Filter (mathematics)

2013-10-21T08:46:43Z

78.250.58.225: /* General definition */ moved condition 3 to comment, for consistency with iff upper set closed under finite meets.

{{For|thermodynamic relations|Maxwell relations}}
{{For|the history of the equations|History of Maxwell's equations}}
{{Electromagnetism|cTopic=Electrodynamics}}

'''Maxwell's equations''' are a set of [[partial differential equation]]s that, together with the [[Lorentz force]] law, form the foundation of [[classical electrodynamics]], classical [[optics]], and [[electric circuit]]s. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how [[electric field|electric]] and [[magnetic field]]s are generated and altered by each other and by [[electric charge|charges]] and [[electric current|currents]]. They are named after the Scottish physicist and mathematician [[James Clerk Maxwell]] who published an early form of those equations between 1861 and 1862.

The equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current, including the complicated charges and currents in materials at the [[atom]]ic scale; it has universal applicability, but may be unfeasible to calculate. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that describe large-scale behavior without having to consider these atomic scale details, but it requires the use of parameters characterizing the electromagnetic properties of the relevant materials.

The term "Maxwell's equations" is often used for [[#Alternative formulations|other forms]] of Maxwell's equations. For example, [[Covariant formulation of classical electromagnetism|space-time formulations]] are commonly used in high energy and gravitational physics. These formulations, defined on [[spacetime|space-time]] rather than space and time separately, are [[manifest covariance|manifestly]]<ref group="note">Maxwell's equations in any form are compatible with relativity. These space-time formulations, though, make that compatibility more readily apparent.</ref> compatible with [[special relativity|special]] and [[general relativity]]. In quantum mechanics, versions of Maxwell's equations based on the [[electric potential|electric]] and [[magnetic potential]]s are preferred.

Since the mid-20th century, it has been understood that Maxwell's equations are not exact laws of the universe, but are a classical approximation to the more accurate and fundamental theory of [[quantum electrodynamics]]. In most cases, though, quantum deviations from Maxwell's equations are immeasurably small. Exceptions occur when the [[photon|particle]] nature of light is important or for very strong electric fields.

{{TOC limit|4}}

==Formulation in terms of electric and magnetic fields==



To describe electromagnetism in this formulation, the powerful language of [[vector calculus]] is used throughout this article. 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]] '''E''', a [[vector field]], and the [[magnetic field]] '''B''', a [[pseudovector]] field, where each generally have time-dependence. The sources of these fields are [[electric charge]]s and [[electric current]]s, which can be expressed as local densities namely [[charge density]] ''ρ'' and [[current density]] '''J'''. A separate law of nature, the [[Lorentz force]] law, describes how the electric and magnetic field act on charged particles and currents. A version of this law was included in the original equations by Maxwell but, by convention, is no longer.

In the electric-magnetic field formulation there are four equations. Two of them describe how the fields vary in space due to sources, if any; electric fields emanating from electric charges in [[Gauss's law]], and magnetic fields as closed field lines ''not due to [[magnetic monopole]]s'' in [[Gauss's law for magnetism]]. The other two describe how the fields "circulate" around their respective sources; the magnetic field "circulates" around electric currents and time varying electric fields in [[Ampère's circuital law|Ampère's law with Maxwell's correction]], while the electric field "circulates" around time varying magnetic fields in [[Faraday's law of induction|Faraday's law]].

The precise formulation of Maxwell's equations depends on the precise definition of the quantities involved. Conventions differ with the unit systems; because various definitions and [[dimensional analysis|dimensions]] are changed by absorbing [[dimensional analysis|dimensionfull]] factors like the [[speed of light]] ''c''. This makes constants come out differently.

==Conventional formulation in SI units==

The equations in this section are given in the convention used with [[SI units]]. Other units commonly used are [[Gaussian units]] based on the [[cgs system]],<ref name=Griffiths>
{{cite book
|author=David J Griffiths
|title=Introduction to electrodynamics
|year= 1999
|edition=Third
|pages=559–562
|publisher=Prentice Hall
|isbn=0-13-805326-X
|url=http://worldcat.org/isbn/013805326X
}}</ref> [[Lorentz–Heaviside units]] (used mainly in [[particle physics]]), and [[Planck units]] (used in [[theoretical physics]]). See [[#Equations in Gaussian units|below]] for the formulation with Gaussian units.

:{| class="wikitable" style="text-align: center;"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | [[Integral]] equations
! scope="col" | [[Partial differential equation|Differential]] equations
|-
! scope="row" | [[Gauss's law]]
| {{oiint
| intsubscpt=<math>{\scriptstyle\partial \Omega }</math>
| integrand=<math>\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>
|-
! scope="row" | [[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>
|-
! scope="row" | 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>
|-
! scope="row" | [[Ampère's circuital law]] (with Maxwell's correction)
| <math>\oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \iint_{\Sigma} \left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf E}{\partial t} \right)\cdot \mathrm{d}\mathbf{S}</math>
| <math>\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) </math>
|-
|}

There are [[universal constant]]s appearing in the equations; in this case the [[permittivity of free space]] ''ε''0 and the [[permeability of free space]] ''μ''0, a general characteristic of fundamental [[field equation]]s.

In the differential equations, a ''local'' description of the fields, the [[nabla symbol]] ∇ denotes the three-dimensional [[gradient operator]], and from it ∇· is the [[divergence]] operator and ∇× the [[curl (mathematics)|curl]] operator. The sources are taken to be as local densities of charge and current.

In the integral equations; a description of the fields within a region of space, Ω is any fixed volume with [[boundary (topology)|boundary]] surface ∂Ω, and Σ is any fixed open surface with boundary curve ∂Σ. Here "fixed" means the volume or surface do not change in time. Although it is possible to formulate Maxwell's equations with time-dependent surfaces and volumes, this is not actually necessary: the equations are correct and complete with time-independent surfaces. The sources are correspondingly the total amounts of charge and current within these volumes and surfaces, found by integration. The [[volume integral]] of the total [[charge density]] ''ρ'' over any fixed volume Ω is the ''total'' [[electric charge]] contained in Ω:

:<math>Q = \iiint_\Omega \rho \, \mathrm{d}V\,,</math>

and the ''net'' [[electrical current]] is the [[surface integral]] of the [[electric current density]] '''J''', passing through any open fixed surface Σ:

:<math>I = \iint_{\Sigma} \mathbf{J} \cdot \mathrm{d} \mathbf{S}\,,</math>

where d'''S''' denotes the [[differential (infinitesimal)|differential]] [[vector area|vector element]] of surface area ''S'' [[Normal (geometry)|normal]] to surface Σ. (Vector area is also denoted by '''A''' rather than '''S''', but this conflicts with the [[magnetic potential]], a separate vector field).

The "total charge or current" refers to including free and bound charges, or [[free current|free]] and [[bound current]]s. These are used in the macroscopic formulation [[#"Microscopic" versus "macroscopic"|below]].

==Relationship between differential and integral formulations==

The differential and integral formulations of the equations are mathematically equivalent, by the [[divergence theorem]] in the case of Gauss's law and Gauss's law for magnetism, and by the [[Kelvin–Stokes theorem]] in the case of Faraday's law and Ampère's law. Both the differential and integral formulations are useful. The integral formulation can often be used to simply and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential formulation is 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=0-471-72070-4 |page=273 |url=http://books.google.com/books?id=-hIG3NZrnd8C&pg=PA273}}</ref>

===Flux and divergence===

[[File:Divergence theorem in EM.svg|thumb|240px|Closed volume Ω and its boundary ∂Ω, enclosing a source (+) and sink (−) of a vector field '''F'''. Here, '''F''' could be the '''E''' field with source electric charges, but ''not'' the '''B''' field which has no magnetic charges as shown. The outward [[unit normal]] is '''n'''.]]

The "fields emanating from the sources" can be inferred from the [[surface integral]]s of the fields through the [[closed surface|closed]] surface ∂Ω, defined as the [[electric flux]] and [[magnetic flux]] respectively:

:{{oiint|
| intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
| integrand = <math> \mathbf{E}\cdot\mathrm{d}\mathbf{S} \,,</math>
}}      {{oiint|
| intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
| integrand = <math> \mathbf{B}\cdot\mathrm{d}\mathbf{S} \,,</math>
}}

as well as their [[divergence]]s:

:<math>\nabla \cdot \mathbf{E}, \quad \nabla \cdot \mathbf{B}\,.</math>

These surface integrals and divergences are connected by the [[divergence theorem]].

===Circulation and curl===

[[File:Curl theorem in EM.svg|thumb|240px|Open surface Σ and boundary ∂Σ. '''F''' could be the '''E''' or '''B''' fields. Again, '''n''' is the [[unit normal]]. (The curl of a vector field doesn't literally look like the "circulations", this is a heuristic depiction).]]

The "circulation of the fields" can be interpreted from the [[line integral]]s of the fields around the closed curve ∂Σ:

:<math>\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}, \quad \oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell}\,,</math>

where d'''{{ell}}''' is the differential vector element of ''path length'' [[tangential]] to the path/curve, as well as their [[curl (mathematics)|curl]]s:

:<math>\nabla \times \mathbf{E}, \quad \nabla \times \mathbf{B}\,.</math>

These line integrals and curls are connected by [[Stokes' theorem]], and 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.

===Time evolution===

The "dynamics" or "[[time evolution]] of the fields" is due to the partial derivatives of the fields with respect to time:

:<math>\frac{\partial\mathbf{E}}{\partial t}, \quad \frac{\partial\mathbf{B}}{\partial t}.</math>

These derivatives are crucial for the prediction of field propagation in the form of [[electromagnetic waves]]. Since the surface is taken to be time-independent, we can make the following transition in Faraday's law:

:<math> \frac{d}{dt} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} = \iint_{\Sigma} \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{S}\,,</math>

see [[differentiation under the integral sign]] for more on this result.

==Conceptual descriptions==

===Gauss's law===

'''[[Gauss's law]]''' describes the relationship between a static [[electric field]] and the [[electric charge]]s that cause it: The static electric field points away from positive charges and towards negative charges. In the field line description, electric field lines begin only at positive electric charges and end only at negative electric charges. 'Counting' the number of field lines passing though a [[closed surface]], therefore, yields the total charge (including bound charge due to polarization of material) enclosed by that surface divided by dielectricity of free space (the [[vacuum permittivity]]). More technically, it relates the [[electric flux]] through any hypothetical [[closed surface|closed]] "[[Gaussian surface]]" to the enclosed electric charge.

[[Image:VFPt dipole magnetic1.svg|right|thumb|250|[[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===

'''[[Gauss's law for magnetism]]''' states that there are no "magnetic charges" (also called [[magnetic monopole]]s), analogous to electric charges.<ref name=VideoGlossary>[http://videoglossary.lbl.gov/2009/maxwells-equations/ J.D. Jackson, "Maxwell's Equations" video glossary entry]</ref> Instead, the magnetic field due to materials is generated by a configuration called a [[dipole]]. Magnetic dipoles are best represented as loops of current but resemble positive and negative 'magnetic charges', inseparably bound together, having no net 'magnetic charge'. In terms of field lines, this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back. In other words, any magnetic field line that enters a given volume must somewhere exit that volume. Equivalent technical statements are that the sum total [[magnetic flux]] through any Gaussian surface is zero, or that the magnetic field is a [[solenoidal vector field]].

===Faraday's law===
[[File:Magnetosphere rendition.jpg|thumb|left|In a [[geomagnetic storm]], a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields in Earth's atmosphere, thus causing surges in electrical [[power grid]]s. Artist's rendition; sizes are not to scale.]]
'''[[Faraday's law of induction#Maxwell–Faraday equation|Faraday's law]]''' describes how a time varying [[magnetic field]] creates ("induces") an [[electric field]].<ref name=VideoGlossary/> This dynamically induced electric field has closed field lines just as the magnetic field, if not superposed by a static (charge induced) electric field. This aspect of [[electromagnetic induction]] is the operating principle behind many [[electric generator]]s: for example, a rotating [[bar magnet]] creates a changing magnetic field, which in turn generates an electric field in a nearby wire. (Note: there are two closely related equations which are called Faraday's law. The form used in Maxwell's equations is always valid but more restrictive than that originally formulated by [[Michael Faraday]].)

===Ampère's law with Maxwell's correction===

[[Image:Magnetic core.jpg|right|thumb|250|[[An Wang]]'s [[magnetic core memory]] (1954) is an application of [[Ampère's law]]. Each [[Magnetic core|core]] stores one [[bit]] of data.]]
'''[[Ampère's circuital law|Ampère's law with Maxwell's correction]]''' states that magnetic fields can be generated in two ways: by [[electrical current]] (this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's correction").

Maxwell's correction to Ampère's law is particularly important: it shows that not only does a changing magnetic field induce an electric field, but also a changing electric field induces a magnetic field.<ref name=VideoGlossary/><ref>[http://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> Therefore, these equations allow self-sustaining "[[electromagnetic waves]]" to travel through empty space (see [[electromagnetic wave equation]]).

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>\scriptstyle{1/\sqrt{\mu_0\varepsilon_0}}</math>, 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 [http://books.google.com/books?id=uwgNAtqSHuQC&pg=PA115 The story of electrical and magnetic measurements: from 500 B.C. to the 1940s, by Joseph F. Keithley, p115]</ref> exactly 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]].

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

[[File:Electromagneticwave3D.gif|thumb|This 3D diagram shows a plane linearly polarized wave propagating from left to right with the same wave equations where {{nowrap|1='''E''' = ''E''0 sin(−ω''t'' + '''k''' ⋅ '''r''')}} and {{nowrap|1='''B''' = ''B''0 sin(−ω''t'' + '''k''' ⋅ '''r''')}}]]

In a region with no charges (''ρ'' {{=}} 0) and no currents ('''J''' {{=}} '''0'''), such as in a vacuum, Maxwell's equations reduce to:

:<math>\begin{align}
\nabla \cdot \mathbf{E} &= 0 \quad
&\nabla \times \mathbf{E} = \ -&\frac{\partial\mathbf B}{\partial t},
\\
\nabla \cdot \mathbf{B} &= 0 \quad
&\nabla \times \mathbf{B} = \frac{1}{c^2} &\frac{\partial \mathbf E}{\partial t}.
\end{align}</math>

Taking the curl (∇×) of the curl equations, and using the [[Vector calculus identities#Curl of the curl|curl of the curl identity]] ∇×(∇×'''X''') = ∇(∇·'''X''') − ∇2'''X''' we obtain the [[wave equation]]s

:<math>
\frac{1}{c^2}\frac{\partial^2 \mathbf E}{\partial t^2} - \nabla^2 \mathbf E = 0\,, \quad
\frac{1}{c^2}\frac{\partial^2 \mathbf B}{\partial t^2} - \nabla^2 \mathbf B = 0\,,
</math>

which identify

:<math>c = \frac{1}{\sqrt{ \mu_0 \varepsilon_0}} = 2.99792458 \times 10^8 \, \mathrm{m~s}^{-1} </math>

with the [[speed of light]] in free space. In materials with [[relative permittivity]] ''εr'' and [[Permeability (electromagnetism)#Relative permeability and magnetic susceptibility|relative permeability]] ''μr'', the [[phase velocity]] of light becomes

:<math>v_p = \frac{1}{\sqrt{ \mu_0\mu_r \varepsilon_0\varepsilon_r }} </math>

which is usually less than ''c''.

In addition, '''E''' and '''B''' are mutually perpendicular to each other and 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 correction to Ampère's law]]. This perpetual cycle allows these waves, now known as [[electromagnetic radiation]], to move through space at velocity ''c''.

=="Microscopic" versus "macroscopic" ==

The ''microscopic'' variant of Maxwell's equation expresses the electric '''E''' field and the magnetic '''B''' field in terms of the ''total charge'' and total ''current'' present including the charges and currents at the atomic level. It is sometimes called the general form of Maxwell's equations or "Maxwell's equations in a vacuum". The macroscopic variant of Maxwell's equation is equally general, however, with the difference being one of bookkeeping.

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

:{| class="wikitable" style="text-align: center;"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | [[Integral]] equations
! scope="col" | [[Partial differential equation|Differential]] equations
|-
! scope="row" | [[Gauss's law]]
| {{oiint
| intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
| integrand = <math>\mathbf{D}\cdot\mathrm{d}\mathbf{S} = \iiint_\Omega \rho_\mathrm{f} \,\mathrm{d}V</math>
}}
| <math>\nabla \cdot \mathbf{D} = \rho_\mathrm{f}</math>
|-
! scope="row" | [[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>
|-
! scope="row" | 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>
|-
! scope="row" | [[Ampère's circuital law]] (with Maxwell's correction)
| <math>\oint_{\partial \Sigma} \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = \iint_{\Sigma} \left( \mathbf{J}_\mathrm{f} + \frac{\partial \mathbf D}{\partial t} \right) \cdot \mathrm{d}\mathbf{S} </math>
| <math>\nabla \times \mathbf{H} = \mathbf{J}_\mathrm{f} + \frac{\partial \mathbf{D}} {\partial t}</math>
|-
|}

Unlike the "microscopic" equations, the "macroscopic" equations factor out the bound charge ''Q''b and current ''I''b to obtain equations that depend only on the free charges ''Q''f and currents ''I''f. This factorization can be made by splitting the total electric charge and current as follows:

:<math>Q = Q_\mathrm{f} + Q_\mathrm{b} = \iiint_\Omega \left(\rho_\mathrm{f} + \rho_\mathrm{b} \right) \, \mathrm{d}V = \iiint_\Omega \rho \,\mathrm{d}V </math>

:<math>I = I_\mathrm{f} + I_\mathrm{b} = \iint_\Sigma \left(\mathbf{J}_\mathrm{f} + \mathbf{J}_\mathrm{b} \right) \cdot \mathrm{d}\mathbf{S} = \iint_\Sigma \mathbf{J} \cdot \mathrm{d}\mathbf{S} </math>

The cost of this factorization is that additional fields, the [[electric displacement field|displacement field]] '''D''' and the [[magnetizing field]]-'''H''', are defined that need to be determined. Phenomenological constituent equations relate the additional fields to the electric field '''E''' and the magnetic '''B'''-field, often through a simple linear relation.

For a detailed description of the differences between the microscopic (''total'' charge and current including material contributes or 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 macroscopic (''free'' charge and current; practical to use on materials) variants of Maxwell's equations, see below.

===Bound charge and current===
{{Main|Current density|Polarization density#Bound charge|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]] '''P''' of the material, its dipole moment per unit volume. If '''P''' is uniform, a macroscopic separation of charge is produced only at the surfaces where '''P''' enter and leave the material. For non-uniform '''P''', a charge is also produced in the bulk.<ref>See {{cite book|author=David J. Griffiths|title=Introduction to Electrodynamics|edition=third|section=4.2.2|publisher=[[Prentice Hall]]|year=1999}} for a good description of how '''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 magnetic moment is traveling a large distance. These ''[[Bound current#Magnetization current|bound currents]]'' can be described using the [[magnetization]] '''M'''.<ref>See {{cite book|author=David J. Griffiths|title=Introduction to Electrodynamics|edition=third|section=6.2.2|publisher=[[Prentice Hall]]|year=1999}} for a good description of how '''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 '''P''' and '''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, the ''Maxwell's macroscopic equations'' ignores 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]]'' (not constitutive relations) of the auxiliary fields are:
:<math>\mathbf{D}(\mathbf{r}, t) = \varepsilon_0 \mathbf{E}(\mathbf{r}, t) + \mathbf{P}(\mathbf{r}, t)</math>
:<math>\mathbf{H}(\mathbf{r}, t) = \frac{1}{\mu_0} \mathbf{B}(\mathbf{r}, t) - \mathbf{M}(\mathbf{r}, t),</math>

where '''P''' is the [[polarization density|polarization]] field and '''M''' is the [[magnetization]] field which are defined in terms of microscopic bound charges and bound current respectively. The macroscopic bound charge density ''ρ''b and bound current density '''J'''b in terms of [[polarization density|polarization]] '''P''' and [[magnetization]] '''M''' are then defined as

::<math>\rho_b = -\nabla\cdot\mathbf{P},</math>
::<math>\mathbf{J}_b = \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}.</math>

If we define the free, bound, and total charge and current density by

::<math>\rho = \rho_\mathrm{b} + \rho_\mathrm{f}, \ </math>
::<math>\mathbf{J} = \mathbf{J}_\mathrm{b} + \mathbf{J}_\mathrm{f},</math>

and use the defining relations above to eliminate '''D''', and '''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]] '''D''' and the electric field '''E''', as well as the [[Magnetic_field#H-field_and_magnetic_materials|magnetizing]] field '''H''' and the magnetic field '''B'''. Equivalently, we have to specify the dependence of the polarisation '''P''' (hence the bound charge) and the magnetisation '''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 for a fuller description.

For materials without polarisation and magnetisation ("vacuum"), the constitutive relations are
:<math>\mathbf{D} = \varepsilon_0\mathbf{E}, \quad \mathbf{H} = \mathbf{B}/\mu_0</math>
for scalar constants <math>\varepsilon_0</math> and <math>\mu_0</math>. Since there is no bound charge, the total and the free charge and current are equal.

More generally, for linear materials the constitutive relations are
:<math>\mathbf{D} = \varepsilon\mathbf{E}\,,\quad \mathbf{H} = \mu^{-1}\mathbf{B} </math>
where ε is the [[permittivity]] and μ the [[permeability (electromagnetism)|permeability]] of the material. Even the linear case can have various complications, however.
*For homogeneous materials, ε and μ are constant throughout the material, while for inhomogeneous materials they depend on [[position vector|location]] within the material (and perhaps time).
*For isotropic materials, ε and μ are scalars, while for anisotropic materials (e.g. due to crystal structure) they are [[tensor]]s.
*Materials are generally [[dispersion (optics)|dispersive]], so ε and μ depend on the [[frequency]] of any incident EM waves.

Even more generally, in the case of non-linear materials (see for example [[nonlinear optics]]), '''D''' and '''P''' are not necessarily proportional to '''E''', similarly '''B''' is not necessarily proportional to '''H''' or '''M'''. In general '''D''' and '''H''' depend on both '''E''' and '''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 '''E''' and '''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 [[Ohms law]] in the form <math>\mathbf J_f = \sigma \mathbf E</math>.

==Equations in Gaussian units==
{{main|Gaussian units}}
Gaussian units are a popular system of units, that is part of the [[centimetre–gram–second system of units]] (cgs). When using cgs units it is conventional to use a slightly different definition of electric field '''E'''cgs = ''c''−1 '''E'''SI. This implies that the modified electric and magnetic field have the same units (in the SI convention this is not the case: e.g. for EM waves in vacuum, |'''E'''SI| = ''c''|'''B'''|, making [[dimensional analysis]] of the equations different). Then it uses a unit of charge [[Centimetre gram second system of units#Alternate derivations of CGS units in electromagnetism|defined]] in such a way that the permittivity of the vacuum ''ε''0 = 1/(4''πc''), hence ''μ''0 = 4''π''/''c''.
Using these different conventions, the Maxwell equations become:<ref name=Littlejohn>
{{cite web
| url=http://bohr.physics.berkeley.edu/classes/221/0708/notes/emunits.pdf
| format=PDF
| title=Gaussian, SI and Other Systems of Units in Electromagnetic Theory
| work=Physics 221A, University of California, Berkeley lecture notes
| author=Littlejohn, Robert
| date=Fall 2007
| accessdate=2008-05-06
}}</ref>
:{| class="wikitable" style="text-align: center;"
|+ Equations in Gaussian units
|-
! scope="col" style="width: 13em;" | Name
! Microscopic equations
! Macroscopic equations
|-
| [[Gauss's law]]
|<math>\nabla \cdot \mathbf{E} = 4\pi\rho </math>
|<math> \nabla \cdot \mathbf{D} = 4\pi\rho_\mathrm{f}</math>
|-
| [[Gauss's law for magnetism]]
| <math>\nabla \cdot \mathbf{B} = 0</math>
| same as microscopic
|-
| Maxwell–Faraday equation ([[Faraday's law of induction]])
| <math>\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}</math>
| same as microscopic
|-
| [[Ampère's law]] (with Maxwell's extension)
|<math>\nabla \times \mathbf{B} = \frac{1}{c} \left(4\pi\mathbf{J} + \frac{\partial \mathbf{E}}{\partial t} \right)</math>
|<math> \nabla \times \mathbf{H} = \frac{1}{c} \left(4\pi\mathbf{J}_\mathrm{f} + \frac{\partial \mathbf{D}} {\partial t} \right)</math>
|}

==Alternative formulations==


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

Following is a summary of some of the numerous other ways to write the microscopic Maxwell's equations, showing they can be formulated using different points of view and mathematical formalisms that describe the same physics. Often, they are also called the Maxwell equations. The direct space-time formulations make manifest that the Maxwell equations are relativistically invariant (in fact studying the hidden symmetry of the vector calculus formulation was a major source of inspiration for relativity theory). In addition, the formulation using potentials was originally introduced as a convenient way to solve the equations but with all the observable physics contained in the fields. The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the fields vanish ([[Aharonov–Bohm effect]]). See the main articles for the details of each formulation. SI units are used throughout.

:{|class="wikitable" style="text-align: center;"
|-
!scope="column" width="160px"|Formalism
!|Formulation
!| Homogeneous equations
!| Non-homogeneous equations
|-
! scope="row" rowspan="3" |[[Vector calculus]]
!Fields
3D Euclidean space + time
||<math>\nabla\cdot\mathbf{B}=0</math>
<math>\nabla\times\mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=0</math>
||<math>\nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_0}</math>
<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>
<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>
<math>\Box\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>
<math>\mathbf E = - \mathbf \nabla \varphi - \frac{\partial \mathbf A}{\partial t}</math>

<math>\mathbf \nabla \cdot \mathbf A + \frac{1}{c^2}\frac{\partial \varphi}{\partial t} = 0</math>
||<math>\Box \varphi = \frac{\rho}{\varepsilon_0}</math>
<math>\Box\mathbf A = \mu_0 \mathbf J</math>
|-
! scope="row" rowspan="6" |[[Tensor calculus]]
![[Covariant formulation of classical electromagnetism#Maxwell's equations in vacuo|Fields]]
[[Minkowski space]]
||<math>\partial_{[\alpha} F_{\beta\gamma]}= 0 </math>
||<math>\partial_\alpha F^{\beta\alpha} = \mu_0 J^\beta </math>
|-
!Potentials (any gauge)
[[Minkowski space]]
||<math>F_{\alpha\beta} = \partial_{[\alpha} A_{\beta]}</math>
||<math>\partial_\alpha \partial^{[\beta} A^{\alpha]} = \mu_0 J^\beta</math>
|-
!Potentials (Lorenz gauge)
[[Minkowski space]]
||<math>F_{\alpha\beta} = \partial_{[\alpha} A_{\beta]}</math>
<math>\partial_\alpha A^\alpha = 0</math>
||<math>\Box A^\alpha = -\mu_0 J^\beta</math>
|-
!Fields
any space-time
||<math>\partial_{[\alpha} F_{\beta\gamma]}= \nabla_{[\alpha} F_{\beta\gamma]} = 0 </math>
||<math>\nabla_\alpha (\sqrt{-g} F^{\beta\alpha}) = \mu_0 J^\beta </math>
|-
!Potentials (any gauge)
any space-time
||<math> F_{\alpha\beta} = \partial_{[\alpha} A_{\beta]} = \nabla_{[\alpha} A_{\beta]}</math>
||<math> \nabla_\alpha (\sqrt{-g}\nabla^{[\beta} A^{\alpha]} ) = \mu_0 J^\beta</math>
|-
!Potentials (Lorenz gauge)
any space-time
||<math> F_{\alpha\beta} = \partial_{[\alpha} A_{\beta]} = \nabla_{[\alpha} A_{\beta]},</math>
<math>\nabla_\alpha A^{\alpha} = 0</math>
||<math> \Box A^{\alpha} - R^{\alpha}_{\ \beta} A^\beta = -\mu_0 J^\alpha</math>
|-
! scope="row" rowspan="3" |[[Exterior calculus|Differential forms]]
!Fields
any space-time
||<math>\mathrm{d} F = 0</math>

||<math>\mathrm{d} * F = \mu_0 J </math>
|-
!Potentials (any gauge)
any space-time
||<math>F = \mathrm{d} A</math>
||<math>\mathrm{d} * \mathrm{d} A = \mu_0 J </math>
|-
!Potentials (Lorenz gauge)
any space-time
||<math>F = \mathrm{d} A</math>
<math> \mathrm{d} \star A = 0 </math>
||<math>\star \Box A = \mu_0 J </math>
|-

|}

where
*In the vector formulation on Euclidean space + time, <math>\varphi</math> is the [[electrical potential]], <math>\mathbf A</math> is the [[vector potential]] and <math>\Box = \frac{1}{c^2} \frac{\partial^2} {\partial t^2}-\nabla^2</math> is the [[D'Alembert operator]].
*In the tensor calculus formulation , the [[electromagnetic tensor]] <math>F_{\alpha\beta}</math> is an antisymmetric covariant rank 2 tensor, the [[four-potential]] <math>A_\alpha</math> is a covariant vector, the current <math>J^\alpha</math> is a vector density, the square bracket [ ] denotes [[Ricci calculus#Symmetric and antisymmetric parts|antisymmetrization of indices]], <math>\partial_\alpha</math> is the derivative with respect to the coordinate <math>x^\alpha</math>. On Minkowski space coordinates are chosen with respect to an [[inertial frame]]; <math>(x^\alpha) = (ct, x, y,z)</math>, so that the [[metric tensor]] used to raise and lower indices is <math>\eta_{\alpha\beta} = \mathrm{diag}(1,-1,-1,-1)</math>. The D'Alembert operator on Minkowskispace is <math>\Box = \partial_\alpha\partial^\alpha </math> as in the vector formulation. On general space-times, the coordinate system <math>x^\alpha</math> is arbitrary, the [[Levi-Civita connection|covariant derivative]] <math>\nabla_\alpha</math>, the Ricci tensor <math>R_{\alpha\beta}</math> and raising and lowering of indices are defined by the Lorentzian metric <math>g_{\alpha\beta}</math> and the D'Alembert operator is defined as <math>\Box = \nabla_\alpha\nabla^\alpha</math>.
*In the [[differential form]] formulation on arbitrary space times, <math>F = F_{\alpha\beta}dx^\alpha\wedge dx^\beta</math> is the electromagnetic tensor considered as two form, <math>A = A_\alpha dx^\alpha</math> is the potential 1 form, <math>J</math> is the current (pseudo) 3 form, d is the [[exterior derivative]], and <math>*,\star</math> are the [[Hodge star]]'s on forms defined by the Lorentzian metric of space-time (the Hodge star <math>*</math> on two forms only depends on the metric up to a local scale i.e. is [[conformal geometry|conformally invariant ]]).  The operator <math>\Box = (-\star \mathrm{d} * \mathrm{d} + \mathrm{d} \star \mathrm{d} \star) </math> is the [[Laplace-Beltrami operator|d'Alembert-Laplace-Beltrami operator]] on 1-forms.

Other formulations include the [[Geometric Algebra#Electrodynamics and special relativity|geometric algebra formulation]] and a [[matrix representation of Maxwell's equations]]. Historically, a [[quaternion]]ic formulation<ref>{{cite article|title=Physical Space as a Quaternion Structure I: Maxwell Equations. A Brief Note.|author=P.M. Jack|location=Toronto, Canada|year=2003|url=http://arxiv.org/abs/math-ph/0307038|arxiv=math-ph/0307038}}</ref><ref>{{cite article|title=On the Notation of Maxwell's Field Equations|author=A. Waser|location=|year=2000|publisher=AW-Verlag|url=http://www.zpenergy.com/downloads/Orig_maxwell_equations.pdf|arXiv=}}</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. In fact, the solutions of these equations encompass all the diverse phenomena in the entire field of [[classical electromagnetism]]. A thorough discussion is far beyond the scope of the article, but some general notes follow.

Like any differential equation, [[boundary conditions]]<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=0-19-850888-3
|url=http://books.google.com/?id=zI7Y1jT9pCwC&pg=PA1&dq=electromagnetism+%22boundary+conditions%22
|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=0-85296-849-3
|url=http://books.google.com/?id=eOofBpuyuOkC&pg=PA261&dq=electromagnetism+%22boundary+conditions%22
|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=0-387-94874-0
|url=http://books.google.com/?id=EdZefkIOR5cC&pg=PA1&dq=electromagnetism+%22boundary+conditions%22
|year=1997
}}</ref> and [[initial conditions]]<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=981-02-1689-0
|url=http://books.google.com/?id=6_CZBHzfhpMC&pg=PA45&dq=electromagnetism+%22initial+conditions%22
|year=1994
}}</ref> are necessary for a unique solution. For example, even with no charges and no currents anywhere in spacetime, many solutions to Maxwell's equations are possible, not just the obvious solution '''E''' = '''B''' = '''0'''. Another solution is '''E''' = constant, '''B''' = constant, while yet other solutions have electromagnetic waves filling spacetime. In some cases, Maxwell's equations are solved through infinite 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=0-486-42567-3
|url=http://books.google.com/?id=bI-ZmZWeyhkC&pg=RA1-PA335&dq=electromagnetism+infinity+boundary+conditions
}}</ref> In other cases, Maxwell's equations are solved in just a finite region of space, with appropriate boundary conditions on that region: For example, the boundary could be 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=3-540-24431-X
|url=http://books.google.com/?id=vSszZ2WuG_IC&pg=PA84&dq=electromagnetism+boundary++-element
|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 (as with a [[waveguide]] or cavity [[resonator]]) the boundary conditions may describe the walls that isolate a small region from the outside world.<ref name=" Mahmoud">
{{cite book
|author=S. F. Mahmoud
|title=Electromagnetic Waveguides: Theory and Applications applications
|page =Chapter 2
|publisher=Institution of Electrical Engineers
|location=London UK
|isbn=0-86341-232-7
|url=http://books.google.com/?id=toehQ7vLwAMC&pg=PA2&dq=Maxwell%27s+equations+waveguides
|nopp=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. Jefimenko's equations are not so helpful 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 approximately solve Maxwell's equations when an exact solution is impossible. These methods usually require a computer, and 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=0-7803-3425-6
|url=http://books.google.com/?id=55q7HqnMZCsC&pg=PA79&dq=electromagnetism+%22boundary+conditions%22
}}</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=0-486-66444-9
|url=http://www.amazon.com/Principles-Techniques-Applied-Mathematics-Friedman/dp/0486664449/ref=sr_1_1?ie=UTF8&s=books&qisbn=1207010487&sr=1-1
}}</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=1-58053-832-0
|url=http://www.amazon.com/gp/reader/1580538320/ref=sib_dp_pop_toc?ie=UTF8&p=S008#reader-link
|nopp=true
}}</ref> For more details, see [[Computational electromagnetics]].

Maxwell's equations ''seem'' [[Overdetermined system|overdetermined]], in that they involve six unknowns (the three components of '''E''' and '''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.<ref>{{cite book|author=H Freistühler & G Warnecke |title=Hyperbolic Problems: Theory, Numerics, Applications |year=2001 |page=605 |url=http://books.google.com/books?id=XXX_mG0vneMC&pg=PA605#v=onepage&q&f=false}}</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 }}</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 }}</ref>

==Limitations for a theory of electromagnetism==

While Maxwell's equations (along with the rest of classical electromagnetism) are extraordinarily successful at explaining and predicting a variety of phenomena, they are not exact laws of the universe, but merely approximations. In some special situations, they can be noticeably inaccurate. Examples include extremely strong fields (see [[Euler–Heisenberg Lagrangian]]) and extremely short distances (see [[vacuum polarization]]). Moreover, various phenomena occur in the world even though Maxwell's equations predicts them to be impossible, such as "[[nonclassical light]]" and [[quantum entanglement]] of electromagnetic fields (see [[quantum optics]]). Finally, any phenomenon involving individual [[photon]]s, such as the [[photoelectric effect]], [[Planck's law]], the [[Duane–Hunt law]], [[Single-photon avalanche diode|single-photon light detectors]], etc., would be difficult or impossible to explain if Maxwell's equations were exactly true, as Maxwell's equations do not involve photons. For the most accurate predictions in all situations, Maxwell's equations have been superseded by [[quantum electrodynamics]].

==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] and [http://www.nature.com/nature/journal/v461/n7266/full/nature08500.html].) 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 '''∇⋅B'''≠0, whereas in these condensed-matter systems, '''∇⋅B'''=0 while only '''∇⋅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>{{cite book|author=J.D. Jackson|title=Classical Electrodynamics|edition=3rd|chapter=6.11|isbn=0-471-43132-X}}</ref><ref>{{cite web|url=http://www.ieeeghn.org/wiki/index.php/Maxwell%27s_Equations |title=IEEEGHN: Maxwell's Equations |publisher=Ieeeghn.org |date= |accessdate=2008-10-19}}</ref>

[[File:Illustration of Maxwell's equations in relation to Genesis 1,3.jpeg|thumbnail|right|And God Said... Maxwell's Equations Tshirt]]
==Popular culture==

Paraphernalia, such as sweatshirts or T-shirts with "And God Said", followed by the Maxwell's equations, are extremely popular among physicists, and geek-types, because of the elegance of these equations that provide a bridge between classical physics and religion. The phrase refers to [[Genesis 1:3]]: and God Said "[[let there be light]]" and there was light, whereas the equations represent the essence of light, which is a form of electromagnetism.

A T-shirt similar to the one pictured was worn by "Alex" in the American Sitcom Modern Family's [[After_the_Fire_(episode)|"After the Fire"]] episode.

==See also==
{{Wikipedia books|Maxwell's equations}}
{{columns-list|2|
* [[Algebra of physical space]]
* [[Fresnel equations]]
* [[Gravitomagnetism]]
* [[Interface conditions for electromagnetic fields]]
* [[Moving magnet and conductor problem]]
* [[Riemann–Silberstein vector|Riemann–Silberstein multivector]]
* [[Spacetime algebra]]
* [[Wheeler–Feynman absorber theory]]
}}

==Notes==

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

==References==

{{Reflist|2}}

:''Further reading can be found in [[list of textbooks in electromagnetism]]''

==Historical publications==

* [[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.)
The developments before relativity
* [[Joseph Larmor]] (1897) "On a dynamical theory of the electric and luminiferous medium", ''Phil. Trans. Roy. Soc.'' '''190''', 205–300 (third and last in a series of papers with the same name).
* [[Hendrik Lorentz]] (1899) "Simplified theory of electrical and optical phenomena in moving systems", ''Proc. Acad. Science Amsterdam'', '''I''', 427–43.
* [[Hendrik Lorentz]] (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", ''Proc. Acad. Science Amsterdam'', '''IV''', 669–78.
* [[Henri Poincaré]] (1900) "La théorie de Lorentz et le Principe de Réaction", ''Archives Néerlandaises'', '''V''', 253–78.
* [[Henri Poincaré]] (1902) ''La Science et l'Hypothèse''
* [[Henri Poincaré]] (1905) [http://www.soso.ch/wissen/hist/SRT/P-1905-1.pdf "Sur la dynamique de l'électron"], ''Comptes rendus de l'Académie des Sciences'', '''140''', 1504–8.

*[http://www.antiquebooks.net/readpage.html#maxwell James Clerk Maxwell, A Treatise on Electricity And Magnetism Vols 1 and 2] 1904—most readable edition with all corrections—Antique Books Collection suitable for free reading online.
*[http://posner.library.cmu.edu/Posner/books/book.cgi?call=537_M46T_1873_VOL._1 Maxwell, J.C., A Treatise on Electricity And Magnetism – Volume 1 – 1873] – Posner Memorial Collection – Carnegie Mellon University
*[http://posner.library.cmu.edu/Posner/books/book.cgi?call=537_M46T_1873_VOL._2 Maxwell, J.C., A Treatise on Electricity And Magnetism – Volume 2 – 1873] – Posner Memorial Collection – Carnegie Mellon University
*[http://blazelabs.com/On%20Faraday's%20Lines%20of%20Force.pdf On Faraday's Lines of Force – 1855/56] Maxwell's first paper (Part 1 & 2) – Compiled by Blaze Labs Research (PDF)
*[[Media:On Physical Lines of Force.pdf|On Physical Lines of Force – 1861]] Maxwell's 1861 paper describing magnetic lines of Force – Predecessor to 1873 Treatise
* Maxwell, James Clerk, "''[[:File:A Dynamical Theory of the Electromagnetic Field.pdf|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.)
*[http://www.electromagnetism.demon.co.uk/z014.htm Catt, Walton and Davidson. "The History of Displacement Current". ''Wireless World'', March 1979.]
* Reprint from Dover Publications (ISBN 0-486-60636-8)
* [http://www.antiquebooks.net/readpage.html#maxwell Full text of 1904 Edition including full text search.]
*[http://books.google.com/books?id=5HE_cmxXt2MC&vid=02IWHrbcLC9ECI_wQx&dq=Proceedings+of+the+Royal+Society+Of+London+Vol+XIII&ie=UTF-8&jtp=531 A Dynamical Theory Of The Electromagnetic Field – 1865] Maxwell's 1865 paper describing his 20 Equations in 20 Unknowns – Predecessor to the 1873 Treatise

==External links==
* {{springer|title=Maxwell equations|id=p/m063140}}
* [http://www.maxwells-equations.com maxwells-equations.com] — An intuitive tutorial of Maxwell's equations.
* Mathematical aspects of Maxwell's equation are discussed on the [http://tosio.math.toronto.edu/wiki/index.php/Main_Page Dispersive PDE Wiki].

===Modern treatments===
* [http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html Electromagnetism], B. Crowell, Fullerton College
* [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].
* [http://ocw.mit.edu/OcwWeb/Physics/8-02Electricity-and-MagnetismSpring2002/VideoAndCaptions/index.htm MIT Video Lecture Series (36 x 50 minute lectures) (in .mp4 format) – Electricity and Magnetism] Taught by Professor [[Walter Lewin]].

===Other===
*[http://uk.arxiv.org/abs/hep-ph/0106235 Feynman's derivation of Maxwell equations and extra dimensions]
*[http://www.nature.com/milestones/milephotons/full/milephotons02.html ''Nature Milestones: Photons'' – ''Milestone 2 (1861) Maxwell's equations'']

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