left(\mu_0 \left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) \right) = \mu_0\left(\nabla\cdot \mathbf{J} + \varepsilon_0\frac{\partial}{\partial t}\nabla\cdot \mathbf{E}\right) = \mu_0\left(\nabla\cdot \mathbf{J} +\frac{\partial \rho}{\partial t}\right)</math>
i.e.,
<math display="block">\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0.</math>
By the Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary:
:{{oiint
| preintegral =<math>\frac{d}{dt}Q_\Omega = \frac{d}{dt} \iiint_{\Omega} \rho \mathrm{d}V = -</math>
| intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
| integrand = <math>\mathbf{J} \cdot {\rm d}\mathbf{S} =  - I_{\partial \Omega}.</math>
}}
In particular, in an isolated system the total charge is conserved.

==Vacuum equations, electromagnetic waves and speed of light==
{{Further|Electromagnetic wave equation|Inhomogeneous electromagnetic wave equation|Sinusoidal plane-wave solutions of the electromagnetic wave equation|Helmholtz equation}}

[[File:Electromagneticwave3D.gif|thumb|This 3D diagram shows a plane linearly polarized wave propagating from left to right, defined by {{math|1='''E''' = '''E'''<sub>0</sub> sin(−''ωt'' + '''k''' ⋅ '''r''')}} and {{math|1='''B''' = '''B'''<sub>0</sub> sin(−''ωt'' + '''k''' ⋅ '''r''')}} The oscillating fields are detected at the flashing point. The horizontal wavelength is ''λ''. {{math|1='''E'''<sub>0</sub> ⋅ '''B'''<sub>0</sub> = 0 = '''E'''<sub>0</sub> ⋅ '''k''' = '''B'''<sub>0</sub> ⋅ '''k'''}}]]

In a region with no charges ({{math|1=''ρ'' = 0}}) and no currents ({{math|1='''J''' = '''0'''}}), such as in a vacuum, Maxwell's equations reduce to:
<math display="block">\begin{align}
  \nabla \cdot \mathbf{E} &= 0, & \nabla \times \mathbf{E} &= -\frac{\partial\mathbf B}{\partial t}, \\
  \nabla \cdot \mathbf{B} &= 0, & \nabla \times \mathbf{B} &= \mu_0\varepsilon_0 \frac{\partial\mathbf E}{\partial t}.
\end{align}</math>

Taking the curl {{math|(∇×)}} of the curl equations, and using the [[Vector calculus identities#Curl of curl|curl of the curl identity]] we obtain

<math display="block">\begin{align}
  \mu_0\varepsilon_0  \frac{\partial^2 \mathbf{E}}{\partial t^2} - \nabla^2 \mathbf{E} = 0, \\
  \mu_0\varepsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} - \nabla^2 \mathbf{B} = 0.
\end{align}</math>

The quantity <math>\mu_0\varepsilon_0</math> has the dimension of (time/length)<sup>2</sup>. Defining
<math>c = (\mu_0 \varepsilon_0)^{-1/2}</math>, the equations above have the form of the standard [[wave equation]]s
<math display="block">\begin{align}
  \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} - \nabla^2 \mathbf{E} = 0, \\
  \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} - \nabla^2 \mathbf{B} = 0.
\end{align}</math>

Already during Maxwell's lifetime, it was found that the known values for <math>\varepsilon_0</math> and <math>\mu_0</math> give <math>c \approx 2.998 \times 10^8~\text{m/s}</math>, then already known to be the [[speed of light]] in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In the [[SI system|old SI system]] of units, the  values of <math>\mu_0 = 4\pi\times 10^{-7}</math> and <math>c = 299\,792\,458~\text{m/s}</math> are defined constants, (which means that by definition <math>\varepsilon_0 = 8.854... \times 10^{-12}~\text{F/m}</math>) that define the ampere and the metre. In the [[new SI]] system, only ''c'' keeps its defined value, and the electron charge gets a defined value.

In materials with [[relative permittivity]], {{math|''ε''<sub>r</sub>}}, and [[Permeability (electromagnetism)#Relative permeability and magnetic susceptibility|relative permeability]], {{math|''μ''<sub>r</sub>}}, the [[phase velocity]] of light becomes

<math display="block">v_\text{p} = \frac{1}\sqrt{\mu_0\mu_\text{r} \varepsilon_0\varepsilon_\text{r}},</math>

which is usually<ref group="note">There are cases ([[anomalous dispersion]]) where the phase velocity can exceed {{math|''c''}}, but the "signal velocity" will still be {{math|< ''c''}}</ref> less than {{math|''c''}}.

In addition, {{math|'''E'''}} and {{math|'''B'''}} are perpendicular to each other and to the direction of wave propagation, and are in [[phase (waves)|phase]] with each other. A [[sinusoidal]] plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through [[Faraday's law of induction|Faraday's law]]. In turn, that electric field creates a changing magnetic field through [[Ampère's circuital law|Maxwell's addition to Ampère's law]]. This perpetual cycle allows these waves, now known as [[electromagnetic radiation]], to move through space at velocity {{math|''c''}}.

==Macroscopic formulation==
The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping.

The microscopic version is sometimes called "Maxwell's equations in a vacuum": this refers to the fact that the material medium is not built into the structure of the equations, but appears only in the charge and current terms. The microscopic version was introduced by Lorentz, who tried to use it to derive the macroscopic properties of bulk matter from its microscopic constituents.<ref name="MiltonSchwinger2006">{{cite book|author1=Kimball Milton|author2=J. Schwinger|title=Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators|date=18 June 2006|publisher=Springer Science & Business Media|isbn=978-3-540-29306-4}}</ref>{{rp|5}}

"Maxwell's macroscopic equations", also known as '''Maxwell's equations in matter''', are more similar to those that Maxwell introduced himself.

{| class="wikitable" 
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | [[Integral]] equations<br/> (SI convention)
! scope="col" | [[Partial differential equation|Differential]] equations<br/> (SI convention)
! scope="col" | Differential equations<br/> (Gaussian convention)
|-
| Gauss's law
| {{oiint
   | intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
   | integrand  = <math>\mathbf{D}\cdot\mathrm{d}\mathbf{S} = \iiint_\Omega \rho_\text{f} \,\mathrm{d}V</math>
  }}
| <math>\nabla \cdot \mathbf{D} = \rho_\text{f}</math>
| <math> \nabla \cdot \mathbf{D} = 4\pi\rho_\text{f}</math>
|-
| Ampère's circuital law (with Maxwell's addition)
| <math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = \\
      & \iint_{\Sigma} \mathbf{J}_\text{f} \cdot \mathrm{d}\mathbf{S} + \frac{d}{dt} \iint_{\Sigma} \mathbf{D} \cdot \mathrm{d}\mathbf{S} \\
\end{align}
</math>
| <math>\nabla \times \mathbf{H} = \mathbf{J}_\text{f} + \frac{\partial \mathbf{D}} {\partial t}</math>
| <math> \nabla \times \mathbf{H} = \frac{1}{c} \left(4\pi\mathbf{J}_\text{f} + \frac{\partial \mathbf{D}} {\partial t} \right)</math>
|-
| Gauss's law for magnetism
| {{oiint
   | intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
   | integrand  = <math>\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0</math>
  }}
| <math>\nabla \cdot \mathbf{B} = 0</math>
| <math>\nabla \cdot \mathbf{B} = 0</math>
|-
| Maxwell–Faraday equation (Faraday's law of induction)
| <math>\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}  = - \frac{d}{dt} \iint_{\Sigma} \mathbf B \cdot \mathrm{d}\mathbf{S} </math>
|<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
|<math>\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}</math>
|-
|}

In the macroscopic equations, the influence of bound charge {{math|''Q''<sub>b</sub>}} and bound current {{math|''I''<sub>b</sub>}} is incorporated into the [[electric displacement field|displacement field]] {{math|'''D'''}} and the [[magnetizing field]] {{math|'''H'''}}, while the equations depend only on the free charges {{math|''Q''<sub>f</sub>}} and free currents {{math|''I''<sub>f</sub>}}. This reflects a splitting of the total electric charge ''Q'' and current ''I'' (and their densities {{mvar|ρ}} and '''J''') into free and bound parts:
<math display="block">\begin{align}
  Q &= Q_\text{f} + Q_\text{b} = \iiint_\Omega \left(\rho_\text{f} + \rho_\text{b} \right) \, \mathrm{d}V = \iiint_\Omega \rho \,\mathrm{d}V, \\
  I &= I_\text{f} + I_\text{b} = \iint_\Sigma \left(\mathbf{J}_\text{f} + \mathbf{J}_\text{b} \right) \cdot \mathrm{d}\mathbf{S} = \iint_\Sigma \mathbf{J} \cdot \mathrm{d}\mathbf{S}.
\end{align}</math>

The cost of this splitting is that the additional fields {{math|'''D'''}} and {{math|'''H'''}} need to be determined through phenomenological constituent equations relating these fields to the electric field {{math|'''E'''}} and the magnetic field {{math|'''B'''}}, together with the bound charge and current.

See below for a detailed description of the differences between the microscopic equations, dealing with ''total'' charge and current including material contributions, useful in air/vacuum;<ref group="note" name="Effective_charge">In some books—e.g., in U. Krey and A. Owen's Basic Theoretical Physics (Springer 2007)—the term ''effective charge'' is used instead of ''total charge'', while ''free charge'' is simply called ''charge''.</ref>
and the macroscopic equations, dealing with ''free'' charge and current, practical to use within materials.

===Bound charge and current===
{{Main|Current density|Polarization density#Polarization density in Maxwell's equations|Magnetization#Magnetization current|l2=Bound charge|l3=Bound current}}
[[File:Polarization and magnetization.svg|thumb|300px|''Left:'' A schematic view of how an assembly of microscopic dipoles produces opposite surface charges as shown at top and bottom. ''Right:'' How an assembly of microscopic current loops add together to produce a macroscopically circulating current loop. Inside the boundaries, the individual contributions tend to cancel, but at the boundaries no cancelation occurs.]]

When an electric field is applied to a [[dielectric|dielectric material]] its molecules respond by forming microscopic [[electric dipole]]s – their [[atomic nucleus|atomic nuclei]] move a tiny distance in the direction of the field, while their [[electron]]s move a tiny distance in the opposite direction. This produces a ''macroscopic'' ''bound charge'' in the material even though all of the charges involved are bound to individual molecules. For example, if every molecule responds the same, similar to that shown in the figure, these tiny movements of charge combine to produce a layer of positive [[Bound charge#Bound charge|bound charge]] on one side of the material and a layer of negative charge on the other side. The bound charge is most conveniently described in terms of the [[polarization density|polarization]] {{math|'''P'''}} of the material, its dipole moment per unit volume. If {{math|'''P'''}} is uniform, a macroscopic separation of charge is produced only at the surfaces where {{math|'''P'''}} enters and leaves the material. For non-uniform {{math|'''P'''}}, a charge is also produced in the bulk.<ref>See {{cite book|author=David J. Griffiths|title=Introduction to Electrodynamics|url=https://archive.org/details/introductiontoel00grif_0|url-access=registration|edition=third|section=4.2.2|publisher=[[Prentice Hall]]|year=1999|isbn=9780138053260|author-link=David J. Griffiths}} for a good description of how {{math|'''P'''}} relates to the [[Bound charge#Bound charge|bound charge]].</ref>

Somewhat similarly, in all materials the constituent atoms exhibit [[magnetic moment#Examples of magnetic moments|magnetic moments]] that are intrinsically linked to the [[gyromagnetic ratio|angular momentum]] of the components of the atoms, most notably their [[electron]]s. The [[magnetic field#Magnetic dipoles|connection to angular momentum]] suggests the picture of an assembly of microscopic current loops. Outside the material, an assembly of such microscopic current loops is not different from a macroscopic current circulating around the material's surface, despite the fact that no individual charge is traveling a large distance. These ''[[Bound current#Magnetization current|bound currents]]'' can be described using the [[magnetization]] {{math|'''M'''}}.<ref>See {{cite book|author=David J. Griffiths|title=Introduction to Electrodynamics|url=https://archive.org/details/introductiontoel00grif_0|url-access=registration|edition=third|section=6.2.2|publisher=[[Prentice Hall]]|year=1999|isbn=9780138053260}} for a good description of how {{math|'''M'''}} relates to the [[bound current]].</ref>

The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of {{math|'''P'''}} and {{math|'''M'''}}, which average these charges and currents on a sufficiently large scale so as not to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material. As such, ''Maxwell's macroscopic equations'' ignore many details on a fine scale that can be unimportant to understanding matters on a gross scale by calculating fields that are averaged over some suitable volume.

===Auxiliary fields, polarization and magnetization===
The ''[[List of electromagnetism equations#Definitions|definitions]]'' of the auxiliary fields are:

<math display="block">\begin{align}
  \mathbf{D}(\mathbf{r}, t) &= \varepsilon_0 \mathbf{E}(\mathbf{r}, t) + \mathbf{P}(\mathbf{r}, t), \\
  \mathbf{H}(\mathbf{r}, t) &= \frac{1}{\mu_0} \mathbf{B}(\mathbf{r}, t) - \mathbf{M}(\mathbf{r}, t),
\end{align}</math>

where {{math|'''P'''}} is the [[polarization density|polarization]] field and {{math|'''M'''}} is the [[magnetization]] field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density {{math|''ρ''<sub>b</sub>}} and bound current density {{math|'''J'''<sub>b</sub>}} in terms of [[polarization density|polarization]] {{math|'''P'''}} and [[magnetization]] {{math|'''M'''}} are then defined as
<math display="block">\begin{align}
        \rho_\text{b} &= -\nabla\cdot\mathbf{P}, \\
  \mathbf{J}_\text{b} &= \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}.
\end{align}</math>

If we define the total, bound, and free charge and current density by
<math display="block">\begin{align}
        \rho &= \rho_\text{b} + \rho_\text{f}, \\
  \mathbf{J} &= \mathbf{J}_\text{b} + \mathbf{J}_\text{f},
\end{align}</math>
and use the defining relations above to eliminate {{math|'''D'''}}, and {{math|'''H'''}}, the "macroscopic" Maxwell's equations reproduce the "microscopic" equations.

===Constitutive relations===
{{main|Constitutive equation#Electromagnetism}}

In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between [[Electric displacement field|displacement field]] {{math|'''D'''}} and the electric field {{math|'''E'''}}, as well as the [[Magnetic field#H-field and magnetic materials|magnetizing]] field {{math|'''H'''}} and the magnetic field {{math|'''B'''}}. Equivalently, we have to specify the dependence of the polarization {{math|'''P'''}} (hence the bound charge) and the magnetization {{math|'''M'''}} (hence the bound current) on the applied electric and magnetic field. The equations specifying this response are called [[constitutive relation]]s. For real-world materials, the constitutive relations are rarely simple, except approximately, and usually determined by experiment. See the main article on constitutive relations for a fuller description.<ref name="Zangwill2013">{{cite book|author=Andrew Zangwill|title=Modern Electrodynamics|year=2013|publisher=Cambridge University Press|isbn=978-0-521-89697-9}}</ref>{{rp|44–45}}

For materials without polarization and magnetization, the constitutive relations are (by definition)<ref name=Jackson/>{{rp|2}}
<math display="block">\mathbf{D} = \varepsilon_0\mathbf{E}, \quad \mathbf{H} = \frac{1}{\mu_0}\mathbf{B},</math>
where {{math|''ε''<sub>0</sub>}} is the [[permittivity]] of free space and {{math|''μ''<sub>0</sub>}} the [[permeability (electromagnetism)|permeability]] of free space. Since there is no bound charge, the total and the free charge and current are equal.

An alternative viewpoint on the microscopic equations is that they are the macroscopic equations ''together'' with the statement that vacuum behaves like a perfect linear "material" without additional polarization and magnetization.
More generally, for linear materials the constitutive relations are<ref name="Zangwill2013"/>{{rp|44–45}}
<math display="block">\mathbf{D} = \varepsilon\mathbf{E}, \quad \mathbf{H} = \frac{1}{\mu}\mathbf{B},</math>
where {{math|''ε''}} is the [[permittivity]] and {{math|''μ''}} the [[permeability (electromagnetism)|permeability]] of the material. For the displacement field {{math|'''D'''}} the linear approximation is usually excellent because for all but the most extreme electric fields or temperatures obtainable in the laboratory (high power pulsed lasers) the interatomic electric fields of materials of the order of 10<sup>11</sup> V/m are much higher than the external field. For the magnetizing field <math>\mathbf{H}</math>, however, the linear approximation can break down in common materials like iron leading to phenomena like [[hysteresis]]. Even the linear case can have various complications, however.
*For homogeneous materials, {{math|''ε''}} and {{math|''μ''}} are constant throughout the material, while for inhomogeneous materials they depend on [[position vector|location]] within the material (and perhaps time).<ref name=Kittel2005>{{citation|last=Kittel|first=Charles|title=[[Introduction to Solid State Physics]]|publisher=John Wiley & Sons, Inc.|year=2005|location=USA|edition=8th|isbn=978-0-471-41526-8}}</ref>{{rp|463}}
*For isotropic materials, {{math|''ε''}} and {{math|''μ''}} are scalars, while for anisotropic materials (e.g. due to crystal structure) they are [[tensor]]s.<ref name="Zangwill2013"/>{{rp|421}}<ref name=Kittel2005/>{{rp|463}}
*Materials are generally [[dispersion (optics)|dispersive]], so {{math|''ε''}} and {{math|''μ''}} depend on the [[frequency]] of any incident EM waves.<ref name="Zangwill2013"/>{{rp|625}}<ref name=Kittel2005/>{{rp|397}}

Even more generally, in the case of non-linear materials (see for example [[nonlinear optics]]), {{math|'''D'''}} and {{math|'''P'''}} are not necessarily proportional to {{math|'''E'''}}, similarly {{math|'''H'''}} or {{math|'''M'''}} is not necessarily proportional to {{math|'''B'''}}. In general {{math|'''D'''}} and {{math|'''H'''}} depend on both {{math|'''E'''}} and {{math|'''B'''}}, on location and time, and possibly other physical quantities.

In applications one also has to describe how the free currents and charge density behave in terms of {{math|'''E'''}} and {{math|'''B'''}} possibly coupled to other physical quantities like pressure, and the mass, number density, and velocity of charge-carrying particles. E.g., the original equations given by Maxwell (see [[History of Maxwell's equations]]) included [[Ohm's law]] in the form
<math display="block">\mathbf{J}_\text{f} = \sigma \mathbf{E}.</math>

==Alternative formulations==
<!--In the table below: Lorenz is the correct name (not Lorentz).-->

{{For|an overview|Mathematical descriptions of the electromagnetic field}}
{{For|the equations in [[quantum field theory]]|Quantum electrodynamics}}

Following is a summary of some of the numerous other mathematical formalisms to write the microscopic Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones involving charge and current. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the [[electrical potential]] {{math|''φ''}} and the [[vector potential]] {{math|'''A'''}}. Potentials were introduced as a convenient way to solve the homogeneous equations, but it was thought that all observable physics was contained in the electric and magnetic fields (or relativistically, the Faraday tensor). The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the electric and magnetic fields vanish ([[Aharonov–Bohm effect]]).

Each table describes one formalism. See the [[Mathematical descriptions of the electromagnetic field|main article]] for details of each formulation.  SI units are used throughout.

{|class="wikitable"
|+ [[Vector calculus]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|-
| Fields
3D Euclidean space + time
| <math>\nabla\cdot\mathbf{B} = 0</math><br />
<math>\nabla\times\mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} = \mathbf{0}</math>
| <math>\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}</math><br />
<math>\nabla\times\mathbf{B} - \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t} = \mu_0\mathbf{J}</math>
|-
| Potentials (any [[Gauge theory|gauge]])
3D Euclidean space + time
| <math>\mathbf B = \mathbf \nabla \times \mathbf A</math><br />
<math>\mathbf E = - \mathbf \nabla \varphi - \frac{\partial \mathbf A}{\partial t}</math>
| <math>-\nabla^2 \varphi - \frac{\partial}{\partial t} \left( \mathbf \nabla \cdot \mathbf A \right) = \frac{\rho}{\varepsilon_0}</math><br />
<math>\left( -\nabla^2  + \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) \mathbf A + \mathbf \nabla \left( \mathbf \nabla \cdot \mathbf A + \frac{1}{c^2} \frac{\partial \varphi}{\partial t} \right) = \mu_0 \mathbf{J}</math>
|-
| Potentials ([[Lorenz gauge]])
3D Euclidean space + time
| <math>\mathbf B = \mathbf \nabla \times \mathbf A</math><br />
<math>\mathbf E = - \mathbf \nabla \varphi - \frac{\partial \mathbf A}{\partial t}</math><br />
<math>\mathbf \nabla \cdot \mathbf A = -\frac{1}{c^2}\frac{\partial \varphi}{\partial t}</math>
| <math>\left( -\nabla^2 +\frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) \varphi = \frac{\rho}{\varepsilon_0}</math><br />
<math>\left( -\nabla^2 +\frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) \mathbf A = \mu_0 \mathbf J</math>
|}

{|class="wikitable"
|+ [[Tensor calculus]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|-
| Fields
space + time

spatial metric independent of time
|<math>
\begin{align}
&\partial_{[i} B_{jk]} = \\
&\qquad \nabla_{[i} B_{jk]} = 0 \\
&\partial_{[i} E_{j]} + \frac{\partial B_{ij}}{\partial t} = \\
&\qquad \nabla_{[i} E_{j]} + \frac{\partial B_{ij}}{\partial t} = 0
\end{align}
</math>
|<math>
\begin{align}
&\frac{1}{\sqrt{h}} \partial_i \sqrt{h} E^i = \\
&\qquad \nabla_i E^i = \frac{\rho}{\varepsilon_0} \\
&{-}\frac{1}{\sqrt{h}}\partial_i \sqrt{h} B^{ij} - \frac{1}{c^2} \frac{\partial}{\partial t}  E^j = \\
&\qquad {-}\nabla_iB^{ij} - \frac{1}{c^2} \frac{\partial E^j}{\partial t} = \mu_0 J^j
\end{align}
</math>
|-
|Potentials
space (with [[#topological restriction|§ topological restriction]]s) + time

spatial metric independent of time
|<math>
\begin{align}
 B_{ij} &= \partial_{[i} A_{j]} \\
        &= \nabla_{[i} A_{j]}
\end{align}</math>
<math>\begin{align}
 E_i &= -\frac{\partial A_i}{\partial t} - \partial_i \varphi \\
     &= -\frac{\partial A_i}{\partial t} - \nabla_i \varphi \\
\end{align}
</math>
|<math>
\begin{align}
& {-}\frac{1}{\sqrt{h}} \partial_i \sqrt{h} \left( \partial^i \varphi + \frac{\partial A^i}{\partial t} \right) = \\
&\qquad -\nabla_i \nabla^i \varphi - \frac{\partial}{\partial t} \nabla_i  A^i = \frac{\rho}{\varepsilon_0} \\
& {-}\frac{1}{\sqrt{h}} \partial_i \left( \sqrt{h}h^{im}h^{jn} \partial_{[m} A_{n]} \right) + \frac{1}{c^2} \frac{\partial}{\partial t} \left( \frac{\partial A^j}{\partial t} + \partial^j \varphi \right) = \\
&\qquad {-}\nabla_i \nabla^i A^j + \frac{1}{c^2} \frac{\partial^2 A^j}{\partial t^2}  + R_i^j A^i + \nabla^j \left( \nabla_i A^i  + \frac{1}{c^2} \frac{\partial \varphi}{\partial t} \right) = \mu_0 J^j
\end{align}
</math>
|-
|Potentials (Lorenz gauge)
space (with topological restrictions) + time

spatial metric independent of time
|<math>
\begin{align}
 B_{ij} &= \partial_{[i} A_{j]} \\
        &= \nabla_{[i} A_{j]}
\end{align}
</math><br />
<math>
\begin{align}
 E_i &= -\frac{\partial A_i}{\partial t} - \partial_i \varphi \\
     &= -\frac{\partial A_i}{\partial t}  - \nabla_i \varphi
\end{align}
</math><br />
<math>
 \nabla_i A^i = -\frac{1}{c^2} \frac{\partial\varphi}{\partial t}
</math>
|<math>
 -\nabla_i \nabla^i \varphi + \frac{1}{c^2} \frac{\partial^2 \varphi }{\partial t^2} = \frac{\rho}{\varepsilon_0}
</math><br />
<math>
 -\nabla_i \nabla^i A^j + \frac{1}{c^2} \frac{\partial^2 A^j}{\partial t^2} + R_i^j A^i = \mu_0 J^j
</math>
|}

{|class="wikitable"
|+ [[Exterior calculus|Differential forms]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|-
|Fields
any space + time
|<math>dB = 0</math><br />
<math>dE + \frac{\partial B}{\partial t} = 0</math>
|<math>d{\star}E = \frac{\rho}{\varepsilon_0}</math><br />
<math>d{\star}B - \frac{1}{c^2} \frac{\partial{\star}E}{\partial t} = \mu_0 J</math>
|-
|Potentials (any gauge)
any space (with [[#topological restriction|§ topological restriction]]s) + time
|<math>B = dA</math><br />
<math>E = -d\varphi - \frac{\partial A}{\partial t}</math>
|<math>
-d{\star}\!\left( d\varphi + \frac{\partial A}{\partial t} \right) = \frac{\rho}{\varepsilon_0}
</math><br />
<math>
d{\star}d A + \frac{1}{c^2} \frac{\partial}{\partial t}{\star}\!\left( d\varphi + \frac{\partial A}{\partial t} \right) = \mu_0 J
</math>
|-
|Potential (Lorenz Gauge)
any space (with topological restrictions) + time

spatial metric independent of time
|<math>B = dA</math><br />
<math>E = -d\varphi - \frac{\partial A}{\partial t}</math><br />
<math>d{\star}A = -{\star}\frac{1}{c^2} \frac{\partial \varphi}{\partial t}</math>
|<math>
{\star}\!\left( -\Delta \varphi + \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \varphi \right) = \frac{\rho}{\varepsilon_0}
</math><br />
<math>
{\star}\!\left( -\Delta A    + \frac{1}{c^2} \frac{\partial^2 A}{\partial^2 t} \right) = \mu_0 J
</math>
|-
<!-- Please don't re-add a geometric calculus version, the table is long enough as it is. For an overview article, the geometric calculus version is not mainstream enough and does not give enough additional physical insight to warrant inclusion in this table. Also it is just one click away as an additional alternative formulation -->
|}

==Relativistic formulations ==
{{For|the equations in [[special relativity]]|Classical electromagnetism and special relativity|Covariant formulation of classical electromagnetism}}
{{For|the equations in [[general relativity]]|Maxwell's equations in curved spacetime}}

The Maxwell equations can also be formulated on a spacetime-like [[Minkowski space]] where space and time are treated on equal footing. The direct spacetime formulations make manifest that the Maxwell equations are [[relativistically invariant]]. Because of this symmetry, the electric and magnetic fields are treated on equal footing and are recognized as components of the [[Faraday tensor]]. This reduces the four Maxwell equations to two, which simplifies the equations, although we can no longer use the familiar vector formulation. In fact the Maxwell equations in the space + time formulation are not [[Galilean transformation|Galileo invariant]] and have Lorentz invariance as a hidden symmetry. This was a major source of inspiration for the development of relativity theory. Indeed,  even the formulation that treats space and time separately is not a non-relativistic approximation and describes the same physics by simply renaming variables. For this reason the relativistic invariant equations are usually called the Maxwell equations as well.

Each table below describes one formalism.

{|class="wikitable"
|+ [[Tensor calculus]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|-
| [[Covariant formulation of classical electromagnetism#Maxwell's equations in vacuum|Fields]]<br/> [[Minkowski space]]
| <math>\partial_{[\alpha} F_{\beta\gamma]} = 0 </math>
| <math>\partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta </math>
|-
| Potentials (any gauge)<br/> [[Minkowski space]]
| <math>F_{\alpha\beta} = 2\partial_{[\alpha} A_{\beta]}</math>
| <math>2\partial_\alpha \partial^{[\alpha} A^{\beta]} = \mu_0 J^\beta</math>
|-
| Potentials (Lorenz&nbsp;gauge)<br/> [[Minkowski space]]
| <math>F_{\alpha\beta} = 2\partial_{[\alpha} A_{\beta]}</math>
<math>\partial_\alpha A^\alpha = 0</math>
| <math>\partial_\alpha\partial^\alpha A^\beta = \mu_0 J^\beta</math>
|-
| Fields<br/> any spacetime
| <math>\begin{align}
& \partial_{[\alpha} F_{\beta\gamma]} = \\
&\qquad \nabla_{[\alpha} F_{\beta\gamma]} = 0
\end{align}</math>
| <math>\begin{align}
& \frac{1}{\sqrt{-g}} \partial_\alpha (\sqrt{-g} F^{\alpha\beta}) = \\
&\qquad \nabla_\alpha F^{\alpha\beta} = \mu_0 J^\beta
\end{align}</math>
|-
| Potentials (any gauge)<br/> any spacetime<br/> (with [[#topological restriction|§topological restriction]]s) 
| <math>F_{\alpha\beta} = 2\partial_{[\alpha} A_{\beta]}</math>
| <math>\begin{align}
& \frac{2}{\sqrt{-g}} \partial_\alpha (\sqrt{-g}g^{\alpha\mu}g^{\beta\nu}\partial_{[\mu}A_{\nu]} ) = \\
&\qquad 2\nabla_\alpha (\nabla^{[\alpha} A^{\beta]}) = \mu_0 J^\beta
\end{align}</math>
|-
| Potentials (Lorenz gauge)<br/> any spacetime<br/> (with topological restrictions)
| <math>F_{\alpha\beta} = 2\partial_{[\alpha} A_{\beta]}</math>
<math>\nabla_\alpha A^{\alpha} = 0</math>
| <math>\nabla_\alpha\nabla^\alpha A^{\beta} - R^{\beta}{}_{\alpha} A^\alpha = \mu_0 J^\beta</math>
|}

{|class="wikitable"
|+ [[Exterior calculus|Differential forms]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|- 
| Fields<br/> any spacetime
| <math>\mathrm{d} F = 0</math>
<!-- We consider the current as a (pseudo) three form rather than a 1 form. A three form can be integrated over a 3D spatial region at a fixed time to get a charge in the region or over 2D spatial surface cross a time interval to get an amount of charge that has flowed through the surface in a certain amount of time. It is therefore closest to the physical interpretation of a current and so makes the form equations much easier to interpret. It also makes Maxwell's equations conformally invariant, because the Hodge star on two forms is-->
| <math>\mathrm{d} {\star} F = \mu_0 J </math>
|-
| Potentials (any gauge)<br/> any spacetime<br/>  (with topological restrictions)
| <math>F = \mathrm{d} A</math>
| <math>\mathrm{d} {\star} \mathrm{d} A = \mu_0 J </math>
|-
| Potentials (Lorenz&nbsp;gauge)<br/> any spacetime<br/>  (with topological restrictions)
| <math>F = \mathrm{d}A</math>
<math>\mathrm{d}{\star} A = 0</math>
| <math>{\star} \Box A = \mu_0 J </math>
|-
<!-- Please don't re-add a geometric calculus version, the table is long enough as it is. For an overview article, the geometric calculus version is not mainstream enough and does not give enough additional physical insight to warrant inclusion in this table. Also it is just one click away as an additional alternative formulation -->
|}

*{{anchor|topological restriction}}In the tensor calculus formulation, the [[electromagnetic tensor]] {{math|''F''{{sub|''αβ''}}}} is an antisymmetric covariant order 2 tensor; the [[four-potential]], {{math|''A''{{sub|''α''}}}}, is a covariant vector; the current, {{math|''J''{{sup|''α''}}}}, is a vector; the square brackets, {{math|[ ]}}, denote [[Ricci calculus#Symmetric and antisymmetric parts|antisymmetrization of indices]]; {{math|∂{{sub|''α''}}}} is the partial derivative with respect to the coordinate, {{math|''x''{{sup|''α''}}}}. In Minkowski space coordinates are chosen with respect to an [[inertial frame]]; {{math|1=(''x''{{sup|''α''}}) = (''ct'',&nbsp;''x'',&nbsp;''y'',&nbsp;''z'')}}, so that the [[metric tensor]] used to raise and lower indices is {{math|1=''η''{{sub|''αβ''}} = diag(1,&nbsp;−1,&nbsp;−1,&nbsp;−1)}}. The [[d'Alembert operator]] on Minkowski space is {{math|1=◻ = ∂{{sub|''α''}}∂{{sup|''α''}}}} as in the vector formulation. In general spacetimes, the coordinate system {{math|''x''{{sup|''α''}}}} is arbitrary, the [[covariant derivative]] {{math|∇{{sub|''α''}}}}, the [[Ricci tensor]], {{math|''R''{{sub|''αβ''}}}} and raising and lowering of indices are defined by the Lorentzian metric, {{math|''g''{{sub|''αβ''}}}} and the d'Alembert operator is defined as {{math|1=◻ = ∇{{sub|''α''}}∇{{sup|''α''}}}}. The topological restriction is that the second real [[cohomology]] group of the space vanishes (see the differential form formulation for an explanation). This is violated for Minkowski space with a line removed, which can model a (flat) spacetime with a point-like monopole on the complement of the line.
*In the [[differential form]] formulation on arbitrary space times, {{math|1=''F'' = {{sfrac|2}}''F''{{sub|''αβ''}}d''x''{{sup|''α''}} ∧ d''x''{{sup|''β''}}}} is the electromagnetic tensor considered as a 2-form, {{math|1=''A'' = ''A''{{sub|''α''}}d''x''{{sup|''α''}}}} is the potential 1-form, <math>J = - J_\alpha {\star}\mathrm{d}x^\alpha</math> is the current 3-form, {{math|d}} is the [[exterior derivative]], and <math>{\star}</math> is the [[Hodge star]] on forms defined (up to its orientation, i.e. its sign) by the Lorentzian metric of spacetime. In the special case of 2-forms such as ''F'', the Hodge star <math>{\star}</math> depends on the metric tensor only for its local scale<!--On signature (1,3) or (3,1) and two forms: δ = −*d* so (d*d − *d*d*) = *(−*d* d + d −*d*) = *Hodge Laplacian -->. This means that, as formulated, the differential form field equations are [[conformal geometry|conformally invariant]], but the Lorenz gauge condition breaks conformal invariance. The operator <math>\Box = (-{\star} \mathrm{d} {\star} \mathrm{d} - \mathrm{d} {\star} \mathrm{d} {\star}) </math> is the [[Laplace–Beltrami operator|d'Alembert–Laplace–Beltrami operator]] on 1-forms on an arbitrary [[pseudo-Riemannian manifold#Lorentzian manifold|Lorentzian spacetime]]. The topological condition is again that the second real cohomology group is 'trivial' (meaning that its form follows from a definition). By the isomorphism with the second [[de Rham cohomology]] this condition means that every closed 2-form is exact.

Other formalisms include the [[Geometric algebra#Spacetime model|geometric algebra formulation]] and a [[matrix representation of Maxwell's equations]]. Historically, a [[quaternion]]ic formulation<ref>{{cite arXiv|title=Physical Space as a Quaternion Structure I: Maxwell Equations. A Brief Note|last=Jack|first=P. M.|year=2003|eprint=math-ph/0307038}}</ref><ref>{{cite news|title=On the Notation of Maxwell's Field Equations|author=A. Waser|year=2000|publisher=AW-Verlag|url=http://www.zpenergy.com/downloads/Orig_maxwell_equations.pdf}}</ref> was used.

==Solutions==
Maxwell's equations are [[partial differential equations]] that relate the electric and magnetic fields to each other and to the electric charges and currents. Often, the charges and currents are themselves dependent on the electric and magnetic fields via the [[Lorentz force|Lorentz force equation]] and the [[#Constitutive relations|constitutive relations]]. These all form a set of coupled partial differential equations which are often very difficult to solve: the solutions encompass all the diverse phenomena of [[classical electromagnetism]]. Some general remarks follow.

As for any differential equation, [[boundary condition]]s<ref name=Monk>
{{cite book
 |author=Peter Monk
 |title=Finite Element Methods for Maxwell's Equations
 |page =1 ff
 |publisher=Oxford University Press
 |location=Oxford UK
 |isbn=978-0-19-850888-5
 |url=https://books.google.com/books?id=zI7Y1jT9pCwC&q=electromagnetism+%22boundary+conditions%22&pg=PA1
 |year=2003
}}</ref><ref name=Volakis>
{{cite book
 |author=Thomas B. A. Senior & John Leonidas Volakis
 |title=Approximate Boundary Conditions in Electromagnetics
 |page =261 ff
 |publisher=Institution of Electrical Engineers
 |location=London UK
 |isbn=978-0-85296-849-9
 |url=https://books.google.com/books?id=eOofBpuyuOkC&q=electromagnetism+%22boundary+conditions%22&pg=PA261
 |date=1995-03-01
}}</ref><ref name=Hagstrom>
{{cite book
 |author=T Hagstrom (Björn Engquist & Gregory A. Kriegsmann, Eds.)
 |title=Computational Wave Propagation
 |page =1 ff
 |publisher=Springer
 |location=Berlin
 |isbn=978-0-387-94874-4
 |url=https://books.google.com/books?id=EdZefkIOR5cC&q=electromagnetism+%22boundary+conditions%22&pg=PA1
 |year=1997
}}</ref> and [[initial condition]]s<ref name=Hussain>
{{cite book
 |author=Henning F. Harmuth & Malek G. M. Hussain
 |title=Propagation of Electromagnetic Signals
 |page =17
 |publisher=World Scientific
 |location=Singapore
 |isbn=978-981-02-1689-4
 |url=https://books.google.com/books?id=6_CZBHzfhpMC&q=electromagnetism+%22initial+conditions%22&pg=PA45
 |year=1994
}}</ref> are necessary for a [[Electromagnetism uniqueness theorem|unique solution]]. For example, even with no charges and no currents anywhere in spacetime, there are the obvious solutions for which '''E''' and '''B''' are zero or constant, but there are also non-trivial solutions corresponding to electromagnetic waves. In some cases, Maxwell's equations are solved over the whole of space, and boundary conditions are given as asymptotic limits at infinity.<ref name=Cook>
{{cite book
 |author=David M Cook
 |title=The Theory of the Electromagnetic Field
 |year=2002
 |page =335 ff
 |publisher=Courier Dover Publications
 |location=Mineola NY
 |isbn=978-0-486-42567-2
 |url=https://books.google.com/books?id=bI-ZmZWeyhkC&q=electromagnetism+infinity+boundary+conditions&pg=RA1-PA335
}}</ref> In other cases, Maxwell's equations are solved in a finite region of space, with appropriate conditions on the boundary of that region, for example an [[Perfectly matched layer|artificial absorbing boundary]] representing the rest of the universe,<ref name=Lourtioz>
{{cite book
 |author=Jean-Michel Lourtioz
 |title=Photonic Crystals: Towards Nanoscale Photonic Devices
 |page =84
 |publisher=Springer
 |location=Berlin
 |isbn=978-3-540-24431-8
 |url=https://books.google.com/books?id=vSszZ2WuG_IC&q=electromagnetism+boundary++-element&pg=PA84
 |date=2005-05-23
}}</ref><ref>S. G. Johnson, [http://math.mit.edu/~stevenj/18.369/pml.pdf Notes on Perfectly Matched Layers], online MIT course notes (Aug. 2007).</ref> or [[periodic boundary conditions]], or walls that isolate a small region from the outside world (as with a [[waveguide]] or cavity [[resonator]]).<ref>
{{cite book
 |author=S. F. Mahmoud
 |title=Electromagnetic Waveguides: Theory and Applications
 |page =Chapter 2
 |publisher=Institution of Electrical Engineers
 |location=London UK
 |isbn=978-0-86341-232-5
 |url=https://books.google.com/books?id=toehQ7vLwAMC&q=Maxwell%27s+equations+waveguides&pg=PA2
 |no-pp=true
 |year=1991
}}</ref>

[[Jefimenko's equations]] (or the closely related [[Liénard–Wiechert potential]]s) are the explicit solution to Maxwell's equations for the electric and magnetic fields created by any given distribution of charges and currents. It assumes specific initial conditions to obtain the so-called "retarded solution", where the only fields present are the ones created by the charges. However, Jefimenko's equations are unhelpful in situations when the charges and currents are themselves affected by the fields they create.

[[Numerical partial differential equations|Numerical methods for differential equations]] can be used to compute approximate solutions of Maxwell's equations when exact solutions are impossible. These include the [[finite element method]] and [[finite-difference time-domain method]].<ref name=Monk/><ref name=Hagstrom/><ref name= Kempel>
{{cite book
 |author=John Leonidas Volakis, Arindam Chatterjee & Leo C. Kempel
 |title=Finite element method for electromagnetics : antennas, microwave circuits, and scattering applications
 |year=1998
 |page =79 ff
 |publisher=Wiley IEEE
 |location=New York
 |isbn=978-0-7803-3425-0
 |url=https://books.google.com/books?id=55q7HqnMZCsC&q=electromagnetism+%22boundary+conditions%22&pg=PA79
}}</ref><ref name= Friedman>
{{cite book
 |author=Bernard Friedman
 |title=Principles and Techniques of Applied Mathematics
 |year= 1990
 |publisher=Dover Publications
 |location=Mineola NY
 |isbn=978-0-486-66444-6
 }}</ref><ref name=Taflove>
{{cite book
 |author=Taflove A & Hagness S C
 |title=Computational Electrodynamics: The Finite-difference Time-domain Method
 |year= 2005
 |page =Chapters 6 & 7
 |publisher=[[Artech House]]
 |location=Boston MA
 |isbn=978-1-58053-832-9
 |no-pp=true
}}</ref> For more details, see [[Computational electromagnetics]].

==Overdetermination of Maxwell's equations==
Maxwell's equations ''seem'' [[Overdetermined system|overdetermined]], in that they involve six unknowns (the three components of {{math|'''E'''}} and {{math|'''B'''}}) but eight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampere's laws). (The currents and charges are not unknowns, being freely specifiable subject to [[charge conservation]].) This is related to a certain limited kind of redundancy in Maxwell's equations: It can be proven that any system satisfying Faraday's law and Ampere's law ''automatically'' also satisfies the two Gauss's laws, as long as the system's initial condition does, and assuming conservation of charge and the nonexistence of magnetic monopoles.<ref>{{cite book|author=H Freistühler & G Warnecke |title=Hyperbolic Problems: Theory, Numerics, Applications |year=2001 |page=605 |publisher=Springer |url=https://books.google.com/books?id=XXX_mG0vneMC&pg=PA605|isbn=9783764367107 }}</ref><ref>{{cite journal |title=Redundancy and superfluity for electromagnetic fields and potentials |journal=American Journal of Physics |author=J Rosen |volume=48 |issue=12 |page=1071 |doi=10.1119/1.12289|bibcode = 1980AmJPh..48.1071R |year=1980 }}</ref> 
This explanation was first introduced by [[Julius Adams Stratton]] in 1941.<ref>{{cite book|author=J. A. Stratton|title=Electromagnetic Theory |url=https://books.google.com/books?id=zFeWdS2luE4C |year=1941 |publisher=McGraw-Hill Book Company |pages=1–6|isbn=9780470131534 }}</ref>

Although it is possible to simply ignore the two Gauss's laws in a numerical algorithm (apart from the initial conditions), the imperfect precision of the calculations can lead to ever-increasing violations of those laws. By introducing dummy variables characterizing these violations, the four equations become not overdetermined after all. The resulting formulation can lead to more accurate algorithms that take all four laws into account.<ref>{{cite journal |title=The Origin of Spurious Solutions in Computational Electromagnetics |author=B Jiang & J Wu & L. A. Povinelli |doi=10.1006/jcph.1996.0082 |year=1996 |journal=Journal of Computational Physics |volume=125 |issue=1 |page=104|bibcode = 1996JCoPh.125..104J |hdl=2060/19950021305 |hdl-access=free }}</ref>

Both identities <math>\nabla\cdot \nabla\times \mathbf{B} \equiv 0, \nabla\cdot \nabla\times \mathbf{E} \equiv 0</math>, which reduce eight equations to six independent ones, are the true reason of overdetermination.<ref>{{cite book | first = Steven | last = Weinberg | title = Gravitation and Cosmology | publisher = John Wiley | date = 1972 | isbn = 978-0-471-92567-5 | pages = [https://archive.org/details/gravitationcosmo00stev_0/page/161 161–162] | url = https://archive.org/details/gravitationcosmo00stev_0/page/161 }}</ref><ref>{{Citation |first1=R. |last1=Courant|author-link=Richard Courant|name-list-style=amp |first2=D. |last2=Hilbert|author2-link=David Hilbert|title=Methods of Mathematical Physics: Partial Differential Equations |volume=II |publisher=Wiley-Interscience |location=New York |year=1962 |pages=15–18 |isbn=9783527617241| url=https://books.google.com/books?id=fcZV4ohrerwC}}</ref>

Equivalently, the overdetermination can be viewed as implying conservation of electric and magnetic charge, as they are required in the derivation described above but implied by the two Gauss's laws.

For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But in differential equations, and especially partial differential equations (PDEs), one needs appropriate boundary conditions, which depend in not so obvious ways on the equations. Even more, if one rewrites them in terms of vector and scalar potential, then the equations are underdetermined because of [[gauge fixing]].

==Maxwell's equations and quantum mechanics ==
Maxwell's equations are valid in both the classical and the quantum realm. In the Heisenberg representation of Quantum Mechanics, the equations of the E and B operators are precisely Maxwell's equations.  Of course since the fields are quantum operators, there are many aspects which differ from the classical fields. For example, the E field acts like the momentum conjugate to the spatial components of the vector potential A. This of course leads to many aspects of the quantum electromagnetic field with differ from them as classical fields but they still obey the same evolution equations as the classical field does. 

Of course once one examines the effects of the  electromagnetic fields on charged matter, and those effects then change the electromagnetic field are examined, the field equations become non-linear, and the quantum behaviour of non-linear field can be very different from the classical behaviour of the non-linear fields. That however does not alter the fact that if one remains in the linear regime, the fields obey Maxwell's equations.

==Variations==
Popular variations on the Maxwell equations as a classical theory of electromagnetic fields are relatively scarce because the standard equations have stood the test of time remarkably well.

===Magnetic monopoles===
{{main|Magnetic monopole}}

Maxwell's equations posit that there is [[electric charge]], but no [[magnetic charge]] (also called [[magnetic monopole]]s), in the universe. Indeed, magnetic charge has never been observed, despite extensive searches,<ref group="note">See [[magnetic monopole]] for a discussion of monopole searches. Recently, scientists have discovered that some types of condensed matter, including [[spin ice]] and [[topological insulator]]s, which display ''emergent'' behavior resembling magnetic monopoles. (See [http://www.sciencemag.org/cgi/content/abstract/1178868 sciencemag.org] and [http://www.nature.com/nature/journal/v461/n7266/full/nature08500.html nature.com].) Although these were described in the popular press as the long-awaited discovery of magnetic monopoles, they are only superficially related. A "true" magnetic monopole is something where {{math|∇ ⋅ '''B''' ≠ 0}}, whereas in these condensed-matter systems, {{math|1=∇ ⋅ '''B''' = 0}} while only {{math|∇ ⋅ '''H''' ≠ 0}}.</ref> and may not exist. If they did exist, both Gauss's law for magnetism and Faraday's law would need to be modified, and the resulting four equations would be fully symmetric under the interchange of electric and magnetic fields.<ref name=Jackson/>{{rp|273–275}}

==See also==
{{Portal|Electronics|Physics}} 
{{columns-list|colwidth=30em|
* [[Algebra of physical space]]
* [[Fresnel equations]]
* [[Gravitoelectromagnetism]]
* [[Interface conditions for electromagnetic fields]]
* [[Moving magnet and conductor problem]]
* [[Riemann–Silberstein vector]]
* [[Spacetime algebra]]
* [[Wheeler–Feynman absorber theory]]
}}

== Explanatory notes ==
{{Reflist|group="note"|1}}

==References==
{{Reflist|30em}}

==Further reading==
{{See also|List of textbooks in electromagnetism}}
* {{citation |last1=Imaeda |first1=K. |year=1995 |chapter=Biquaternionic Formulation of Maxwell's Equations and their Solutions |editor-last=Ablamowicz |editor-first=Rafał |editor-last2=Lounesto |editor-first2=Pertti |title=Clifford Algebras and Spinor Structures |pages=265–280 |publisher=Springer |doi=10.1007/978-94-015-8422-7_16 |isbn=978-90-481-4525-6 }}

===Historical publications===
* [https://web.archive.org/web/20081217035457/http://blazelabs.com/On%20Faraday%27s%20Lines%20of%20Force.pdf On Faraday's Lines of Force]&nbsp;– 1855/56. Maxwell's first paper (Part&nbsp;1 & 2)&nbsp;– Compiled by Blaze Labs Research (PDF).
* [//upload.wikimedia.org/wikipedia/commons/b/b8/On_Physical_Lines_of_Force.pdf On Physical Lines of Force]&nbsp;– 1861. Maxwell's 1861 paper describing magnetic lines of force&nbsp;– Predecessor to 1873 Treatise.
* [[James Clerk Maxwell]], "[[A Dynamical Theory of the Electromagnetic Field]]", ''Philosophical Transactions of the Royal Society of London'' '''155''', 459–512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
** [https://books.google.com/books?id=5HE_cmxXt2MC&q=Proceedings+of+the+Royal+Society+Of+London+Vol+XIII A Dynamical Theory Of The Electromagnetic Field]&nbsp;– 1865. Maxwell's 1865 paper describing his 20 equations, link from [[Google Books]].
* J. Clerk Maxwell (1873), "[[A Treatise on Electricity and Magnetism]]":
** Maxwell, J. C., "A Treatise on Electricity And Magnetism" – [http://posner.library.cmu.edu/Posner/books/book.cgi?call=537_M46T_1873_VOL._1 Volume 1] – 1873&nbsp;– Posner Memorial Collection&nbsp;– Carnegie Mellon University.
** Maxwell, J. C., "A Treatise on Electricity And Magnetism" – [http://posner.library.cmu.edu/Posner/books/book.cgi?call=537_M46T_1873_VOL._2 Volume 2] – 1873&nbsp;– Posner Memorial Collection&nbsp;– Carnegie Mellon University.

;The developments before relativity<nowiki>:</nowiki>

* {{cite journal | author = Larmor Joseph  | year = 1897 | title = [[s:Dynamical Theory of the Electric and Luminiferous Medium III|On a dynamical theory of the electric and luminiferous medium. Part&nbsp;3, Relations with material media]] | url = | journal = Phil. Trans. R. Soc. | volume = 190 | issue = | pages = 205–300 }}
* {{cite journal | author = Lorentz Hendrik  | year = 1899 | title = [[s:Simplified Theory of Electrical and Optical Phenomena in Moving Systems|Simplified theory of electrical and optical phenomena in moving systems]] | url = | journal = Proc. Acad. Science Amsterdam | volume = I | issue = | pages = 427–443 }}
* {{cite journal | author = Lorentz Hendrik  | year = 1904 | title = [[s:Electromagnetic phenomena|Electromagnetic phenomena in a system moving with any velocity less than that of light]] | url = | journal = Proc. Acad. Science Amsterdam | volume = IV | issue = | pages = 669–678 }}
* [[Henri Poincaré]] (1900) "La théorie de Lorentz et le Principe de Réaction" {{in lang|fr}}, ''Archives Néerlandaises'', '''V''', 253–278.
* [[Henri Poincaré]] (1902) "[[La Science et l'Hypothèse]]" {{in lang|fr}}.
* [[Henri Poincaré]] (1905) [http://www.soso.ch/wissen/hist/SRT/P-1905-1.pdf "Sur la dynamique de l'électron"] {{in lang|fr}}, ''Comptes Rendus de l'Académie des Sciences'', '''140''', 1504–1508.
* Catt, Walton and Davidson. [http://www.electromagnetism.demon.co.uk/z014.htm "The History of Displacement Current"] {{Webarchive|url=https://web.archive.org/web/20080506120012/http://www.electromagnetism.demon.co.uk/z014.htm |date=2008-05-06 }}. ''Wireless World'', March 1979.

==External links==
{{Commons category}}
{{wikiquote}}
{{sister project|project=Wikiversity|text=[[v:MyOpenMath/Solutions/Maxwell's integral equations|Wikiversity discusses basic Maxwell integrals for students.]]}}
* {{springer|title=Maxwell equations|id=p/m063140}}
* [http://www.maxwells-equations.com maxwells-equations.com] — An intuitive tutorial of Maxwell's equations.
* [https://feynmanlectures.caltech.edu/II_18.html The Feynman Lectures on Physics Vol. II Ch. 18: The Maxwell Equations]
* [[wikiversity:Maxwell's equations|Wikiversity Page on Maxwell's Equations]]

===Modern treatments===
* [http://lightandmatter.com/area1sn.html Electromagnetism (ch. 11)], B. Crowell, Fullerton College
* [https://web.archive.org/web/20030803151533/http://farside.ph.utexas.edu/~rfitzp/teaching/jk1/lectures/node6.html Lecture series: Relativity and electromagnetism], R. Fitzpatrick, University of Texas at Austin
* [http://www.physnet.org/modules/pdf_modules/m210.pdf ''Electromagnetic waves from Maxwell's equations''] on [http://www.physnet.org Project PHYSNET].
* [https://web.archive.org/web/20090324084439/http://ocw.mit.edu/OcwWeb/Physics/8-02Electricity-and-MagnetismSpring2002/VideoAndCaptions/index.htm MIT Video Lecture Series (36 × 50 minute lectures) (in .mp4 format) – Electricity and Magnetism] Taught by Professor [[Walter Lewin]].

===Other===
*{{cite journal |arxiv=hep-ph/0106235|last1=Silagadze|first1=Z. K.|title=Feynman's derivation of Maxwell equations and extra dimensions|journal=Annales de la Fondation Louis de Broglie|volume=27|pages=241–256|year=2002|bibcode=2001hep.ph....6235S}}
*[http://www.nature.com/milestones/milephotons/full/milephotons02.html ''Nature Milestones: Photons''&nbsp;– ''Milestone 2 (1861) Maxwell's equations'']

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