Maxwell's equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.

Maxwell's four equations express, respectively, how electric charges produce electric fields (Gauss' law), the experimental absence of magnetic monopoles, how currents and changing electric fields produce magnetic fields (the Ampere-Maxwell law), and how changing magnetic fields produce electric fields (Faraday's law of induction).

Historical development of Maxwell's equations

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Maxwell, in 1864, was the first to put all four equations together and to notice that a correction was required to Ampere's law: changing electric fields act like currents, likewise producing magnetic fields. (This additional term is called the displacement current.) The most common modern notation for these equations was developed by Oliver Heaviside.

Furthermore, Maxwell showed that waves of oscillating electric and magnetic fields travel through empty space at a speed that could be predicted from simple electrical experiments—using the data available at the time, Maxwell obtained a velocity of 310,740,000 m/s. Maxwell (1865) wrote:

This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

Maxwell was correct in this conjecture, though he did not live to see the first experimental confirmation by Heinrich Hertz in 1888. Maxwell's quantitative explanation of light as an electromagnetic wave is considered one of the great triumphs of 19th-century physics. (Actually, Michael Faraday had postulated a similar picture of light in 1846, but had not been able to give a quantitative description or predict the velocity.) Moreover, it laid the foundation for many future developments in physics, such as special relativity and its unification of electric and magnetic fields as a single tensor quantity, and Kaluza and Klein's unification of electromagnetism with gravity and general relativity.

Maxwell's 1865 formulation was in terms of 20 equations in 20 variables, which included several equations now considered to be auxiliary to what are now called "Maxwell's equations":

• The corrected Ampere's law (three component equations).
• Gauss' law for charge (one equation).
• The relationship between total and displacement current densities (three component equations).
• The relationship between magnetic field and the vector potential (three component equations, which imply the absence of magnetic charge).
• The relationship between electric field and the scalar and vector potentials (three component equations, which imply Faraday's law).
• The relationship between the electric and displacement fields (three component equations).
• Ohm's law relating current density and electric field (three component equations).
• The continuity equation relating current density and charge density (one equation).

The modern mathematical formulation of Maxwell's equations is due to Heaviside and Willard Gibbs, who in 1884 reformulated Maxwell's original system of equations to a far simpler representation using vector calculus. (In 1873 Maxwell also published a quaternion-based notation that ultimately proved unpopular.) The change to the vector notation produced a symmetric mathematical representation that reinforced the perception of physical symmetries between the various fields. This highly symmetrical formulation would directly inspire later developments in fundamental physics.

Links to relativity

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In the late 19th century, because of the appearance of a velocity,

$c=\backslash frac\{1\}\{\backslash sqrt\{\backslash varepsilon\_0\backslash mu\_0\}\}$

in the equations, Maxwell's equations were only thought to express electromagnetism in the rest frame of the luminiferous aether (the postulated medium for light, whose interpretation was considerably debated). The symbols represent the permittivity and permeability of free space. When the Michelson-Morley experiment, conducted by Edward Morley and Albert Abraham Michelson, produced a null result for the change of the velocity of light due to the Earth's motion through the hypothesized aether, however, alternative explanations were sought by George FitzGerald, Joseph Larmor and Hendrik Lorentz. Both Larmor (1897) and Lorentz (1899, 1904) derived the Lorentz transformation (so named by Henri Poincaré) as one under which Maxwell's equations were invariant. Poincaré (1900) analysed the coordination of moving clocks by exchanging light signals. He also established the group property of the Lorentz transformation (Poincaré 1905). This culminated in Einstein's theory of special relativity, which postulated the absence of any absolute rest frame, dismissed the ether as unnecessary, and established the invariance of Maxwell's equations in all inertial frames of reference.

The electromagnetic field equations have an intimate link with special relativity: the magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities. Einstein motivated the special theory by noting that a description of a conductor moving with respect to a magnet must generate a consistent set of fields irrespective of whether the frame is the magnet frame or the conductor frame..^*

In relativity, the equations are written in an even more compact, "manifestly covariant" form, in terms of the rank-2 antisymmetric field-strength 4-tensor that unifies the electric and magnetic fields into a single object.

Kaluza and Klein showed in the 1920s that Maxwell's equations can be derived by extending general relativity into five dimensions. This strategy of using higher dimensions to unify different forces is an active area of research in particle physics.

Summary of the equations

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Symbols in bold represent vector quantities, whereas symbols in italics represent scalar quantities.

General case

{d \over dt} \int_S \mathbf{D} \cdot d \mathbf{A}

Name	Differential form	Integral form
Gauss's law:	$\backslash nabla\; \backslash cdot\; \backslash mathbf\{D\}\; =\; \backslash rho$	$\backslash oint\_S\; \backslash mathbf\{D\}\; \backslash cdot\; d\backslash mathbf\{A\}\; =\; \backslash int\_V\; \backslash rho\; dV$
Gauss' law for magnetism (absence of magnetic monopoles):	$\backslash nabla\; \backslash cdot\; \backslash mathbf\{B\}\; =\; 0$	$\backslash oint\_S\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}\; =\; 0$
Faraday's law of induction:	$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$	$\backslash oint\_C\; \backslash mathbf\{E\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; -\; \backslash \; \{\; d\; \backslash over\; dt\; \}\; \backslash int\_S\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}$
Ampère's law (with Maxwell's extension):	$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{D\}\}\; \{\backslash partial\; t\}$	$\backslash oint\_C\; \backslash mathbf\{H\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; \backslash int\_S\; \backslash mathbf\{J\}\; \backslash cdot\; d\; \backslash mathbf\{A\}\; +$

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

small magnitude and direction normal to surface S

Symbol	Meaning	SI Unit of Measure
$\backslash mathbf\{E\}$	electric field	volt per meter
$\backslash mathbf\{H\}$	magnetic field also called the auxiliary field	ampere per meter
$\backslash mathbf\{D\}$	electric displacement field also called the electric flux density	coulomb per square meter
$\backslash mathbf\{B\}$	magnetic flux density also called the magnetic induction also called the magnetic field	tesla, or equivalently, weber per square meter
$\backslash \; \backslash rho\; \backslash$	free electric charge density, not including dipole charges bound in a material	coulomb per cubic meter
$\backslash mathbf\{J\}$	free current density, not including polarization or magnetization currents bound in a material	ampere per square meter
$d\backslash mathbf\{A\}$	differential vector element of surface area A, with infinitesimally	square meters
$dV\; \backslash$	differential element of volume V enclosed by surface S	cubic meters
$d\; \backslash mathbf\{l\}$	differential vector element of path length tangential to contour C enclosing surface S	meters
$\backslash nabla\; \backslash cdot$	the divergence operator	per meter
$\backslash nabla\; \backslash times$	the curl operator	per meter

Although SI units are given here for the various symbols, Maxwell's equations will hold unchanged in many different unit systems (and with only minor modifications in all others). The most commonly used systems of units are SI units, used for engineering, electronics and most practical physics experiments, and Planck units (also known as "natural units"), used in theoretical physics, quantum physics and cosmology. An older system of units, the cgs system, is sometimes also used.

The second equation is equivalent to the statement that magnetic monopoles do not exist. The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:

$\backslash mathbf\{F\}\; =\; q\; (\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}),$

where $q\; \backslash$ is the charge on the particle and $\backslash mathbf\{v\}\; \backslash$ is the particle velocity. This is slightly different when expressed in the cgs system of units below.

Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo quantum mechanical effects as well). It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material, below (the microscopic Maxwell's equations, ignoring quantum effects, are simply those of a vacuum — but one must include all atomic charges and so on, which is normally an intractable problem).

In linear materials

In linear materials, the polarization density P (in coulombs per square meter) and magnetization density M (in amperes per meter) are given by:

$\backslash mathbf\{P\}\; =\; \backslash chi\_e\; \backslash varepsilon\_0\; \backslash mathbf\{E\}$

$\backslash mathbf\{M\}\; =\; \backslash chi\_m\; \backslash mathbf\{H\}$

and the D and B fields are related to E and H by:

$\backslash mathbf\{D\}\; \backslash \; \backslash \; =\; \backslash \; \backslash \; \backslash varepsilon\_0\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{P\}\; \backslash \; \backslash \; =\; \backslash \; \backslash \; (1\; +\; \backslash chi\_e)\; \backslash varepsilon\_0\; \backslash mathbf\{E\}\; \backslash \; \backslash$

= \ \ \varepsilon \mathbf{E}

$\backslash mathbf\{B\}\; \backslash \; \backslash \; =\; \backslash \; \backslash \; \backslash mu\_0\; (\; \backslash mathbf\{H\}\; +\; \backslash mathbf\{M\}\; )\; \backslash \; \backslash \; =\; \backslash \; \backslash \; (1\; +\; \backslash chi\_m)\; \backslash mu\_0\; \backslash mathbf\{H\}\; \backslash \; \backslash$

= \ \ \mu \mathbf{H}

where:

$\backslash chi\_e$ is the electrical susceptibility of the material,

$\backslash chi\_m$ is the magnetic susceptibility of the material,

ε is the electrical permittivity of the material, and

μ is the magnetic permeability of the material

(This can actually be extended to handle nonlinear materials as well, by making ε and μ depend upon the field strength; see e.g. the Kerr and Pockels effects.)

In non-dispersive, isotropic media, ε and μ are time-independent scalars, and Maxwell's equations reduce to

$\backslash nabla\; \backslash cdot\; \backslash varepsilon\; \backslash mathbf\{E\}\; =\; \backslash rho$

$\backslash nabla\; \backslash cdot\; \backslash mu\; \backslash mathbf\{H\}\; =\; 0$

$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\; \backslash mu\; \backslash frac\{\backslash partial\; \backslash mathbf\{H\}\}\; \{\backslash partial\; t\}$

$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}\; +\; \backslash varepsilon\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}$

In a uniform (homogeneous) medium, ε and μ are constants independent of position, and can thus be furthermore interchanged with the spatial derivatives.

More generally, ε and μ can be rank-2 tensors (3×3 matrices) describing birefringent (anisotropic) materials. Also, although for many purposes the time/frequency-dependence of these constants can be neglected, every real material exhibits some material dispersion by which ε and/or μ depend upon frequency (and causality constrains this dependence to obey the Kramers-Kronig relations).

In vacuum, without charges or currents

The vacuum is a linear, homogeneous, isotropic, dispersionless medium, and the proportionality constants in the vacuum are denoted by ε₀ and μ₀ (neglecting very slight nonlinearities due to quantum effects).

$\backslash mathbf\{D\}\; =\; \backslash varepsilon\_0\; \backslash mathbf\{E\}$

$\backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash mathbf\{H\}$

Since there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:

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

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{H\}\; =\; 0$

$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\; \backslash mu\_0\; \backslash frac\{\backslash partial\backslash mathbf\{H\}\}\; \{\backslash partial\; t\}$

$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash \; \backslash \; \backslash varepsilon\_0\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}$

These equations have a simple solution in terms of travelling sinusoidal plane waves, with the electric and magnetic field directions orthogonal to one another and the direction of travel, and with the two fields in phase, travelling at the speed

$c\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash mu\_0\; \backslash varepsilon\_0\}\}$

Maxwell discovered that this quantity c is simply the speed of light in vacuum, and thus that light is a form of electromagnetic radiation. The currently accepted values for the speed of light, the permittivity, and the permeability are summarized in the following table:

Symbol	Name	Numerical Value	SI Unit of Measure	Type
$c\; \backslash$	Speed of light	$2.998\; \backslash times\; 10^\{8\}$	meters per second	defined
$\backslash \; \backslash varepsilon\_0$	Permittivity	$8.854\; \backslash times\; 10^\{-12\}$	farads per meter	derived
$\backslash \; \backslash mu\_0\; \backslash$	Permeability	$4\; \backslash pi\; \backslash times\; 10^\{-7\}$	henries per meter	defined

Detail

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Charge density and the electric field

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{D\}\; =\; \backslash rho$,

where $\{\backslash rho\}$ is the free electric charge density (in units of C/m³), not including dipole charges bound in a material, and $\backslash mathbf\{D\}$ is the electric displacement field (in units of C/m²). This equation corresponds to Coulomb's law for stationary charges in vacuum.

The equivalent integral form (by the divergence theorem), also known as Gauss' law, is:

$\backslash oint\_S\; \backslash mathbf\{D\}\; \backslash cdot\; d\backslash mathbf\{A\}\; =\; Q\_\backslash mathrm\{enclosed\}$

where $d\backslash mathbf\{A\}$ is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and $Q\_\backslash mathrm\{enclosed\}$ is the free charge enclosed by the surface.

In a linear material, $\backslash mathbf\{D\}$ is directly related to the electric field $\backslash mathbf\{E\}$ via a material-dependent constant called the permittivity, $\backslash epsilon$:

$\backslash mathbf\{D\}\; =\; \backslash varepsilon\; \backslash mathbf\{E\}$.

Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as $\backslash epsilon\_0$, and appears in:

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash rho\_t\}\{\backslash varepsilon\_0\}$

where, again, $\backslash mathbf\{E\}$ is the electric field (in units of V/m), $\backslash rho\_t$ is the total charge density (including bound charges), and $\backslash epsilon\_0$ (approximately 8.854 pF/m) is the permittivity of free space. $\backslash epsilon$ can also be written as $\backslash varepsilon\_0\; \backslash cdot\; \backslash varepsilon\_r$, where $\backslash epsilon\_r$ is the material's relative permittivity or its dielectric constant.

Compare Poisson's equation.

The structure of the magnetic field

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

$\backslash mathbf\{B\}$ is the magnetic flux density (in units of teslas, T), also called the magnetic induction.

Equivalent integral form:

$\backslash oint\_S\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}\; =\; 0$

$d\backslash mathbf\{A\}$ is the area of a differential square on the surface $A$ with an outward facing surface normal defining its direction.

Like the electric field's integral form, this equation only works if the integral is done over a closed surface.

This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, this is the mathematical formulation of the asumption that there are no magnetic monopoles.

A changing magnetic flux and the electric field

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

Equivalent integral Form:

$\backslash oint\_\{C\}\; \backslash mathbf\{E\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; -\; \backslash frac\; \{d\backslash Phi\_\{\backslash mathbf\{B\}\}\}\; \{dt\}$ where $\backslash Phi\_\{\backslash mathbf\{B\}\}\; =\; \backslash int\_\{S\}\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}$

where

Φ_B is the magnetic flux through the area A described by the second equation

E is the electric field generated by the magnetic flux

l is a closed path in which current is induced, such as a wire.

The electromotive force (sometimes denoted $\backslash mathcal\{E\}$, not to be confused with the permittivity above) is equal to the value of this integral.

This law corresponds to the Faraday's law of electromagnetic induction.

Some textbooks show the right hand sign of the Integral form with an N (representing the number of coils of wire that are around the edge of A) in front of the flux derivative. The N can be taken care of in calculating A (multiple wire coils means multiple surfaces for the flux to go through), and it is an engineering detail so it has been omitted here.

The negative sign is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's law.

This equation relates the electric and magnetic fields, but it also has a lot of practical applications, too. This equation describes how electric motors and electric generators work. Specifically, it demonstrates that a voltage can be generated by varying the magnetic flux passing through a given area over time, such as by uniformly rotating a loop of wire through a fixed magnetic field. In a motor or generator, the fixed excitation is provided by the field circuit and the varying voltage is measured across the armature circuit. In some types of motors/generators, the field circuit is mounted on the rotor and the armature circuit is mounted on the stator, but other types of motors/generators employ the reverse configuration.

Maxwell's equations apply to a right-handed coordinate system. To apply them unmodified to a left handed system would mean a reversal of polarity of magnetic fields (not inconsistent, but confusingly against convention).

The source of the magnetic field

$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}\; +\; \backslash frac\; \{\backslash partial\; \backslash mathbf\{D\}\}\; \{\backslash partial\; t\}$

where H is the magnetic field strength (in units of A/m), related to the magnetic flux B by a constant called the permeability, μ (B = μH), and J is the current density, defined by: $\backslash begin\{matrix\}\backslash mathbf\{J\}\; =\; \backslash rho\_q\backslash mathbf\{v\}\backslash end\{matrix\}$ where v is a vector field called the drift velocity that describes the velocities of the charge carriers which have a density described by the scalar function ρ_q.

In free space, the permeability μ is the permeability of free space, μ₀, which is defined to be exactly 4π×10^-7 W/A·m. Also, the permittivity becomes the permittivity of free space ε₀. Thus, in free space, the equation becomes:

$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash mu\_0\; \backslash mathbf\{J\}\; +\; \backslash mu\_0\backslash varepsilon\_0\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\{\backslash partial\; t\}$

Equivalent integral form:

$\backslash oint\_C\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; \backslash mu\_0\; I\_\backslash mathrm\{encircled\}\; +\; \backslash mu\_0\backslash varepsilon\_0\; \backslash int\_S\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\{\backslash partial\; t\}\; \backslash cdot\; d\; \backslash mathbf\{A\}$

$\backslash mathbf\{l\}$ is the edge of the open surface A (any surface with the curve $\backslash mathbf\{l\}$ as its edge will do), and I_encircled is the current encircled by the curve $\backslash mathbf\{l\}$ (the current through any surface is defined by the equation: $\backslash begin\{matrix\}I\_\{\backslash mathrm\{through\}\backslash \; A\}\; =\; \backslash int\_S\; \backslash mathbf\{J\}\backslash cdot\; d\backslash mathbf\{A\}\backslash end\{matrix\}$).In some situations, this integral form of Ampere-Maxwell Law appears in:

$\backslash oint\_C\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; \backslash mu\_0\; (I\_\backslash mathrm\{enc\}\; +\; I\_\backslash mathrm\{d,enc\})$

for

$\backslash varepsilon\_0\; \backslash int\_S\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\{\backslash partial\; t\}\; \backslash cdot\; d\; \backslash mathbf\{A\}$

is sometimes called displacement current

If the electric flux density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's law.

Maxwell's equations in CGS units

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The above equations are given in the International System of Units, or SI for short. In a related unit system, called cgs (short for centimeter-gram-second), the equations take the following form:

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; 4\backslash pi\backslash rho$

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

$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$

$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\; \{\backslash partial\; t\}\; +\; \backslash frac\{4\backslash pi\}\{c\}\; \backslash mathbf\{J\}$

Where c is the speed of light in a vacuum. For the electromagnetic field in a vacuum, the equations become:

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

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

$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$

$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\{\backslash partial\; t\}$

The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:

$\backslash mathbf\{F\}\; =\; q\; (\backslash mathbf\{E\}\; +\; \backslash frac\{\backslash mathbf\{v\}\}\{c\}\; \backslash times\; \backslash mathbf\{B\}),$

where $q\; \backslash$ is the charge on the particle and $\backslash mathbf\{v\}\; \backslash$ is the particle velocity. This is slightly different from the SI-unit expression above. For example, here the magnetic field $\backslash mathbf\{B\}\; \backslash$ has the same units as the electric field $\backslash mathbf\{E\}\; \backslash$.

Formulation of Maxwell's equations in special relativity

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In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form (cgs units):

$\{\; 4\; \backslash pi\; \backslash over\; c\; \}J^\; b\; =\; \{\backslash partial\; F^\{ab\}\; \backslash over\; \{\backslash partial\; x^a\}\; \}\; \backslash equiv\; \backslash partial\_a\; F^\{ab\}\; \backslash equiv\; \{F^\{ab\}\}\_\{,a\}\; \backslash ,\backslash !$,

and

$0\; =\; \backslash partial\_c\; F\_\{ab\}\; +\; \backslash partial\_b\; F\_\{ca\}\; +\; \backslash partial\_a\; F\_\{bc\}\; \backslash equiv\; \{F\_\{ab\}\}\_\{,c\}\; +\; \{F\_\{ca\}\}\_\{,b\}\; +\{F\_\{bc\}\}\_\{,a\}\; \backslash equiv\; \backslash epsilon\_\{dabc\}\; \{F^\{bc\}\}\_\{,a\}$

where $\backslash ,\; J^a$ is the 4-current, $\backslash ,\; F^\{ab\}$ is the field strength tensor, $\backslash ,\; \backslash epsilon\_\{abcd\}$ is the Levi-Civita symbol, and

$\{\; \backslash partial\; \backslash over\; \{\; \backslash partial\; x^a\; \}\; \}\; \backslash equiv\; \backslash partial\_a\; \backslash equiv\; \{\}\_\{,a\}\; \backslash equiv\; (\backslash partial/\backslash partial\; ct,\; \backslash nabla)$

is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations.

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and Ampere's law with Maxwell's correction. The second equation is an expression of the homogenous equations, Faraday's law of induction and the absence of magnetic monopoles.

Maxwell's equations in terms of differential forms

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In a vacuum, where ε and μ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. Maxwell's equations then reduce to the Bianchi identity

$d\backslash bold\{F\}=0$

where d denotes the exterior derivative - a differential operator acting on forms - and the source equation

$d\; *\; \{\backslash bold\{F\}\}=\backslash bold\{J\}$

where the (dual) Hodge star operator * is a linear transformation from the space of 2 forms to the space of 4-2 forms defined by the metric in Minkowski space (or in four dimensions by its conformal class), and the fields are in natural units where $1/4\backslash pi\backslash epsilon\_0=1$. Here, the 3-form J is called the "electric current" or "current (3-)form" satisfying the continuity equation

$d\{\backslash bold\{J\}\}=0$

As the exterior derivative is defined on any manifold, this formulation of electromagnetism works for any 4-dimensional oriented manifold with a Lorentz metric, e.g. on the curved space-time of general relativity.

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call

$C:\backslash Lambda^2\backslash ni\backslash bold\{F\}\backslash mapsto\; \backslash bold\{G\}\backslash in\backslash Lambda^\{(4-2)\}$

the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become:

$d\backslash bold\{F\}\; =\; 0$

$d\backslash bold\{G\}\; =\; \backslash bold\{J\}$

where the tree current form J still satisfies the continuity equation dJ= 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms $\backslash bold\{\backslash theta\}^p$,

$\backslash bold\{F\}\; =\; F\_\{pq\}\backslash bold\{\backslash theta\}^p\backslash wedge\backslash bold\{\backslash theta\}^q$.

the constitutive relation takes the form

$G\_\{pq\}\; =\; C\_\{pq\}^\{mn\}F\_\{mn\}$

where the field coefficient functions are antisymmetric in the indices and the constitutive coefficients are antisymmetric in the corresponding pairs. The Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking

$C\_\{pq\}^\{mn\}\; =\; g^\{ma\}g^\{nb\}\; \backslash epsilon\_\{abpq\}\; \backslash sqrt\{-g\}$

which up to scaling is the only invariant tensor of this type that can be defined with the metric.

In this formulation, electromagnetism generalises immediately to any 4 dimensional oriented manifold or with small adaptations any manifold, requiring not even a metric. Thus the expression of Maxwell's equations in terms of differential forms leads to a further notational simplification. Whereas Maxwell's Equations could be written as two tensor equations instead of eight scalar equations, from which the propagation of electromagnetic disturbances and the continuity equation could be derived with a little effort, using differential forms leads to an even simpler derivation of these results. The price one pays for this simplification, however, is a need for knowledge of more technical mathematics.

Conceptual insight from this formulation

On the conceptual side, from a point of view of physics, this shows that the second and third Maxwell equations should be grouped together, be called the homogeneous ones, and be seen as geometric identities expressing nothing else that the field F derives from a more "fundamental" potential A, while the first and last one should be seen as the dynamical equations of motion, obtained via the Lagrangian principle of least action, from the "interaction term" A J (introduced through gauge covariant derivatives), coupling the field to matter.

Often, the time derivative in the third law motivates calling this equation "dynamical", which is somehow 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 Fadeev-Popov ghosts.

Classical electrodynamics as the curvature of a line bundle

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An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or principal bundles with fibre U(1). The connection $\backslash nabla$ on the line bundle has a curvature $\backslash bold\{F\}\; =\; \backslash nabla^2$ which is a two form that automatically satisfies $d\backslash bold\{F\}\; =\; 0$ and can be interpreted as a field strength. If the line bundle is trivial with flat reference connection d we can write $\backslash nabla\; =\; d+\backslash bold\{A\}$ and F = d A with A the 1-form comprised of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. Some feel that this formulation allows a more natural description of the Aharonov-Bohm effect. In this system a magnetic field shielded by a long super conducting tube defines a flat (F = 0) but non trivial connection outside of the tube. The connection has a non trivial holonomy along a curve encircling the tube which corresponds to a phase shift for electrons waves travelling either side of the tube. This can be detected by a double split electron diffraction experiment by changing the magnetic field. The magnetic field remains constant zero outside of the tube so is undectable classically. (See Michael Murray, Line Bundles, 2002 (PDF web link) for a simple mathematical review of this formulation. See also R. Bott, On some recent interactions between mathematics and physics, Canadian Mathematical Bulletin, 28 (1985) )no. 2 pp 129-164.)''

Maxwell's equations in curved spacetime

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Traditional formulation

Matter and energy generate curvature in spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. Electrodynamics also generates curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. The sourced and source-free equations become (cgs units):

$\{\; 4\; \backslash pi\; \backslash over\; c\; \}J^\; b\; =\; \backslash partial\_a\; F^\{ab\}\; +\; \{\backslash Gamma^a\}\_\{\backslash mu\; a\}\; F^\{\backslash mu\; b\}\; +\; \{\backslash Gamma^b\}\_\{\backslash mu\; a\}\; F^\{a\; \backslash mu\}\; \backslash equiv\; D\_a\; F^\{ab\}\; \backslash equiv\; \{F^\{ab\}\}\_\{;a\}\; \backslash ,\backslash !$,

and

$0\; =\; \backslash partial\_c\; F\_\{ab\}\; +\; \backslash partial\_b\; F\_\{ca\}\; +\; \backslash partial\_a\; F\_\{bc\}\; =\; D\_c\; F\_\{ab\}\; +\; D\_b\; F\_\{ca\}\; +\; D\_a\; F\_\{bc\}$.

Here,

$\{\backslash Gamma^a\}\_\{\backslash mu\; b\}\; \backslash !$

is a Christoffel symbol that characterizes the curvature of spacetime and $D\_c$ is the covariant derivative.

Formulation in terms of differential forms

The above formulation is related to the differential form formulation of the Maxwell equations as follows. We have implicitly chosen local coordinates x^a and therefore have a basis of 1-forms d x^a in every point of the open set where the coordinates are defined. Using this basis we have:

• The field form

$\backslash bold\{F\}\; =\; F\_\{ab\}\; d\backslash ,x^a\; \backslash wedge\; d\backslash ,x^b$

• The current form

$\backslash bold\{J\}\; =\; \{4\; \backslash pi\; \backslash over\; c\; \}\; J^a\; \backslash sqrt\{-g\}\; \backslash ,\; \backslash epsilon\_\{abcd\}\; d\backslash ,x^b\; \backslash wedge\; d\backslash ,x^c\; \backslash wedge\; d\backslash ,x^d$

• the Bianchi identity

$d\backslash bold\{F\}\; =\; 2(\backslash partial\_c\; F\_\{ab\}\; +\; \backslash partial\_b\; F\_\{ca\}\; +\; \backslash partial\_a\; F\_\{bc\})d\backslash ,x^a\backslash wedge\; d\backslash ,x^b\; \backslash wedge\; d\backslash ,x^c\; =\; 0$

• the source equation

$d\; *\; \backslash bold\{F\}\; =\; \{F^\{ab\}\}\_\{;a\}\backslash sqrt\{-g\}\; \backslash ,\; \backslash epsilon\_\{bcde\}d\backslash ,x^c\; \backslash wedge\; d\backslash ,x^d\; \backslash wedge\; d\backslash ,x^e\; =\; \backslash bold\{J\}$

• the continuity equation

$d\backslash bold\{J\}\; =\; \{\; 4\; \backslash pi\; \backslash over\; c\; \}\; \{J^a\}\_\{;a\}\; \backslash sqrt\{-g\}\; \backslash ,\; \backslash epsilon\_\{abcd\}d\backslash ,x^a\backslash wedge\; d\backslash ,x^b\; \backslash wedge\; d\backslash ,x^c\; \backslash wedge\; d\backslash ,x^d\; =\; 0$

Here g is as usual the determinant of the metric tensor g^ab.

See also

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• Electromagnetic wave equation
• Nonhomogeneous electromagnetic wave equation
• Finite-difference time-domain method
• Jefimenko's equations
• Abraham-Lorentz force
• Catt, Walton and Davidson. "The History of Displacement Current". Wireless World, March 1979.

References

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Journal articles

• 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

• Larmor, J. (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).
• Lorentz, H. A. (1899) "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
• Lorentz, H. A. (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
• Poincaré, H. (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaies, V, 253-78.
• Poincaré, H (1901) Science and Hypothesis
• Poincaré, H. (1905) "Sur la dynamique de l'electron", Comptes Rendues, 140, 1504-8.

see

• Macrossan, M. N. (1986) note on relativity before Einstein", Brit. J. Phil. Sci., 37, 232-234

University level textbooks

Undergraduate

• Hoffman, Banesh, 1983. Relativity and Its Roots. W. H. Freeman.
• Lounesto, Pertti, 1997. Clifford Algebras and Spinors. Cambridge Univ. Press. Chpt. 8 sets out several variants of the equations, using exterior algebra and differential forms.
• .
• Stevens, Charles F., 1995. The Six Core Theories of Modern Physics. MIT Press. ISBN 0-262-69188-4.

Graduate

• Landau, L. D., 1987. The Classical Theory of Fields (Course of Theoretical Physics: Volume 2). Oxford: Butterworth-Heinemann.
• James Clerk Maxwell, 1954. A Treatise on Electricity and Magnetism. Dover. ISBN 0486606376.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, 1973. Gravitation. W.H. Freeman. ISBN 0-7167-0344-0. Sets out the equations using differential forms.

External links

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• Fitzpatrick, Richard, "Lecture series: Relativity and electromagnetism". Advanced Classical Electromagnetism, PHY387K. University of Texas at Austin, Fall 1996.
• MISN-0-513 Maxwell's Equations: Electricity and Magnetism (PDF file) by R. Young for Project PHYSNET.
• Feynman's derivation of Maxwell equations and extra dimensions.
• A Treatise on Electricity And Magnetism - Volume 1 - 1873 - Posner Memorial Collection - Carnegie Mellon University
• A Treatise on Electricity And Magnetism - Volume 2 - 1873 - Posner Memorial Collection - Carnegie Mellon University
• Original Maxwell Equations - Maxwell's 20 Equations in 20 Unknowns - PDF
• On Physical Lines of Force - 1861 Maxwell's 1861 paper describing Faraday's lines of Force - Predecessor to 1873 Treatise
• A Dynamical Theory Of The Electromagnetic Field - 1865 Maxwell's 1865 paper describing his 20 Equations in 20 Unknowns - Predecessor to the 1873 Treatise

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