Maxwell's equations, orMaxwell–Heaviside equations, are a set of coupledpartial differential equations that, together with theLorentz force law, form the foundation ofclassical electromagnetism, classicaloptics,electric andmagnetic 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 howelectric andmagnetic fields are generated bycharges,currents, and changes of the fields.^{[note 1]} The equations are named after the physicist and mathematicianJames 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 toOliver Heaviside.^[1]

Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum,c (299792458 m/s^[2]). Known aselectromagnetic radiation, these waves occur at various wavelengths to produce aspectrum of radiation fromradio waves togamma rays.

Inpartial differential equation form and acoherent system of units, Maxwell's microscopic equations can be written as (top to bottom: Gauss's law, Gauss's law for magnetism, Faraday's law, Ampère-Maxwell law)With${\displaystyle \mathbf{E}}$ the electric field,${\displaystyle \mathbf{B}}$ the magnetic field,${\displaystyle \rho }$ theelectric charge density and${\displaystyle \mathbf{J}}$ thecurrent density.${\displaystyle {\epsilon }_0}$ is thevacuum permittivity and${\displaystyle {\mu }_0}$ thevacuum permeability.

The equations have two major variants:

• Themicroscopic 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 theatomic scale.
• Themacroscopic 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 forequivalent alternative formulations. Versions of Maxwell's equations based on theelectric andmagnetic scalar potentials are preferred for explicitly solving the equations as aboundary value problem,analytical mechanics, or for use inquantum mechanics. Thecovariant formulation (onspacetime rather than space and time separately) makes the compatibility of Maxwell's equations withspecial relativitymanifest.Maxwell's equations in curved spacetime, commonly used inhigh-energy andgravitational physics, are compatible withgeneral relativity.^{[note 2]} 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 theunification 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 aclassical limit of the more precise theory ofquantum electrodynamics.

Gauss's law describes the relationship between anelectric field andelectric charges: an electric field points away from positive charges and towards negative charges, and the netoutflow of the electric field through aclosed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is thepermittivity of free space.

Gauss's law for magnetism states that electric charges have no magnetic analogues, calledmagnetic monopoles; no north or south magnetic poles exist in isolation.^[3] Instead, the magnetic field of a material is attributed to adipole, 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 totalmagnetic flux through a Gaussian surface is zero, and the magnetic field is asolenoidal vector field.^{[note 3]}

TheMaxwell–Faraday version ofFaraday's law of induction describes how a time-varyingmagnetic field corresponds tocurl of anelectric field.^[3] 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.

Theelectromagnetic induction is the operating principle behind manyelectric generators: for example, a rotatingbar magnet creates a changing magnetic field and generates an electric field in a nearby wire.

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

Maxwell's modification of Ampère's circuital law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields.^{[4][clarification needed]} As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field.^[3][5] A further consequence is the existence of self-sustainingelectromagnetic waves whichtravel through empty space.

The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents,^{[note 4]} matches thespeed of light; indeed,lightis one form ofelectromagnetic radiation (as areX-rays,radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories ofelectromagnetism andoptics.

In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separatelaw of nature, theLorentz force law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. Thevector calculus formalism below, the work ofOliver Heaviside,^[6][7] has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations inx,y andz components. Therelativistic formulations are more symmetric and Lorentz invariant. For the same equations expressed using tensor calculus or differential forms (see§ 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 purelylocal and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example usingfinite element analysis.^[8]

Symbols inbold representvector quantities, and symbols initalics representscalar quantities, unless otherwise indicated.The equations introduce theelectric field,E, avector field, and themagnetic field,B, apseudovector field, each generally having a time and location dependence.The sources are

• the total electriccharge density (total charge per unit volume),ρ, and
• the total electriccurrent density (total current per unit area),J.

Theuniversal constants appearing in the equations (the first two ones explicitly only in the SI formulation) are:

• thepermittivity of free space,ε₀, and
• thepermeability of free space,μ₀, and
• thespeed of light,${\displaystyle c=({\epsilon }_0{\mu }_0{)}^{-1/2}}$ 

In the differential equations,

• thenabla symbol,∇, denotes the three-dimensionalgradient operator,del,
• the∇⋅ symbol (pronounced "del dot") denotes thedivergence operator,
• the∇× symbol (pronounced "del cross") denotes thecurl operator.

In the integral equations,

• Ω is any volume with closedboundary surface∂Ω, and
• Σ is any surface with closed boundary curve∂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 thedifferentiation under the integral sign in Faraday's law:$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}{\iint }_{\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}={\iint }_{\mathrm{\Sigma }}\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\cdot \mathrm{d}\mathbf{S},}$$ Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss' and Stokes' theorems as appropriate.

•  ${\displaystyle {\phantom{\int }}_{\mathrm{\partial }\mathrm{\Omega }}}$  is asurface integral over the boundary surface∂Ω, with the loop indicating the surface is closed
• ${\displaystyle {\iiint }_{\mathrm{\Omega }}}$  is avolume integral over the volumeΩ,
• ${\displaystyle {\oint }_{\mathrm{\partial }\mathrm{\Sigma }}}$  is aline integral around the boundary curve∂Σ, with the loop indicating the curve is closed.
• ${\displaystyle {\iint }_{\mathrm{\Sigma }}}$  is asurface integral over the surfaceΣ,
• Thetotalelectric chargeQ enclosed inΩ is thevolume integral overΩ of thecharge densityρ (see the "macroscopic formulation" section below):$${\displaystyle Q={\iiint }_{\mathrm{\Omega }}\rho \mathrm{d}V,}$$  wheredV is thevolume element.
• Thenetmagnetic fluxΦ_B is thesurface integral of the magnetic fieldB passing through a fixed surface,Σ:$${\displaystyle {\mathrm{\Phi }}_B={\iint }_{\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathbf{S},}$$ 
• Thenetelectric fluxΦ_E is the surface integral of the electric fieldE passing throughΣ:$${\displaystyle {\mathrm{\Phi }}_E={\iint }_{\mathrm{\Sigma }}\mathbf{E}\cdot \mathrm{d}\mathbf{S},}$$ 
• Thenetelectric currentI is the surface integral of theelectric current densityJ passing throughΣ:$${\displaystyle I={\iint }_{\mathrm{\Sigma }}\mathbf{J}\cdot \mathrm{d}\mathbf{S},}$$  wheredS denotes the differentialvector element of surface areaS,normal to surfaceΣ. (Vector area is sometimes denoted byA rather thanS, but this conflicts with the notation formagnetic vector potential).

The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbingdimensioned factors ofε₀ andμ₀ into the units (and thus redefining these). With a corresponding change in the values of the quantities for theLorentz force law this yields the same physics, i.e. trajectories of charged particles, orwork 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 theelectromagnetic tensor: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension.^[9]: vii Such modified definitions are conventionally used with the Gaussian (CGS) units. Using these definitions, colloquially "in Gaussian units",^[10]the Maxwell equations become:^[11]

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

Further changes are possible by absorbing factors of4π. This process, called rationalization, affects whetherCoulomb's law orGauss's law includes such a factor (seeHeaviside–Lorentz units, used mainly inparticle physics).

The equivalence of the differential and integral formulations are a consequence of theGauss divergence theorem and theKelvin–Stokes theorem.

According to the (purely mathematical)Gauss divergence theorem, theelectric flux through theboundary surface∂Ω can be rewritten as

 ${\displaystyle {\phantom{\oint }}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{E}\cdot \mathrm{d}\mathbf{S}={\iiint }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot \mathbf{E}\mathrm{d}V}$ 

The integral version of Gauss's equation can thus be rewritten as$${\displaystyle {\iiint }_{\mathrm{\Omega }}\left(\mathrm{\nabla }\cdot \mathbf{E}-\frac{\rho }{{\epsilon }_0}\right)\mathrm{d}V=0}$$ SinceΩ is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfiedif and only if the integrand is zero everywhere. This is the differential equations formulation of Gauss equation up to a trivial rearrangement.

Similarly rewriting themagnetic flux in Gauss's law for magnetism in integral form gives

 ${\displaystyle {\phantom{\oint }}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}={\iiint }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot \mathbf{B}\mathrm{d}V=0.}$ 

which is satisfied for allΩ if and only if${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$  everywhere.

By theKelvin–Stokes theorem we can rewrite theline integrals of the fields around the closed boundary curve∂Σ to an integral of the "circulation of the fields" (i.e. theircurls) over a surface it bounds, i.e.$${\displaystyle {\oint }_{\mathrm{\partial }\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathit{\ell }={\iint }_{\mathrm{\Sigma }}(\mathrm{\nabla }\times \mathbf{B})\cdot \mathrm{d}\mathbf{S},}$$ Hence theAmpère–Maxwell law, the modified version of Ampère's circuital law, in integral form can be rewritten as$${\displaystyle {\iint }_{\mathrm{\Sigma }}\left(\mathrm{\nabla }\times \mathbf{B}-{\mu }_0\left(\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\right)\cdot \mathrm{d}\mathbf{S}=0.}$$ SinceΣ can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that the integrand is zeroif and only if the Ampère–Maxwell 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 classicalfluid dynamics: thecirculation of a fluid is the line integral of the fluid'sflow velocity field around a closed loop, and thevorticity of the fluid is the curl of the velocity field.

The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the Ampère–Maxwell law has zero divergence by thediv–curl identity. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives:$${\displaystyle 0=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{B})=\mathrm{\nabla }\cdot \left({\mu }_0\left(\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\right)={\mu }_0\left(\mathrm{\nabla }\cdot \mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\nabla }\cdot \mathbf{E}\right)={\mu }_0\left(\mathrm{\nabla }\cdot \mathbf{J}+\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}\right)}$$ i.e.,$${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{J}=0.}$$ 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:

${\displaystyle \frac{d}{dt}{Q}_{\mathrm{\Omega }}=\frac{d}{dt}{\iiint }_{\mathrm{\Omega }}\rho \mathrm{d}V=-}$  ${\displaystyle {\phantom{\oint }}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{J}\cdot \mathrm{d}\mathbf{S}=-I_{\mathrm{\partial }\mathrm{\Omega }}.}$ 

In particular, in an isolated system the total charge is conserved.

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

Taking the curl(∇×) of the curl equations, and using thecurl of the curl identity we obtain$${\displaystyle \begin{array}{r}{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{E}=0,\\ {\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{B}=0.\end{array}}$$ 

The quantity${\displaystyle {\mu }_0{\epsilon }_0}$  has the dimension (T/L)². Defining${\displaystyle c=({\mu }_0{\epsilon }_0{)}^{-1/2}}$ , the equations above have the form of the standardwave equations$${\displaystyle \begin{array}{r}\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{E}=0,\\ \frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{B}=0.\end{array}}$$ 

Already during Maxwell's lifetime, it was found that the known values for${\displaystyle {\epsilon }_0}$  and${\displaystyle {\mu }_0}$  give${\displaystyle c\approx 2.998\times {10}^8\text{m/s}}$ , then already known to be thespeed of light in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In theold SI system of units, the values of${\displaystyle {\mu }_0=4\pi \times {10}^{-7}}$  and${\displaystyle c=299792458\text{m/s}}$  are defined constants, (which means that by definition${\displaystyle {\epsilon }_0=8.854187\mathrm{8...}\times {10}^{-12}\text{F/m}}$ ) that define the ampere and the metre. In thenew SI system, onlyc keeps its defined value, and the electron charge gets a defined value.

In materials withrelative permittivity,ε_r, andrelative permeability,μ_r, thephase velocity of light becomes$${\displaystyle v_{\text{p}}=\frac{1}{\sqrt{{\mu }_0{\mu }_{\text{r}}{\epsilon }_0{\epsilon }_{\text{r}}}},}$$ which is usually^{[note 5]} less thanc.

In addition,E andB are perpendicular to each other and to the direction of wave propagation, and are inphase with each other. Asinusoidal 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 throughFaraday's law. In turn, that electric field creates a changing magnetic field throughMaxwell's modification of Ampère's circuital law. This perpetual cycle allows these waves, now known aselectromagnetic radiation, to move through space at velocityc.

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 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.^[12]: 5

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

In the macroscopic equations, the influence of bound chargeQ_b and bound currentI_b is incorporated into thedisplacement fieldD and themagnetizing fieldH, while the equations depend only on the free chargesQ_f and free currentsI_f. This reflects a splitting of the total electric chargeQ and currentI (and their densitiesρ andJ) into free and bound parts: 

The cost of this splitting is that the additional fieldsD andH need to be determined through phenomenological constituent equations relating these fields to the electric fieldE and the magnetic fieldB, together with the bound charge and current.

See below for a detailed description of the differences between the microscopic equations, dealing withtotal charge and current including material contributions, useful in air/vacuum;^{[note 6]}and the macroscopic equations, dealing withfree charge and current, practical to use within materials.

When an electric field is applied to adielectric material its molecules respond by forming microscopicelectric dipoles – theiratomic nuclei move a tiny distance in the direction of the field, while theirelectrons move a tiny distance in the opposite direction. This produces amacroscopicbound 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 positivebound 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 thepolarizationP of the material, its dipole moment per unit volume. IfP is uniform, a macroscopic separation of charge is produced only at the surfaces whereP enters and leaves the material. For non-uniformP, a charge is also produced in the bulk.^[13]

Somewhat similarly, in all materials the constituent atoms exhibitmagnetic moments that are intrinsically linked to theangular momentum of the components of the atoms, most notably theirelectrons. Theconnection 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. Thesebound currents can be described using themagnetizationM.^[14]

The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms ofP andM, 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.

Thedefinitions of the auxiliary fields are: whereP is thepolarization field andM is themagnetization field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge densityρ_b and bound current densityJ_b in terms ofpolarizationP andmagnetizationM are then defined as 

If we define the total, bound, and free charge and current density by and use the defining relations above to eliminateD, andH, the "macroscopic" Maxwell's equations reproduce the "microscopic" equations.

In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations betweendisplacement fieldD and the electric fieldE, as well as themagnetizing fieldH and the magnetic fieldB. Equivalently, we have to specify the dependence of the polarizationP (hence the bound charge) and the magnetizationM (hence the bound current) on the applied electric and magnetic field. The equations specifying this response are calledconstitutive relations. 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.^{[15]: 44–45}

For materials without polarization and magnetization, the constitutive relations are (by definition)^[9]: 2$${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E},\mathbf{H}=\frac{1}{{\mu }_0}\mathbf{B},}$$ whereε₀ is thepermittivity of free space andμ₀ thepermeability 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 equationstogether 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^{[15]: 44–45}$${\displaystyle \mathbf{D}=\epsilon \mathbf{E},\mathbf{H}=\frac{1}{\mu }\mathbf{B},}$$ whereε is thepermittivity andμ thepermeability of the material. For the displacement fieldD 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 10¹¹ V/m are much higher than the external field. For the magnetizing field${\displaystyle \mathbf{H}}$ , however, the linear approximation can break down in common materials like iron leading to phenomena likehysteresis. Even the linear case can have various complications, however.

• For homogeneous materials,ε andμ are constant throughout the material, while for inhomogeneous materials they depend onlocation within the material (and perhaps time).^[16]: 463
• For isotropic materials,ε andμ are scalars, while for anisotropic materials (e.g. due to crystal structure) they aretensors.^{[15]: 421 [16]: 463}
• Materials are generallydispersive, soε andμ depend on thefrequency of any incident EM waves.^{[15]: 625 [16]: 397}

Even more generally, in the case of non-linear materials (see for examplenonlinear optics),D andP are not necessarily proportional toE, similarlyH orM is not necessarily proportional toB. In generalD andH depend on bothE andB, 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 ofE andB 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 (seeHistory of Maxwell's equations) includedOhm's law in the form$${\displaystyle {\mathbf{J}}_{\text{f}}=\sigma \mathbf{E}.}$$ 

Following are some of the several other mathematical formalisms of Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of theelectrical potentialφ and thevector potentialA. 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 themain article for details of each formulation.

The direct spacetime formulations make manifest that the Maxwell equations arerelativistically invariant, where space and time are treated on equal footing. Because of this symmetry, the electric and magnetic fields are treated on equal footing and are recognized as components of theFaraday tensor. This reduces the four Maxwell equations to two, which simplifies the equations, although we can no longer use the familiar vector formulation. Maxwell equations in formulation that do not treat space and time manifestly on the same footing 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.

• In the tensor calculus formulation, theelectromagnetic tensorF_αβ is an antisymmetric covariant order 2 tensor; thefour-potential,A_α, is a covariant vector; the current,J^α, is a vector; the square brackets,[ ], denoteantisymmetrization of indices;∂_α is the partial derivative with respect to the coordinate,x^α. In Minkowski space coordinates are chosen with respect to aninertial frame;(x^α) = (ct, x, y, z), so that themetric tensor used to raise and lower indices isη_αβ = diag(1, −1, −1, −1). Thed'Alembert operator on Minkowski space is◻ = ∂_α∂^α as in the vector formulation. In general spacetimes, the coordinate systemx^α is arbitrary, thecovariant derivative∇_α, theRicci tensor,R_αβ and raising and lowering of indices are defined by the Lorentzian metric,g_αβ and the d'Alembert operator is defined as◻ = ∇_α∇^α. The topological restriction is that the second realcohomology 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 thedifferential form formulation on arbitrary space times,F =⁠1/2⁠F_αβ‍dx^α ∧ dx^β is the electromagnetic tensor considered as a 2-form,A =A_αdx^α is the potential 1-form,${\displaystyle J=-J_{\alpha }\star \mathrm{d}{x}^{\alpha }}$  is the current 3-form,d is theexterior derivative, and${\displaystyle \star }$  is theHodge 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 asF, the Hodge star${\displaystyle \star }$  depends on the metric tensor only for its local scale. This means that, as formulated, the differential form field equations areconformally invariant, but theLorenz gauge condition breaks conformal invariance. The operator${\displaystyle ◻=(-\star \mathrm{d}\star \mathrm{d}-\mathrm{d}\star \mathrm{d}\star )}$  is thed'Alembert–Laplace–Beltrami operator on 1-forms on an arbitraryLorentzian 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 secondde Rham cohomology this condition means that every closed 2-form is exact.

Other formalisms include thegeometric algebra formulation and amatrix representation of Maxwell's equations. Historically, aquaternionic formulation^[17][18] was used.

Maxwell's equations arepartial 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 theLorentz force equation and theconstitutive 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 ofclassical electromagnetism. Some general remarks follow.

As for any differential equation,boundary conditions^[19][20][21] andinitial conditions^[22] are necessary for aunique solution. For example, even with no charges and no currents anywhere in spacetime, there are the obvious solutions for whichE andB 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.^[23] 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 anartificial absorbing boundary representing the rest of the universe,^[24][25] orperiodic boundary conditions, or walls that isolate a small region from the outside world (as with awaveguide or cavityresonator).^[26]

Jefimenko's equations (or the closely relatedLiénard–Wiechert potentials) 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 methods for differential equations can be used to compute approximate solutions of Maxwell's equations when exact solutions are impossible. These include thefinite element method andfinite-difference time-domain method.^{[19][21][27][28][29]} For more details, seeComputational electromagnetics.

Maxwell's equationsseemoverdetermined, in that they involve six unknowns (the three components ofE andB) but eight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampère's circuital laws). (The currents and charges are not unknowns, being freely specifiable subject tocharge 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 Ampère's circuital lawautomatically 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.^[30][31] This explanation was first introduced byJulius Adams Stratton in 1941.^[32]

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.^[33]

Both identities${\displaystyle \mathrm{\nabla }\cdot \mathrm{\nabla }\times \mathbf{B}\equiv 0,\mathrm{\nabla }\cdot \mathrm{\nabla }\times \mathbf{E}\equiv 0}$ , which reduce eight equations to six independent ones, are the true reason of overdetermination.^[34][35]

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 ofgauge fixing.

Maxwell's equations and the Lorentz force law (along with the rest of classical electromagnetism) are extraordinarily successful at explaining and predicting a variety of phenomena. However, they do not account for quantum effects, and so their domain of applicability is limited. Maxwell's equations are thought of as the classical limit ofquantum electrodynamics (QED).

Some observed electromagnetic phenomena cannot be explained with Maxwell's equations if the source of the electromagnetic fields are the classical distributions of charge and current. These includephoton–photon scattering and many other phenomena related tophotons orvirtual photons, "nonclassical light" andquantum entanglement of electromagnetic fields (seeQuantum optics). E.g.quantum cryptography cannot be described by Maxwell theory, not even approximately. The approximate nature of Maxwell's equations becomes more and more apparent when going into the extremely strong field regime (seeEuler–Heisenberg Lagrangian) or to extremely small distances.

Finally, Maxwell's equations cannot explain any phenomenon involving individualphotons interacting with quantum matter, such as thephotoelectric effect,Planck's law, theDuane–Hunt law, andsingle-photon light detectors. However, many such phenomena may be explained using a halfway theory of quantum matter coupled to a classical electromagnetic field, either as external field or with the expected value of the charge current and density on the right hand side of Maxwell's equations. This is known as semiclassical theory or self-field QED and was initially discovered by de Broglie and Schrödinger and later fully developed by E.T. Jaynes and A.O. Barut.

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.

Maxwell's equations posit that there iselectric charge, but nomagnetic charge (also calledmagnetic monopoles), in the universe. Indeed, magnetic charge has never been observed, despite extensive searches,^{[note 7]} 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.^{[9]: 273–275}

• Imaeda, K. (1995), "Biquaternionic Formulation of Maxwell's Equations and their Solutions", in Ablamowicz, Rafał; Lounesto, Pertti (eds.),Clifford Algebras and Spinor Structures, Springer, pp. 265–280,doi:10.1007/978-94-015-8422-7_16,ISBN 978-90-481-4525-6

• On Faraday's Lines of Force – 1855/56. Maxwell's first paper (Part 1 & 2) – Compiled by Blaze Labs Research (PDF).
• On Physical Lines of Force – 1861. Maxwell's 1861 paper describing magnetic lines of force – Predecessor to 1873 Treatise.
• James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field",Philosophical Transactions of the Royal Society of London155, 459–512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
  • A Dynamical Theory Of The Electromagnetic Field – 1865. Maxwell's 1865 paper describing his 20 equations, link fromGoogle Books.
• J. Clerk Maxwell (1873), "A Treatise on Electricity and Magnetism":
  • Maxwell, J. C., "A Treatise on Electricity And Magnetism" –Volume 1 – 1873 – Posner Memorial Collection – Carnegie Mellon University.
  • Maxwell, J. C., "A Treatise on Electricity And Magnetism" –Volume 2 – 1873 – Posner Memorial Collection – Carnegie Mellon University.

Developments before the theory of relativity

• Larmor Joseph (1897)."On a dynamical theory of the electric and luminiferous medium. Part 3, Relations with material media" .Phil. Trans. R. Soc.190:205–300.
• Lorentz Hendrik (1899)."Simplified theory of electrical and optical phenomena in moving systems" .Proc. Acad. Science Amsterdam.I:427–443.
• Lorentz Hendrik (1904)."Electromagnetic phenomena in a system moving with any velocity less than that of light" .Proc. Acad. Science Amsterdam.IV:669–678.
• Henri Poincaré (1900) "La théorie de Lorentz et le Principe de Réaction"(in French),Archives Néerlandaises,V, 253–278.
• Henri Poincaré (1902) "La Science et l'Hypothèse"(in French).
• Henri Poincaré (1905)"Sur la dynamique de l'électron"(in French),Comptes Rendus de l'Académie des Sciences,140, 1504–1508.
• Catt, Walton and Davidson."The History of Displacement Current"Archived 2008-05-06 at theWayback Machine.Wireless World, March 1979.

• "Maxwell equations",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• maxwells-equations.com — An intuitive tutorial of Maxwell's equations.
• The Feynman Lectures on Physics Vol. II Ch. 18: The Maxwell Equations
• Wikiversity Page on Maxwell's Equations

• Electromagnetism (ch. 11), B. Crowell, Fullerton College
• Lecture series: Relativity and electromagnetism, R. Fitzpatrick, University of Texas at Austin
• Electromagnetic waves from Maxwell's equations onProject PHYSNET.
• MIT Video Lecture Series (36 × 50 minute lectures) (in .mp4 format) – Electricity and Magnetism Taught by ProfessorWalter Lewin.

• Silagadze, Z. K. (2002). "Feynman's derivation of Maxwell equations and extra dimensions".Annales de la Fondation Louis de Broglie.27:241–256.arXiv:hep-ph/0106235.
• Nature Milestones: Photons –Milestone 2 (1861) Maxwell's equations