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Maxwell's equations are a set of partial differential equations that, together with theLorentz force law, form the foundation of classical electrodynamics, classical optics, andelectric circuits. These fields in turn underlie modern electrical and communications technologies.

Maxwell's equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents.

Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, "On Physical Lines of Force", which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper.

It is often useful to write Maxwell's equations in other forms; these representations are still formally termed "Maxwell's equations". A relativistic formulation in terms of covariant field tensors is used in special relativity, while in quantum mechanics, a version based on theelectric and magnetic potentials is preferred.

While Maxwell's equations are consistent within special and general relativity, there are some quantum mechanical situations in which Maxwell's equations are significantly inaccurate: including 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 photons, such as the photoelectric effect, Planck's law, the Duane–Hunt law, 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. Maxwell's equations are usually an extremely accurate approximation to the more accurate theory of quantum electrodynamics.

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[edit]Conceptual description

Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. (See below for a mathematical description of these laws.) Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges. (For the magnetic field there is no magnetic charge and therefore magnetic fields lines neither begin nor end anywhere.) The other two equations describe how the fields 'circulate' around their respective sources; the magnetic field 'circulates' around electric currents and time varying electric field in Ampère's law with Maxwell's correction, while the electric field 'circulates' around time varying magnetic fields in Faraday's law.

[edit]Gauss's law

Main article: Gauss's law

Gauss's law describes the relationship between an electric field and the electric charges that cause it: The 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 in a closed surface, therefore, yields the total charge enclosed by that surface. More technically, it relates the electric flux through any hypothetical closed "Gaussian surface" to the enclosed electric charge.

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.

[edit]Gauss's law for magnetism

Main article: Gauss's law for magnetism

Gauss's law for magnetism states that there are no "magnetic charges" (also calledmagnetic monopoles), analogous to electric charges.^[1] 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 totalmagnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

[edit]Faraday's law

Main article: Faraday's law

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 our electrical power grids. Artist's rendition; sizes are not to scale.

Faraday's law describes how a time varyingmagnetic field creates ("induces") an electric field.^[1] This aspect of electromagnetic induction is the operating principle behind many electric generators: for example, a rotatingbar 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.)

[edit]Ampère's law with Maxwell's correction

Main article: Ampère's law with Maxwell's correction

An Wang's magnetic core memory (1954) is an application of Ampère's law. Each corestores one bit of data.

Ampère's law with Maxwell's correctionstates 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.^[1]^[2] 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,^{[note 1]} exactly matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories ofelectromagnetism and optics.

[edit]Equations (SI units)

Maxwell's equations vary with the unit system used. Though the general form remains the same, various definitions become changed and different constants appear at different places. (This may seem strange at first, but this is because some unit systems, e.g. variants of cgs, define their units in such a way that certain physical constants are fixed, dimensionless constants, e.g. 1, so these constants disappear from the equations). The equations in this section are given in SI units. Other units commonly used are Gaussian units(based on the cgs system^[3]), Lorentz–Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). See below for CGS-Gaussian units.

For a detailed description of the differences between the microscopic (total charge and current)^{[note 2]} and macroscopic (free charge and current) variants of Maxwell's equations, see below.

In the following equations, symbols in bold represent vector quantities, and symbols in italics represent scalar quantities. The definitions of terms used in the two tables of equations are given in another table immediately following.

**Integral form**
Name	"Microscopic" equations	"Macroscopic" equations
Gauss's law	$\oiint$ ${\scriptstyle\partial \Omega }$ $\mathbf{E}\cdot\mathrm{d}\mathbf{S} = \frac{Q(V)}{\varepsilon_0}$	$\oiint$ ${\scriptstyle \partial \Omega }$ $\mathbf{D}\cdot\mathrm{d}\mathbf{S} = Q_{f}(V)$
Gauss's law for magnetism	$\oiint$ ${\scriptstyle \partial \Omega }$ $\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0$
Maxwell–Faraday equation (Faraday's law of induction)	$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = -\iint_{\Sigma} \frac{\partial \mathbf B}{\partial t} \cdot \mathrm{d}\mathbf{S}$
Ampère's circuital law (with Maxwell's correction)	$\oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 I + \mu_0 \varepsilon_0 \iint_{\Sigma} \frac{\partial \mathbf E}{\partial t} \cdot \mathrm{d}\mathbf{S}$	$\oint_{\partial \Sigma} \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = I_f + \iint_{\Sigma} \frac{\partial \mathbf D}{\partial t} \cdot \mathrm{d}\mathbf{S}$

**Differential form**
Name	"Microscopic" equations	"Macroscopic" equations
Gauss's law	$\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$	$\nabla \cdot \mathbf{D} = \rho_f$
Gauss's law for magnetism	$\nabla \cdot \mathbf{B} = 0$
Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$
Ampère's circuital law (with Maxwell's correction)	$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\$	$\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}$

[edit]Table of terms used in Maxwell's equations

The following table provides the meaning of each symbol and the SI unit of measure:

Definitions and units
Symbol	Meaning (first term is the most common)	SI Unit of measure
Differential operators
$\mathbf{\nabla \cdot}$	the divergence operator	per meter (factor contributed by applying either operator)
$\mathbf{\nabla \times}$	the curl operator	per meter (factor contributed by applying either operator)
$\frac {\partial}{\partial t}$	partial derivative with respect to time	per second (factor contributed by applying the operator)
Fields
E	electric field, also called the electric field intensity	volt per meter or, equivalently, newton per coulomb
B	magnetic field, also called: the magnetic induction the magnetic field density the magnetic flux density	tesla, or equivalently, weber per square meter, volt-second per square meter
D	electric displacement field, also called: the electric induction the electric flux density	coulombs per square meter or equivalently, newton per volt-meter
H	magnetizing field, also called: auxiliary magnetic field magnetic field intensity magnetic field	ampere per meter
ε₀	permittivity of free space, also called the electric constant, a universal constant	farads per meter
μ₀	permeability of free space, also called the magnetic constant, a universal constant	henries per meter, or newtons per ampere squared
Charges and currents
Q_f(V)	net free electric charge within the three-dimensional volume V (not including bound charge)	coulombs
Q(V)	net electric charge within the three-dimensional volume V (including both free and bound charge)	coulombs
ρ_f	free charge density (not including bound charge)	coulombs per cubic meter
ρ	total charge density (including both free and bound charge)	coulombs per cubic meter
J_f	free current density (not including bound current)	amperes per square meter
J	total current density (including both free and bound current)	amperes per square meter
Line and surface integrals
Σ and ∂Σ	Σ is any surface, and ∂Σ is its boundary curve. The surface is fixed (unchanging in time).
dℓ	differential vector element of path length tangential to the path/curve	meters
$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}$	line integral of the electric field along the boundary ∂Σ of a surface Σ (∂Σ is always a closed curve).	joules per coulomb
$\oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell}$	line integral of the magnetic field over the closed boundary ∂Σ of the surface Σ	tesla-meters
Ω and ∂Ω	Ω is any three-dimensional volume, and ∂Ω is its boundary surface. The volume is fixed (unchanging in time).
dS	differential vector element of surface area S, with infinitesimally small magnitude and direction normal to surface Σ (also denoted by Arather than S, but this conflicts with the magnetic potential)	square meters
$\oiint$ ${\scriptstyle \partial \Omega }$ $\mathbf{E}\cdot\mathrm{d}\mathbf{S}$	the electric flux (surface integral of the electric field) through the (closed) surface ∂Ω (the boundary of the volume Ω)	joule-meter per coulomb
$\oiint$ ${\scriptstyle \partial \Omega }$ $\mathbf{B}\cdot\mathrm{d}\mathbf{S}$	the magnetic flux (surface integral of the magnetic B-field) through the (closed) surface ∂Ω (the boundary of the volume Ω)	tesla meters-squared or webers
$\oiint$ ${\scriptstyle \partial \Omega }$ $\mathbf{D}\cdot\mathrm{d}\mathbf{S}$	flux of electric displacement field through the (closed) surface ∂Ω (the boundary of the volume Ω)	coulombs
$\iint_\Sigma \mathbf{J}_f \cdot \mathrm{d} \mathbf{S} = I_f$	net free electrical current passing through the surface Σ (not including bound current)	amperes
$\iint_\Sigma \mathbf{J} \cdot \mathrm{d} \mathbf{S} = I$	net electrical current passing through the surface Σ (including bothfree and bound current)	amperes

[edit]Relationship between differential and integral forms

The differential and integral forms 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. In addition the following relations are used:

Definition of bound charge density ρ_b and bound current density J_b in terms of polarization P and magnetization M:

$\rho_b = -\nabla\cdot\mathbf{P},$

$\mathbf{J}_b = \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}.$

Relations between D and E and between B and H:

$\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P},$

$\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M}),$

Relations between free, bound, and total charge and current density:

$\rho = \rho_b + \rho_f, \$

$\mathbf{J} = \mathbf{J}_b + \mathbf{J}_f,$

Substituting all these equations into the "macroscopic" Maxwell's equations gives the "microscopic" equations. Both the differential and integral forms are useful. The integral forms can often be used to simply and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential forms are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis.^[4]

[edit]Maxwell's "microscopic" equations

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". Both variants of Maxwell's equations are equally general, though, as they are mathematically equivalent. The microscopic equations are most useful in waveguides for example, when there are no dielectric or magnetic materials nearby.

[edit]With neither charges nor currents

Further information: Electromagnetic wave equation and Sinusoidal plane-wave solutions of the electromagnetic wave equation

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

$\nabla \cdot \mathbf{E} = 0$

$\nabla \cdot \mathbf{B} = 0$

$\nabla \times \mathbf{E} = - \frac{\partial\mathbf{B}} {\partial t}$

$\nabla \times \mathbf{B} = \ \ \mu_0\varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}.$

These equations lead directly to E and B satisfying the wave equation for which the solutions are linear combinations of plane wavestraveling at the speed of light,

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}. \$

In addition, E and B are mutually perpendicular to each other and the direction of wave propagation, and are in phase with each other. Asinusoidal plane wave is one special solution of these equations.

In fact, 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. In turn, that electric field creates a changing magnetic field through 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.

[edit]Maxwell's "macroscopic" equations

Unlike the "microscopic" equations, "Maxwell's macroscopic equations", also known as Maxwell's equations in matter, factor out the bound charge and current to obtain equations that depend only on the free charges and currents. These equations are more similar to those that Maxwell himself introduced. The cost of this factorization is that additional fields need to be defined: the displacement field Dwhich is defined in terms of the electric field E and the polarization P of the material, and the magnetic-H field, which is defined in terms of the magnetic-B field and the magnetization M of the material.

[edit]Bound charge and current

Main articles: Bound charge, Bound current, and Bound current#Magnetization current

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 material its molecules respond by forming microscopic electric dipoles – their atomic nuclei move a tiny distance in the direction of the field, while their electrons 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 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 a polarization, P, in the material. If P is uniform, a macroscopic separation of charge is produced only at the surfaces where Penter and leave the material. For non-uniform P, a charge is also produced in the bulk.^[5]

Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the angular momentum of the components of the atoms, most notably their electrons. The 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 currents can be described using the magnetization M.^[6]

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 suitabe volume.

[edit]Constitutive relations

Main article: constitutive equation

In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between displacement field D and E, and the magnetic H-field H and B. These equations specify the response of bound charge and current to the applied fields and are calledconstitutive relations. For real-world materials, the constitutive relations are rarely simple, except approximately, see the main article for a fuller description.

The definitions (not constitutive relations) of the auxiliary fields are:

$\mathbf{D}(\mathbf{r}, t) = \varepsilon_0 \mathbf{E}(\mathbf{r}, t) + \mathbf{P}(\mathbf{r}, t)$

$\mathbf{H}(\mathbf{r}, t) = \frac{1}{\mu_0} \mathbf{B}(\mathbf{r}, t) - \mathbf{M}(\mathbf{r}, t),$

where P is the polarization field and M is the magnetization field which are defined in terms of microscopic bound charges and bound current respectively.

The macroscopic forms of Maxwell's equations for different materials are presented below. In each case, Faraday's law of induction andGauss's law for magnetism are always the same (not for monopoles, see below).

No materials (vacuum)

The constitutive relations are

$\mathbf{D} = \varepsilon_0\mathbf{E}, \quad \mathbf{H} = \mathbf{B}/\mu_0$

The currents and charges are free, not total (expected since there are no bound charges nor currents);

Gauss's law:	$\nabla\cdot\mathbf{D} = \rho_f$
Ampère's circuital law:	$\nabla\times\mathbf{H} = \mathbf{J}_f + \frac{\partial\mathbf{D}}{\partial t}$

Linear materials

The constitutive relations are

$\mathbf{D} = \varepsilon\mathbf{E}\,,\quad \mathbf{H} = \mathbf{B}/\mu$

where ε is the permittivity and μ the permeability of the material.

For homogeneous materials, ε and μ are constant throughout the material, for inhomogeneous they depend on location within the material (and perhaps time).
For isotropic materials, ε and μ are independent of the directions of the applied fields to the material, for anisotropic they are tensors(incorporating directional dependence of the medium).
Materials are generally dispersive, so ε and μ depend on the frequency of any incident EM waves.

The vacuum permittivity and permeability are replaced by those of the material, the charges and currents are free (not total);

Gauss's law:	$\nabla \cdot (\varepsilon \mathbf{E}) = \rho_f$
Ampère's circuital law:	$\nabla \times (\mathbf{B} / \mu) = \mathbf{J}_f + \varepsilon \frac{\partial \mathbf{E}} {\partial t}.$

In the case of non-linear materials (see for example nonlinear optics), D and P are not proportional to E, similarly B is not proportional to H or M.

[edit]Equations (Gaussian units)

Main article: Gaussian units

Gaussian units are a popular electromagnetism variant of the centimetre gram second system of units (cgs), in which case Maxwell's equations become:^[7]

Name	Microscopic equations	Macroscopic equations
Gauss's law	$\nabla \cdot \mathbf{E} = 4\pi\rho_{\mathrm{tot}}$	$\nabla \cdot \mathbf{D} = 4\pi\rho_\mathrm{f}$
Gauss's law for magnetism	$\nabla \cdot \mathbf{B} = 0$
Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}$
Ampère's law (with Maxwell's extension)	$\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t} + \frac{4\pi}{c}\mathbf{J}_{\mathrm{tot}}$	$\nabla \times \mathbf{H} = \frac{1}{c} \frac{\partial \mathbf{D}} {\partial t} + \frac{4\pi}{c} \mathbf{J}_\mathrm{f}$

One result of Gaussian units is that the magnetic field B has the same units as the electric field E – in SI units this doesn't happen (since for EM waves in vacuum, |E| = c|B|), making dimensional analysis of the equations different. See SI and Gaussian units for how to convert between them.

[edit]History

[edit]Relation between electricity, magnetism, and the speed of light

The relation between electricity, magnetism, and the speed of light can be summarized by the modern equation:

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \ .$

The left-hand side is the speed of light, and the right-hand side is a quantity related to the equations governing electricity and magnetism. Although the right-hand side has units of velocity, it can be inferred from measurements of electric and magnetic forces, which involve no physical velocities. Therefore, establishing this relationship provided convincing evidence that light is an electromagnetic phenomenon.

The discovery of this relationship started in 1855, when Wilhelm Eduard Weber and Rudolf Kohlrausch determined that there was a quantity related to electricity and magnetism, "the ratio of the absolute electromagnetic unit of charge to the absolute electrostatic unit of charge" (in modern language, the value $1/\sqrt{\mu_0 \varepsilon_0}$ ), and determined that it should have units of velocity. They then measured this ratio by an experiment which involved charging and discharging a Leyden jar and measuring the magnetic force from the discharge current, and found a value 3.107×10⁸ m/s,^[8] remarkably close to the speed of light, which had recently been measured at3.14×10⁸ m/s by Hippolyte Fizeau in 1848 and at 2.98×10⁸ m/s by Léon Foucault in 1850.^[8] However, Weber and Kohlrausch did not make the connection to the speed of light.^[8] Towards the end of 1861 while working on part III of his paper On Physical Lines of Force, Maxwell travelled from Scotland to London and looked up Weber and Kohlrausch's results. He converted them into a format which was compatible with his own writings, and in doing so he established the connection to the speed of light and concluded that light is a form of electromagnetic radiation.^[9]

[edit]The term Maxwell's equations

The four modern Maxwell's equations can be found individually throughout his 1861 paper, derived theoretically using a molecular vortex model of Michael Faraday's "lines of force" and in conjunction with the experimental result of Weber and Kohlrausch. But it wasn't until 1884 that Oliver Heaviside,^[10] concurrently with similar work by Willard Gibbs and Heinrich Hertz,^[11] grouped the four together into a distinct set. This group of four equations was known variously as the Hertz-Heaviside equations and the Maxwell-Hertz equations,^[10]and are sometimes still known as the Maxwell–Heaviside equations.^[12]

Maxwell's contribution to science in producing these equations lies in the correction he made to Ampère's circuital law in his 1861 paper On Physical Lines of Force. He added the displacement current term to Ampère's circuital law and this enabled him to derive theelectromagnetic wave equation in his later 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrate the fact that light is an electromagnetic wave. This fact was then later confirmed experimentally by Heinrich Hertz in 1887. The physicist Richard Feynman predicted that, "The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade."^[13]

The concept of fields was introduced by, among others, Faraday. Albert Einstein wrote:

The precise formulation of the time-space laws was the work of Maxwell. Imagine his feelings when the differential equations he had formulated proved to him that electromagnetic fields spread in the form of polarised waves, and at the speed of light! To few men in the world has such an experience been vouchsafed ... it took physicists some decades to grasp the full significance of Maxwell's discovery, so bold was the leap that his genius forced upon the conceptions of his fellow-workers

—(Science, May 24, 1940)

Heaviside worked to eliminate the potentials (electric potential and magnetic potential) that Maxwell had used as the central concepts in his equations;^[10] this effort was somewhat controversial,^[14] though it was understood by 1884 that the potentials must propagate at the speed of light like the fields, unlike the concept of instantaneous action-at-a-distance like the then conception of gravitational potential.^[11] Modern analysis of, for example, radio antennas, makes full use of Maxwell's vector and scalar potentials to separate the variables, a common technique used in formulating the solutions of differential equations. However, the potentials can be introduced by algebraic manipulation of the four fundamental equations.

[edit]On Physical Lines of Force

Main article: On Physical Lines of Force

The four modern day Maxwell's equations appeared throughout Maxwell's 1861 paper On Physical Lines of Force:

Equation (56) in Maxwell's 1861 paper is ∇ • B = 0.
Equation (112) is Ampère's circuital law with Maxwell's displacement current added. It is the addition of displacement currentthat is the most significant aspect of Maxwell's work in electromagnetism, as it enabled him to later derive the electromagnetic wave equation in his 1865 paper A Dynamical Theory of the Electromagnetic Field, and hence show that light is an electromagnetic wave. It is therefore this aspect of Maxwell's work which gives the equations their full significance. (Interestingly, Kirchhoff derived the telegrapher's equations in 1857 without using displacement current. But he did use Poisson's equation and the equation of continuity which are the mathematical ingredients of the displacement current. Nevertheless, Kirchhoff believed his equations to be applicable only inside an electric wire and so he is not credited with having discovered that light is an electromagnetic wave).
Equation (115) is Gauss's law.
Equation (54) is an equation that Oliver Heaviside referred to as 'Faraday's law'. This equation caters for the time varying aspect of electromagnetic induction, but not for the motionally induced aspect, whereas Faraday's original flux law caters for both aspects.^[15]^[16] Maxwell deals with the motionally dependent aspect of electromagnetic induction, v × B, at equation (77). Equation (77) which is the same as equation (D) in the original eight Maxwell's equations listed below, corresponds to all intents and purposes to the modern day force law F = q( E + v × B ) which sits adjacent to Maxwell's equations and bears the nameLorentz force, even though Maxwell derived it when Lorentz was still a young boy.

The difference between the B and the H vectors can be traced back to Maxwell's 1855 paper entitled On Faraday's Lines of Force which was read to the Cambridge Philosophical Society. The paper presented a simplified model of Faraday's work, and how the two phenomena were related. He reduced all of the current knowledge into a linked set of differential equations.

Figure of Maxwell's molecular vortex model. For a uniform magnetic field, the field lines point outward from the display screen, as can be observed from the black dots in the middle of the hexagons. The vortex of each hexagonal molecule rotates counter-clockwise. The small green circles are clockwise rotating particles sandwiching between the molecular vortices.

It is later clarified in his concept of a sea of molecular vortices that appears in his 1861 paper On Physical Lines of Force. Within that context, H represented pure vorticity (spin), whereas B was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell considered magnetic permeability µ to be a measure of the density of the vortex sea. Hence the relationship,

Magnetic induction current causes a magnetic current density B = μH was essentially a rotational analogy to the linear electric current relationship,
Electric convection current J = ρ v where ρ is electric charge density.B was seen as a kind of magnetic current of vortices aligned in their axial planes, with H being the circumferential velocity of the vortices. With µ representing vortex density, it follows that the product of µ with vorticity H leads to the magnetic field denoted as B.

The electric current equation can be viewed as a convective current of electric charge that involves linear motion. By analogy, the magnetic equation is an inductive current involving spin. There is no linear motion in the inductive current along the direction of the B vector. The magnetic inductive current represents lines of force. In particular, it represents lines of inverse square lawforce.

The extension of the above considerations confirms that where B is to H, and where J is to ρ, then it necessarily follows from Gauss's law and from the equation of continuity of charge that E is to D. i.e. B parallels with E, whereasH parallels with D.

[edit]A Dynamical Theory of the Electromagnetic Field

Main article: A Dynamical Theory of the Electromagnetic Field

In 1864 Maxwell published A Dynamical Theory of the Electromagnetic Field in which he showed that light was an electromagnetic phenomenon. Confusion over the term "Maxwell's equations" sometimes arises because it has been used for a set of eight equations that appeared in Part III of Maxwell's 1864 paper A Dynamical Theory of the Electromagnetic Field, entitled "General Equations of the Electromagnetic Field",^[17] and this confusion is compounded by the writing of six of those eight equations as three separate equations (one for each of the Cartesian axes), resulting in twenty equations and twenty unknowns. (As noted above, this terminology is not common: Modern references to the term "Maxwell's equations" refer to the Heaviside restatements.)

The eight original Maxwell's equations can be written in modern vector notation as follows:

(A) The law of total currents	$\mathbf{J}_\mathrm{tot} = \mathbf{J} + \frac{\partial\mathbf{D}}{\partial t}$
(B) The equation of magnetic force	$\mu \mathbf{H} = \nabla \times \mathbf{A}$
(C) Ampère's circuital law	$\nabla \times \mathbf{H} = \mathbf{J}_\mathrm{tot}$
(D) Electromotive force created by convection, induction, and by static electricity. (This is in effect the Lorentz force)	$\mathbf{E} = \mu \mathbf{v} \times \mathbf{H} - \frac{\partial\mathbf{A}}{\partial t}-\nabla \phi$
(E) The electric elasticity equation	$\mathbf{E} = \frac{1}{\varepsilon} \mathbf{D}$
(F) Ohm's law	$\mathbf{E} = \frac{1}{\sigma} \mathbf{J}$
(G) Gauss's law	$\nabla \cdot \mathbf{D} = \rho$
(H) Equation of continuity	$\nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\partial t}$ or $\nabla \cdot \mathbf{J}_\mathrm{tot} = 0$

Notation

H is the magnetizing field, which Maxwell called the magnetic intensity.

J is the current density (withJ_tot being the total current including displacement current).^{[note 3]}

D is the displacement field (called the electric displacement by Maxwell).

ρ is the free charge density (called the quantity of free electricity by Maxwell).

A is the magnetic potential (called the angular impulse by Maxwell).

E is called the electromotive force by Maxwell. The term electromotive force is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern term electric field.

φ is the electric potential (which Maxwell also called electric potential).

σ is the electrical conductivity (Maxwell called the inverse of conductivity the specific resistance, what is now called the resistivity).

It is interesting to note the μv × H term that appears in equation D. Equation D is therefore effectively the Lorentz force, similarly to equation (77) of his 1861 paper (see above).

When Maxwell derives the electromagnetic wave equation in his 1865 paper, he uses equation D to cater for electromagnetic inductionrather than Faraday's law of induction which is used in modern textbooks. (Faraday's law itself does not appear among his equations.) However, Maxwell drops the μv × H term from equation D when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame.

[edit]A Treatise on Electricity and Magnetism

Main article: A Treatise on Electricity and Magnetism

English Wikisource has original text related to this article:

A Treatise on Electricity and Magnetism

In A Treatise on Electricity and Magnetism, an 1873 treatise on electromagnetism written byJames Clerk Maxwell, eleven general equations of the electromagnetic field are listed and these include the eight that are listed in the 1865 paper.^[18]

[edit]Maxwell's equations and relativity

[edit]Modified to include magnetic monopoles

Main article: Magnetic monopole

Maxwell's equations posit electric charge, but not magnetic charge, which has never been observed^[21] and may not exist. Nevertheless, Maxwell's equations including magnetic charge (and magnetic current) are of some theoretical interest.^[22]

For one reason, Maxwell's equations can be made fully symmetric under interchange of electric and magnetic fields by allowing for the possibility of magnetic charges with magnetic charge density ρ_m and currents with magnetic current density J_m.^[23] The extended Maxwell's equations (in cgs-Gaussian units) are:

Name	Without magnetic monopoles	With magnetic monopoles (hypothetical)
Gauss's law	$\nabla \cdot \mathbf{E} = 4 \pi \rho_\mathrm{e}$
Gauss's law for magnetism	$\nabla \cdot \mathbf{B} = 0$	$\nabla \cdot \mathbf{B} = 4 \pi \rho_\mathrm{m}$
Maxwell–Faraday equation (Faraday's law of induction)	$-\nabla \times \mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}$	$-\nabla \times \mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{B}}{\partial t} + \frac{4 \pi}{c} \mathbf{j}_\mathrm{m}$
Ampère's law (with Maxwell's extension)	$\nabla \times \mathbf{B} = \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t} + \frac{4 \pi}{c} \mathbf{j}_\mathrm{e}$

If magnetic charges do not exist, or if they exist but not in the region studied, then the new variables are zero, and the symmetric equations reduce to the conventional equations of electromagnetism such as ∇ • B = 0. Further, if every particle has the same ratio of electric to magnetic charge, then an E and a B field can be defined that obeys the normal Maxwell's equation (having no magnetic charges or currents) with its own charge and current densities.^[24]

[edit]Solving Maxwell's equations

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 equation and the 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^[25]^[26]^[27] and initial conditions^[28] 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.^[29] 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 a artificial absorbing boundary representing the rest of the universe,^[30]^[31] 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.^[32]
Jefimenko's equations (or the closely related Liénard–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. Jefimenko's equations are not so helpful in situations when the charges and currents are themselves affected by the fields they create.
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.^[25]^[27]^[33]^[34]^[35] For more details, see Computational electromagnetics.
Maxwell's equations seem 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.) In fact, Maxwell's equations are somewhat redundant; 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. Therefore numerical algorithms often ignore the two Gauss's laws, although more accurate algorithms take them into account in order to minimize numerical inaccuracy.^[36]^[37]

[edit]Alternative formulations of Maxwell's equations

Main articles: Mathematical descriptions of the electromagnetic field and classical electromagnetism and special relativity

Following is a summary of the numerous other ways to write the equations (in SI units, not Gaussian), showing they can be collected together in simpler and more unified formulae, though in terms of more complicated mathematics. See the main articles for the details of each formulation.

Formulation	Homogeneous equations		Nonhomogeneous equations
Vector calculus(fields)	$\nabla\cdot\mathbf{B}=0$	$\nabla\times\mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=0$	$\nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_0}$	$\nabla\times\mathbf{B}-\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}=\mu_0\mathbf{J}$
Vector calculus (potentials, any gauge)	identities		$\nabla^2 \varphi + \frac{\partial}{\partial t} \left ( \mathbf \nabla \cdot \mathbf A \right ) = - \frac{\rho}{\varepsilon_0}$	$\Box\mathbf A+\mathbf \nabla \left ( \mathbf \nabla \cdot \mathbf A + \frac{1}{c^2} \frac{\partial \varphi}{\partial t} \right ) = \mu_0 \mathbf J$
QED, vector calculus (potentials,Lorenz gauge)	identities		$\Box \varphi = -\frac{1}{\varepsilon_0} e \psi^{\dagger} \psi$	$\Box \mathbf A = -\mu_0 e \psi^{\dagger} \boldsymbol{\alpha} \psi$
Tensor calculus(potentials, Lorenz gauge)	identities		$\Box A^\mu = \mu_0 J^\mu$
Tensor calculus (fields)	$\dfrac{\partial F_{\alpha\beta}}{\partial x^\gamma} + \dfrac{\partial F_{\gamma\alpha}}{\partial x^\beta} + \dfrac{\partial F_{\beta\gamma}}{\partial x^\alpha} = 0$		$\dfrac{\partial F^{\beta\alpha}}{\partial x^\alpha}=\mu_0 J^\beta$
Differential forms(fields)	$\mathrm{d}\bold{F}=0$		$\mathrm{d} \star \bold{F}=\bold{J}$
Geometric algebra(fields)	$\nabla F = \mu_0 c J$
Algebra of physical space(fields)	$\left(\frac{1}{c}\dfrac{\partial }{\partial t} + \boldsymbol{\nabla}\right)F = \mu_0 c J$

where

$\Box = \frac{1}{c^2} \frac{\partial^2} {\partial t^2}-\nabla^2$

is the D'Alembert operator. Following are the reasons for using such formulations:

Potential formulation approach: In advanced classical mechanics it is often useful, and in quantum mechanics frequently essential, to express Maxwell's equations in a potential formulation involving the electric potential (also called scalar potential) φ, and the magnetic potential A, (also called vector potential). These are defined such that:

$\mathbf E = - \mathbf \nabla \varphi - \frac{\partial \mathbf A}{\partial t}\,,\quad \mathbf B = \mathbf \nabla \times \mathbf A.$

Many different choices of A and φ are consistent with a given E and B, making these choices physically equivalent – a flexibility known as gauge freedom. Suitable choice of A and φ can simplify these equations, or can adapt them to suit a particular situation.

Manifestly covariant (tensor) approach: Maxwell's equations are exactly consistent with special relativity—i.e., if they are valid in one inertial reference frame, then they are automatically valid in every other inertial reference frame. In fact, Maxwell's equations were crucial in the historical development of special relativity. However, in the usual formulation Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation that involves a seemingly-miraculous cancellation of different terms.

For example, consider a conductor moving in the field of a magnet.^[38] In the frame of the magnet, that conductor experiences amagnetic force. But in the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an electricfield. The motion is exactly consistent in these two different reference frames, but it mathematically arises in quite different ways.

For this reason and others, it is often useful to rewrite Maxwell's equations in a way that is "manifestly covariant"—i.e. obviouslyconsistent with special relativity, even with just a glance at the equations—using covariant and contravariant four-vectors and tensors. This can be done using the EM tensor F, or the 4-potential A, with the 4-current J - see covariant formulation of classical electromagnetism.

Differential forms approach: Gauss's law for magnetism and the Faraday-Maxwell law can be grouped together since the equations are homogeneous, and be seen as geometric identities expressing the field F (a 2-form), which can be derived from the 4-potential A. Gauss's law for electricity and the Ampere-Maxwell law could be seen as the dynamical equations of motion of the fields, obtained via the Lagrangian principle of least action, from the "interaction term" A J (introduced through gauge covariant derivatives), coupling the field to matter. For the field formulation of Maxwell's equations in terms of a principle of extremal action, see electromagnetic tensor.

Often, the time derivative in the Faraday-Maxwell equation motivates calling this equation "dynamical", which is somewhat misleading in the sense of the preceding analysis. This is rather an artifact of breaking relativistic covariance by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a kinetic term F *F for A; and take into account the non-physical degrees of freedom which can be removed by gauge transformation A → A' = A − dα. See also gauge fixing and Faddeev–Popov ghosts.