Maxwell's equations

For thermodynamic relations, see Maxwell relations.

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric 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 the electric and magnetic potentials is preferred.

Maxwell's equations are an approximation—usually an extremely close approximation—to the more accurate theory of quantum electrodynamics.

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.

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.

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 called magnetic monopoles), analogous to electric charges. Instead, the magnetic field due to materials is generated by a configuration called a dipole. Magnetic dipoles are best represented as loops of current but resemble positive and negative 'magnetic charges', inseparably bound together, having no net 'magnetic charge'. In terms of field lines, this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back. In other words, any magnetic field line that enters a given volume must somewhere exit that volume. Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

Faraday's law

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 varying magnetic field creates ("induces") an electric field. This aspect of electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field, which in turn generates an electric field in a nearby wire. (Note: there are two closely related equations which are called Faraday's law. The form used in Maxwell's equations is always valid but more restrictive than that originally formulated by Michael Faraday.)

Ampère's law with Maxwell's correction

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

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

Ampère's law with Maxwell's correction states that magnetic fields can be generated in two ways: by electrical current (this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's correction").

Maxwell's correction to Ampère's law is particularly important: it shows that not only a changing magnetic field induces an electric field, but also a changing electric field induces a magnetic field. 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, 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 of electromagnetism and optics.

Units and summary of equations

Maxwell's equations vary with the unit system used. Though the general form remains the same, various definitions get 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), Lorentz–Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). See below for CGS-Gaussian units.

For a description of the difference between the microscopic and macroscopic variants of Maxwell's equations see the relevant sections below.

In the equations given below, 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.

Table of the "microscopic" equations

Formulation in terms of total charge and current

Name	Differential form	Integral form
Gauss's law	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$	$\scriptstyle \partial V$ $\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {Q(V)}{\varepsilon _{0}}}$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	$\scriptstyle \partial V$ $\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\iint _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$
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}}\$	$\oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}I_{S}+\mu _{0}\varepsilon _{0}\iint _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

Table of the "macroscopic" equations

Formulation in terms of free charge and current

Name	Differential form	Integral form
Gauss's law	$\nabla \cdot \mathbf {D} =\rho _{f}$	$\scriptstyle \partial V$ $\mathbf {D} \cdot \mathrm {d} \mathbf {A} =Q_{f}(V)$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	$\scriptstyle \partial V$ $\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\iint _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$
Ampère's circuital law (with Maxwell's correction)	$\nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}$	$\oint _{\partial S}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =I_{f,S}+\iint _{S}{\frac {\partial \mathbf {D} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

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
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 also called the magnetic field density also called 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 also called the electric flux density	coulombs per square meter or equivalently, newton per volt-meter
H	magnetizing field also called auxiliary magnetic field also called magnetic field intensity also called magnetic field	ampere per meter
$\mathbf {\nabla \cdot }$	the divergence operator	per meter (factor contributed by applying either operator)
$\mathbf {\nabla \times }$	the curl operator
${\frac {\partial }{\partial t}}$	partial derivative with respect to time	per second (factor contributed by applying the operator)
S and ∂S	S is any surface, and ∂S is its boundary curve. The surface is fixed (unchanging in time).
V and ∂V	V is any three-dimensional volume, and ∂V is its boundary surface. The volume is fixed (unchanging in time).
dA	differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S	square meters
dl	differential vector element of path length tangential to the path/curve	meters
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
ρf	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
Jf	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
$\,Q_{f}(V)$	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
$\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l}$	line integral of the electric field along the boundary ∂S of a surface S (∂S is always a closed curve).	joules per coulomb
$\oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l}$	line integral of the magnetic field over the closed boundary ∂S of the surface S	tesla-meters
$\scriptstyle \partial V$ $\mathbf {E} \cdot \mathrm {d} \mathbf {A}$	the electric flux (surface integral of the electric field) through the (closed) surface ∂V (the boundary of the volume V)	joule-meter per coulomb
$\scriptstyle \partial V$ $\mathbf {B} \cdot \mathrm {d} \mathbf {A}$	the magnetic flux (surface integral of the magnetic B-field) through the (closed) surface ∂V (the boundary of the volume V)	tesla meters-squared or webers
$\scriptstyle \partial V$ $\mathbf {D} \cdot \mathrm {d} \mathbf {A}$	flux of electric displacement field through the (closed) surface ∂V (the boundary of the volume V)	coulombs
$\int \!\!\!\!\int _{S}\mathbf {J} _{f}\cdot \mathrm {d} \mathbf {A} =I_{f,s}$	net free electrical current passing through the surface S (not including bound current)	amperes
$\int \!\!\!\!\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {A} =I_{S}$	net electrical current passing through the surface S (including both free and bound current)	amperes

Proof that the two general formulations are equivalent

The two alternate general formulations of Maxwell's equations given above are mathematically equivalent and related by the following relations:

• 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.

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. 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.

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.

Formulation in terms of total charge and current

Name	Differential form	Integral form
Gauss's law	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$	$\scriptstyle \partial V$ $\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {Q(V)}{\varepsilon _{0}}}$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	$\scriptstyle \partial V$ $\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\iint _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$
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}}\$	$\oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}I_{S}+\mu _{0}\varepsilon _{0}\iint _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

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 waves traveling 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 motion and are in phase with each other. A sinusoidal 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.

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 D which 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.

Bound charge and current

Main articles: Bound charge and Bound 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 P enter and leave the material. For non-uniform P, a charge is also produced in the bulk.

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.

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.

Equations

Formulation in terms of free charge and current

Name	Differential form	Integral form
Gauss's law	$\nabla \cdot \mathbf {D} =\rho _{f}$	$\scriptstyle \partial V$ $\mathbf {D} \cdot \mathrm {d} \mathbf {A} =Q_{f}(V)$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	$\scriptstyle \partial V$ $\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\iint _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$
Ampère's circuital law (with Maxwell's correction)	$\nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}$	$\oint _{\partial S}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =I_{f,S}+\iint _{S}{\frac {\partial \mathbf {D} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

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 called constitutive relations.

Determining the constitutive relationship between the auxiliary fields D and H and the E and B fields starts with the definition of the auxiliary fields themselves:

$\mathbf {D} (\mathbf {r} ,t)=\varepsilon _{0}\mathbf {E} (\mathbf {r} ,t)+\mathbf {P} (\mathbf {r} ,t)$

$\mathbf {H} (\mathbf {r} ,t)={\frac {1}{\mu _{0}}}\mathbf {B} (\mathbf {r} ,t)-\mathbf {M} (\mathbf {r} ,t),$

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. Before getting to how to calculate M and P it is useful to examine some special cases, though.

Without magnetic or dielectric materials

In the absence of magnetic or dielectric materials, the constitutive relations are simple:

$\mathbf {D} =\varepsilon _{0}\mathbf {E} ,\;\;\;\mathbf {H} =\mathbf {B} /\mu _{0}$

where ε₀ and μ₀ are two universal constants, called the permittivity of free space and permeability of free space, respectively. Substituting these back into Maxwell's macroscopic equations lead directly to Maxwell's microscopic equations, except that the currents and charges are replaced with free currents and free charges. This is expected since there are no bound charges nor currents.

Isotropic linear materials

In an (isotropic) linear material, where P is proportional to E, and M is proportional to B, the constitutive relations are also straightforward. In terms of the polarization P and the magnetization M they are:

$\mathbf {P} =\varepsilon _{0}\chi _{e}\mathbf {E} ,\;\;\;\mathbf {M} =\chi _{m}\mathbf {H} ,$

where χ_e and χ_m are the electric and magnetic susceptibilities of a given material respectively. In terms of D and H the constitutive relations are:

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

where ε and μ are constants (which depend on the material), called the permittivity and permeability, respectively, of the material. These are related to the susceptibilities by:

$\varepsilon =\varepsilon _{0}(1+\chi _{e})\;\;\;\mu =\mu _{0}(1+\chi _{m})$

Substituting in the constitutive relations above into Maxwell's equations in linear, dispersionless, time-invariant materials (differential form only) are:

$\nabla \cdot (\varepsilon \mathbf {E} )=\rho _{f}$

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

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

$\nabla \times (\mathbf {B} /\mu )=\mathbf {J} _{f}+\varepsilon {\frac {\partial \mathbf {E} }{\partial t}}.$

These are formally identical to the general formulation in terms of E and B (given above), except that the permittivity of free space was replaced with the permittivity of the material, the permeability of free space was replaced with the permeability of the material, and only free charges and currents are included (instead of all charges and currents). Unless that material is homogeneous in space, ε and μ cannot be factored out of the derivative expressions on the left sides.

General case

For real-world materials, the constitutive relations are not linear, except approximately. Calculating the constitutive relations from first principles involves determining how P and M are created from a given E and B. These relations may be empirical (based directly upon measurements), or theoretical (based upon statistical mechanics, transport theory or other tools of condensed matter physics). The detail employed may be macroscopic or microscopic, depending upon the level necessary to the problem under scrutiny.

In general, though the constitutive relations can usually still be written:

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

but ε and μ are not, in general, simple constants, but rather functions. Examples are:

• Dispersion and absorption where ε and μ are functions of frequency. (Causality does not permit materials to be nondispersive; see, for example, Kramers–Kronig relations). Neither do the fields need to be in phase which leads to ε and μ being complex. This also leads to absorption.
• Bi-(an)isotropy where H and D depend on both B and E:

$D=\varepsilon E+\xi H\;\;\;B=\mu H+\zeta E.$

• Nonlinearity where ε and μ are functions of E and B.
• Anisotropy (such as birefringence or dichroism) which occurs when ε and μ are second-rank tensors,

$D_{i}=\sum _{j}\varepsilon _{ij}E_{j}\;\;\;B_{i}=\sum _{j}\mu _{ij}H_{j}.$

• Dependence of P and M on E and B at other locations and times. This could be due to spatial inhomogeneity; for example in a domained structure, heterostructure or a liquid crystal, or most commonly in the situation where there are simply multiple materials occupying different regions of space). Or it could be due to a time varying medium or due to hysteresis. In such cases P and M can be calculated as:

$\mathbf {P} (\mathbf {r} ,t)=\varepsilon _{0}\int {\rm {d}}^{3}\mathbf {r} '{\rm {d}}t'\;{\hat {\chi }}_{e}(\mathbf {r} ,\mathbf {r} ',t,t';\mathbf {E} )\,\mathbf {E} (\mathbf {r} ',t')$

$\mathbf {M} (\mathbf {r} ,t)={\frac {1}{\mu _{0}}}\int {\rm {d}}^{3}\mathbf {r} '{\rm {d}}t'\;{\hat {\chi }}_{m}(\mathbf {r} ,\mathbf {r} ',t,t';\mathbf {B} )\,\mathbf {B} (\mathbf {r} ',t'),$

in which the permittivity and permeability functions are replaced by integrals over the more general electric and magnetic susceptibilities.

In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects. For example: optical nonlinearities can be neglected for low field strengths; material dispersion is unimportant when frequency is limited to a narrow bandwidth; material absorption can be neglected for wavelengths for which a material is transparent; and metals with finite conductivity often are approximated at microwave or longer wavelengths as perfect metals with infinite conductivity (forming hard barriers with zero skin depth of field penetration).

It may be noted that man-made materials can be designed to have customized permittivity and permeability, such as metamaterials and photonic crystals.

Calculation of constitutive relations

See also: Computational electromagnetics

In general, the constitutive equations are theoretically determined by calculating how a molecule responds to the local fields through the Lorentz force. Other forces may need to be modeled as well such as lattice vibrations in crystals or bond forces. Including all of the forces leads to changes in the molecule which are used to calculate P and M as a function of the local fields.

The local fields differ from the applied fields due to the fields produced by the polarization and magnetization of nearby material; an effect which also needs to be modeled. Further, real materials are not continuous media; the local fields of real materials vary wildly on the atomic scale. The fields need to be averaged over a suitable volume to form a continuum approximation.

These continuum approximations often require some type of quantum mechanical analysis such as quantum field theory as applied to condensed matter physics. See, for example, density functional theory, Green–Kubo relations and Green's function. Various approximate transport equations have evolved, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier–Stokes equations. Some examples where these equations are applied are magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, plasma modeling. An entire physical apparatus for dealing with these matters has developed. A different set of homogenization methods (evolving from a tradition in treating materials such as conglomerates and laminates) are based upon approximation of an inhomogeneous material by a homogeneous effective medium (valid for excitations with wavelengths much larger than the scale of the inhomogeneity).

The theoretical modeling of the continuum-approximation properties of many real materials often rely upon measurement as well, for example, ellipsometry measurements.

History

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, remarkably close to the speed of light, which had recently been measured at 3.14×10⁸ m/s by Hippolyte Fizeau in 1848 and at 2.98×10⁸ m/s by Léon Foucault in 1850. However, Weber and Kohlrausch did not make the connection to the speed of light. 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.

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, concurrently with similar work by Willard Gibbs and Heinrich Hertz, 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, and are sometimes still known as the Maxwell–Heaviside equations.

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 the electromagnetic 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."

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; this effort was somewhat controversial, 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. 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.

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:

1. Equation (56) in Maxwell's 1861 paper is ∇ ⋅ B = 0.
2. Equation (112) is Ampère's circuital law with Maxwell's displacement current added. It is the addition of displacement current that 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).
3. Equation (115) is Gauss's law.
4. 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. 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 name Lorentz 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,

1. Magnetic induction current causes a magnetic current density B = μ H was essentially a rotational analogy to the linear electric current relationship,
2. 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 law force.

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, whereas H parallels with D.

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", 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).

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 induction rather 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.

A Treatise on Electricity and Magnetism

Main article: A Treatise on Electricity and Magnetism

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

Maxwell's equations and relativity

See also: History of special relativity

Maxwell's original equations are based on the idea that light travels through a sea of molecular vortices known as the "luminiferous aether", and that the speed of light has to be respective to the reference frame of this aether. Measurements designed to measure the speed of the Earth through the aether conflicted with this notion, though.

A more theoretical approach was suggested by Hendrik Lorentz along with George FitzGerald and Joseph Larmor. 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) analyzed the coordination of moving clocks by exchanging light signals. He also established the mathematical group property of the Lorentz transformation (Poincaré 1905). Sometimes this transformation is called the FitzGerald - Lorentz Transformation or even the FitzGerald - Lorentz - Einstein Transformation.

Albert Einstein dismissed the notion of the aether as an unnecessary one, and he concluded that Maxwell's equations predicted the existence of a fixed speed of light, independent of the velocity of the observer. Hence, he used the Maxwell's equations as the starting point for his Special Theory of Relativity. In doing so, he established that the FitzGerald - Lorentz Transformation is valid for all matter and space, and not just Maxwell's Equations. Maxwell's Equations played a key role in Einstein's groundbreaking scientic paper on Special Relativity (1905). For example, in the opening paragraph of his paper, he began his theory by noting that a description of an electric conductor moving with respect to a magnet must generate a consistent set of fields regardless of whether the force is calculated in the rest frame of the magnet or that of the conductor.

The General Theory of Relativity has also has a close relationship with Maxwell's equations. For example, Theodor Kaluza and Oskar Klein showed duirng the 1920s that Maxwell's equations could be derived by extending General Relativity into five physical dimensions. This strategy of using additional dimensions to unify different forces remains an active area of research in physics.

Modified to include magnetic monopoles

Main article: magnetic monopole

Maxwell's equations provide for an electric charge, but posit no magnetic charge. Magnetic charge has never been seen and may not exist. Nevertheless, Maxwell's equations including magnetic charge (and magnetic current) are of some theoretical interest.

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

Name	Without magnetic monopoles	With magnetic monopoles (hypothetical)
Gauss's law:	$\nabla \cdot \mathbf {E} =4\pi \rho _{\mathrm {e} }$	$\nabla \cdot \mathbf {E} =4\pi \rho _{\mathrm {e} }$
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):	$\nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {j} _{\mathrm {e} }$	$\nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {j} _{\mathrm {e} }$

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.

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 and initial conditions 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. 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, 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.
• 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. 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 uniquely determined from E and B via the Lorentz force law.) 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.

Gaussian units

Main article: Gaussian units

Gaussian units is a popular electromagnetism variant of the centimetre gram second system of units (cgs). In gaussian units, Maxwell's equations are:

$\nabla \cdot \mathbf {D} =4\pi \rho _{\mathrm {f} }$

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

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

$\nabla \times \mathbf {H} ={\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {J} _{\mathrm {f} }$

where c is the speed of light in a vacuum. The microscopic equations are:

$\nabla \cdot \mathbf {E} =4\pi \rho _{\mathrm {tot} }$

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

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

$\nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {J} _{\mathrm {tot} }.$

The relation between electric displacement field, electric field and polarization density is:

$\mathbf {D} =\mathbf {E} +4\pi \mathbf {P} .$

And likewise the relation between magnetic induction, magnetic field and total magnetization is:

$\mathbf {B} =\mathbf {H} +4\pi \mathbf {M} .$

In the linear approximation, the electric susceptibility and magnetic susceptibility are defined so that:

$\mathbf {P} =\chi _{\mathrm {e} }\mathbf {E}$,     $\mathbf {M} =\chi _{\mathrm {m} }\mathbf {H} .$

(Note: although the susceptibilities are dimensionless numbers in both cgs and SI, they differ in value by a factor of 4π.) The permittivity and permeability are:

$\ \varepsilon =1+4\pi \chi _{\mathrm {e} }$,     $\ \mu =1+4\pi \chi _{\mathrm {m} },$

so that

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

In vacuum, ε = μ = 1, therefore D = E, and B = H.

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

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

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

Some equations in the article are given in Gaussian units but not SI or vice-versa. Fortunately, there are general rules to convert from one to the other; see the article Gaussian units for details.

Alternative formulations of Maxwell's equations

Main article: Mathematical descriptions of the electromagnetic field

In terms of a minimum action principle

For the field formulation of Maxwell's equations in terms of a principle of extremal action, see the article on the electromagnetic tensor.

Potential formulation

Main article: Mathematical descriptions of the electromagnetic field

In advanced classical mechanics it is often useful, and in quantum mechanics it is often 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:

$\mathbf {E} =-\mathbf {\nabla } \varphi -{\frac {\partial \mathbf {A} }{\partial t}},$

$\mathbf {B} =\mathbf {\nabla } \times \mathbf {A} .$

With these definitions, the two homogeneous Maxwell's equations (Faraday's Law and Gauss's law for magnetism) are automatically satisfied and the other two (inhomogeneous) equations give the following equations (for "Maxwell's microscopic equations"):

Maxwell's equations (Potential formulation)

$\nabla ^{2}\varphi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon _{0}}}$

$\left(\nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)-\mathbf {\nabla } \left(\mathbf {\nabla } \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=-\mu _{0}\mathbf {J}$

These equations, taken together, are as powerful and complete as Maxwell's equations. Moreover, the mathematics is often simplified, because the electric and magnetic fields each have three vector components that need to be calculated at each point, or six numbers altogether, while the electric and magnetic potentials have only four components altogether.

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 formulations

Main articles: Covariant formulation of classical electromagnetism and Classical electromagnetism and special relativity

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. In the frame of the magnet, that conductor experiences a magnetic force. But in the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an electric field. 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. obviously consistent with special relativity, even with just a glance at the equations—using covariant and contravariant four-vectors and tensors.

(This section uses Einstein notation, including Einstein summation convention. See also Ricci calculus for a summary of tensor index notations, and raising and lowering indices for definition of superscript and subscript indices, and how to switch between them. The Minkowski metric tensor here is "−+++".)

One ingredient in this formulation is the four-current:

$J^{\alpha }=(c\rho ,\mathbf {J} )$

where ρ is the charge density and J is the current density.

The other ingredient is the electromagnetic tensor, a rank-2 covariant antisymmetric tensor combining the electric and magnetic fields:

$F^{\alpha \beta }=\left({\begin{matrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&B_{z}&-B_{y}\\-E_{y}/c&-B_{z}&0&B_{x}\\-E_{z}/c&B_{y}&-B_{x}&0\end{matrix}}\right).$

With these ingredients, Maxwell's equations can be written:

Maxwell's equations (Covariant formulation)

${\dfrac {\partial F^{\beta \alpha }}{\partial x^{\alpha }}}=\mu _{0}J^{\beta }$

${\dfrac {\partial F_{\alpha \beta }}{\partial x^{\gamma }}}+{\dfrac {\partial F_{\gamma \alpha }}{\partial x^{\beta }}}+{\dfrac {\partial F_{\beta \gamma }}{\partial x^{\alpha }}}=0$

Notice the cyclic permutation of indices in the second equation: ${\begin{array}{rc}&\scriptstyle {\alpha \,\,\longrightarrow \,\,\beta }\\&\nwarrow _{\gamma }\swarrow \end{array}}$.

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss's law and Ampère's law with Maxwell's correction. The second equation is an expression of the two homogeneous equations, Faraday's law of induction and Gauss's law for magnetism.

An alternative manifestly-covariant formulation uses potentials (as in the previous section) in the Lorenz gauge. This involves the electromagnetic four-potential defined by

$A^{\mu }=\left(\varphi /c,\mathbf {A} \right)$

formed from the vector potential A and the scalar potential φ. The resulting single equation, due to Arnold Sommerfeld, a generalization of an equation due to Bernhard Riemann and known as the Riemann–Sommerfeld equation or the covariant form of the Maxwell equations, is:

Maxwell's equations (Covariant Lorenz gauge formulation)

$\Box A^{\mu }=\mu _{0}J^{\mu }$

where $\Box =-{\frac {\partial }{\partial x^{\alpha }}}{\frac {\partial }{\partial x_{\alpha }}}=\left({1 \over c^{2}}{\partial ^{2} \over \partial t^{2}}-\nabla ^{2}\right)$ is the d'Alembertian operator, or four-Laplacian, sometimes written $\Box ^{2}$, or $\Box \cdot \Box$, where $\Box$ is the four-gradient.

Differential geometric formulations

In free space, where ε = ε₀ and μ = μ₀ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. In what follows, cgs-Gaussian units, not SI units are used. (To convert to SI, see here.) 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 and the source equation, respectively:

Maxwell's equations (Differential forms)

$\mathrm {d} {\mathbf {F}}=0$

$\mathrm {d} \,{*{\mathbf {F}}}={\mathbf {J}}$

where d denotes the exterior derivative — a natural coordinate and metric independent differential operator acting on forms, and 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 (in four dimensions even by any metric conformal to this metric). The fields are in natural units where 1/4πε₀ = 1. Here, the 3-form J is called the electric current form or current 3-form satisfying the continuity equation

$\mathrm {d} {\mathbf {J}}=0.$

The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any manifold, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. In particular the differential form version of the Maxwell equations are a convenient and intuitive formulation of the Maxwell equations in 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

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

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:

$\mathrm {d} {\mathbf {F}}=0$

$\mathrm {d} {\mathbf {G}}={\mathbf {J}}$

where the current 3-form J still satisfies the continuity equation dJ = 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms θ^p,

${\mathbf {F}}={\frac {1}{2}}F_{pq}{\mathbf {\theta }}^{p}\wedge {\mathbf {\theta }}^{q}.$

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. In particular, the Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking

$C_{pq}^{mn}={\frac {1}{2}}g^{ma}g^{nb}\epsilon _{abpq}{\sqrt {-g}}$

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 and conceptual 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.

Conceptual insight from this formulation

On the conceptual side, from the 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 than: 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 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.

Geometric algebra (GA) formulation

Main article: Mathematical descriptions of the electromagnetic field

In geometric algebra, Maxwell's equations are reduced to a single equation,

Maxwell's equations (GA formulation)

$\left({\frac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }}\right)F=\mu _{0}cJ$

where F and J are multivectors

$F={\mathbf {E}}+Ic{\mathbf {B}}$

and

$J=c\rho +{\mathbf {J}}.$

with the unit pseudoscalar I² = −1.

The GA spatial gradient operator ∇ acts on a vector field, such that

${\boldsymbol {\nabla }}{\mathbf {F}}={\boldsymbol {\nabla }}\cdot {\mathbf {F}}+I{\boldsymbol {\nabla }}\times {\mathbf {F}},$

In spacetime algebra using the same geometric product the equation is simply

$\nabla F=\mu _{0}cJ,$

the spacetime derivative of the electromagnetic field is its source. Here the (non-bold) spacetime gradient

$\nabla =\gamma ^{\mu }\partial _{\mu }$

is a four vector, as is the current density

$J=\gamma _{\mu }J^{\mu }=\gamma _{0}c\rho +J^{k}\gamma _{k}=(c\rho +{\mathbf {J}})\gamma _{0}.$

For a demonstration that the equations given reproduce Maxwell's equations see the main article.

Classical electrodynamics as the curvature of a line bundle

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 ∇ on the line bundle has a curvature F = ∇² which is a two-form that automatically satisfies dF = 0 and can be interpreted as a field-strength. If the line bundle is trivial with flat reference connection d we can write ∇ = d + A and F = dA with A the 1-form composed of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov–Bohm effect. In this experiment, a static magnetic field runs through a long magnetic wire (e.g., an iron wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment. This means by definition that the connection ∇ is flat there.

However, as mentioned, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern.

Curved spacetime

Main article: Maxwell's equations in curved spacetime

Traditional formulation

Matter and energy generate curvature of spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum 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. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs-Gaussian units):

${4\pi \over c}j^{\beta }=\partial _{\alpha }F^{\alpha \beta }+{\Gamma ^{\alpha }}_{\mu \alpha }F^{\mu \beta }+{\Gamma ^{\beta }}_{\mu \alpha }F^{\alpha \mu }\ {\stackrel {\mathrm {def} }{=}}\ \nabla _{\alpha }F^{\alpha \beta }\ {\stackrel {\mathrm {def} }{=}}\ {F^{\alpha \beta }}_{;\alpha }\,\!$

and

$0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }=\nabla _{\gamma }F_{\alpha \beta }+\nabla _{\beta }F_{\gamma \alpha }+\nabla _{\alpha }F_{\beta \gamma }.\,$

Here,

${\Gamma ^{\alpha }}_{\mu \beta }\!$

is a Christoffel symbol that characterizes the curvature of spacetime and ∇_α is the covariant derivative.

Formulation in terms of differential forms

The formulation of the Maxwell equations in terms of differential forms can be used without change in general relativity. The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates x^α which gives a basis of 1-forms dx^α in every point of the open set where the coordinates are defined. Using this basis and cgs-Gaussian units we define

• The antisymmetric infinitesimal field tensor F_αβ, corresponding to the field 2-form F

${\mathbf {F}}:={\frac {1}{2}}F_{\alpha \beta }\,\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }.$

• The current-vector infinitesimal 3-form J

${\mathbf {J}}:={4\pi \over c}\left({\frac {1}{6}}j^{\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }.\right)$

Here g is as usual the determinant of the metric tensor g_αβ. A small computation that uses the symmetry of the Christoffel symbols (i.e., the torsion-freeness of the Levi-Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:

• the Bianchi identity

$\mathrm {d} {\mathbf {F}}=2(\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma })\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }=0,$

• the source equation

$\mathrm {d} \,{*{\mathbf {F}}}={\frac {1}{6}}{F^{\alpha \beta }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\beta \gamma \delta \eta }\mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }\wedge \mathrm {d} \,x^{\eta }={\mathbf {J}},$

• the continuity equation

$\mathrm {d} {\mathbf {J}}={4\pi \over c}{j^{\alpha }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }=0.$

See also

2

Further reading

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

• Joseph Larmor (1897) "On a dynamical theory of the electric and luminiferous medium", Phil. Trans. Roy. Soc. 190, 205-300 (third and last in a series of papers with the same name).
• Hendrik Lorentz (1899) "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
• Hendrik Lorentz (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
• Henri Poincaré (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Néerlandaises, V, 253-78.
• Henri Poincaré (1901) Science and Hypothesis
• Henri Poincaré (1905) "Sur la dynamique de l'électron", Comptes rendus de l'Académie des Sciences, 140, 1504-8.

see

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

University level textbooks

Undergraduate

• Feynman, Richard P. (2005). The Feynman Lectures on Physics. Vol. 2 (2nd ed.). Addison-Wesley. ISBN 978-0805390650.
• Fleisch, Daniel (2008). A Student's Guide to Maxwell's Equations. Cambridge University Press. ISBN 978-0521877619.
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Hoffman, Banesh (1983). Relativity and Its Roots. W. H. Freeman.
• Krey, U.; Owen, A. (2007). Basic Theoretical Physics: A Concise Overview. Springer. ISBN 978-3-540-36804-5. See especially part II.
• Pollack, Gerald L.; Stump, Daniel R. (2002). Electromagnetism. Addison Wesley. ISBN 0-8053-8567-3.
• Purcell, Edward Mills (1985). Electricity and Magnetism. McGraw-Hill. ISBN 0-07-004908-4.
• Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (2008). Foundations of Electromagnetic Theory (4th ed.). Addison Wesley. ISBN 978-0321581747.
• Sadiku, Matthew N. O. (2006). Elements of Electromagnetics (4th ed.). Oxford University Press. ISBN 0-19-5300483.
• Schwarz, Melvin (1987). Principles of Electrodynamics. Dover. ISBN 0-486-65493-1.
• Stevens, Charles F. (1995). The Six Core Theories of Modern Physics. MIT Press. ISBN 0-262-69188-4.
• Tipler, Paul; Mosca, Gene (2007). Physics for Scientists and Engineers. Vol. 2 (6th ed.). W. H. Freeman. ISBN 978-1429201339.
• Ulaby, Fawwaz T. (2007). Fundamentals of Applied Electromagnetics (5th ed.). Pearson Education. ISBN 0-13-241326-4.
• I.S. Grant, W.R. Phillips (2008). Electromagnetism (2nd ed.). Manchester Physics, John Wiley & Sons. ISBN 9-780471-927129.

Graduate

• Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.
• Panofsky, Wolfgang K. H.; Phillips, Melba (2005). Classical Electricity and Magnetism (2nd ed.). Dover. ISBN 978-0486439242.

Older classics

• Lifshitz, Evgeny (1980). The Classical Theory of Fields (4th ed.). Butterworth-Heinemann. ISBN 0750627689. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Lifshitz, Evgeny (1984). Electrodynamics of Continuous Media (2nd ed.). Butterworth-Heinemann. ISBN 0750626348. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Maxwell, James Clerk (1873). A Treatise on Electricity and Magnetism. Dover. ISBN 0-486-60637-6.
• Misner, Charles W. (1973). Gravitation. W. H. Freeman. ISBN 0-7167-0344-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) Sets out the equations using differential forms.

Computational techniques

• Chew, W. C. (2001). Fast and Efficient Algorithms in Computational Electromagnetics. Artech House. ISBN 1-58053-152-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Harrington, R. F. (1993). Field Computation by Moment Methods. Wiley-IEEE Press. ISBN 0-78031-014-4.
• Jin, J. (2002). The Finite Element Method in Electromagnetics (2nd ed.). Wiley-IEEE Press. ISBN 0-47143-818-9.
• Lounesto, Pertti (1997). Clifford Algebras and Spinors. Cambridge University Press. ISBN 0521599164. Chapter 8 sets out several variants of the equations using exterior algebra and differential forms.
• Taflove, Allen (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd ed.). Artech House. ISBN 1-58053-832-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Mathematical aspects of Maxwell's equation are discussed on the Dispersive PDE Wiki.

Modern treatments

• Electromagnetism, B. Crowell, Fullerton College
• Lecture series: Relativity and electromagnetism, R. Fitzpatrick, University of Texas at Austin
• Electromagnetic waves from Maxwell's equations on Project PHYSNET.
• MIT Video Lecture Series (36 x 50 minute lectures) (in .mp4 format) - Electricity and Magnetism Taught by Professor Walter Lewin.

Historical

• James Clerk Maxwell, A Treatise on Electricity And Magnetism Vols 1 and 2 1904—most readable edition with all corrections—Antique Books Collection suitable for free reading online.
• 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
• 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
• Maxwell, James Clerk, "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.)
• Catt, Walton and Davidson. "The History of Displacement Current". Wireless World, March 1979.
• Reprint from Dover Publications (ISBN 0-486-60636-8)
• Full text of 1904 Edition including full text search.
• A Dynamical Theory Of The Electromagnetic Field – 1865 Maxwell's 1865 paper describing his 20 Equations in 20 Unknowns – Predecessor to the 1873 Treatise

Other

• Feynman's derivation of Maxwell equations and extra dimensions
• Nature Milestones: Photons – Milestone 2 (1861) Maxwell's equations