1st  Top partial differential equation topics 
Maxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave. Individually, the equations are known as Gauss's law, Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction. The set of equations is named after James Clerk Maxwell.
These four equations, together with the Lorentz force law are the complete set of laws of classical electromagnetism. The Lorentz force law itself was actually derived by Maxwell under the name of Equation for Electromotive Force and was one of an earlier set of eight equations by Maxwell.
Contents

This section will conceptually describe each of the four Maxwell's equations, and also how they link together to explain the origin of electromagnetic radiation such as light. The exact equations are set out in later sections of this article.
Maxwell's correction to Ampère's law is particularly important: It means that a changing magnetic field creates an electric field, and a changing electric field creates a magnetic field.^{[1]}^{[2]} Therefore, these equations allow selfsustaining "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,^{[3]} exactly matches the speed of light; indeed, light is one form of electromagnetic radiation (as are Xrays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1864, thereby unifying the previouslyseparate fields of electromagnetism and optics.
The equations in this section are given in SI units. Unlike the equations of mechanics (for example), Maxwell's equations are not unchanged in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places. Other than SI (used in engineering), the units commonly used are Gaussian units (based on the cgs system and considered to have some theoretical advantages over SI^{[4]}), LorentzHeaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). See below for CGSGaussian units.
Two equivalent, general formulations of Maxwell's equations follow. The first separates bound charge and bound current (which arise in the context of dielectric and/or magnetized materials) from free charge and free current (the more conventional type of charge and current). This separation is useful for calculations involving dielectric or magnetized materials. The second formulation treats all charge equally, combining free and bound charge into total charge (and likewise with current). This is the more fundamental or microscopic point of view, and is particularly useful when no dielectric or magnetic material is present. More details, and a proof that these two formulations are mathematically equivalent, are given in section 4.
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.
Name  Differential form  Integral form 

Gauss's law  
Gauss's law for magnetism  
Maxwell–Faraday equation (Faraday's law of induction) 

Ampère's circuital law (with Maxwell's correction) 
Name  Differential form  Integral form 

Gauss's law  
Gauss's law for magnetism  
Maxwell–Faraday equation (Faraday's law of induction) 

Ampère's circuital law (with Maxwell's correction) 
The following table provides the meaning of each symbol and the SI unit of measure:
Symbol  Meaning (first term is the most common)  SI Unit of Measure 

electric field  volt per meter or, equivalently, newton per coulomb 

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, voltsecond per square meter 

electric displacement field also called the electric induction also called the electric flux density 
coulombs per square meter or equivalently, newton per voltmeter 

magnetizing field also called auxiliary magnetic field also called magnetic field intensity also called magnetic field 
ampere per meter  
the divergence operator  per meter (factor contributed by applying either operator)  
the curl operator  
partial derivative with respect to time  per second (factor contributed by applying the operator)  
differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S  square meters  
differential vector element of path length tangential to the path/curve  meters  
permittivity of free space, also called the electric constant, a universal constant  farads per meter  
permeability of free space, also called the magnetic constant, a universal constant  henries per meter, or newtons per ampere squared  
free charge density (not including bound charge)  coulombs per cubic meter  
total charge density (including both free and bound charge)  coulombs per cubic meter  
free current density (not including bound current)  amperes per square meter  
total current density (including both free and bound current)  amperes per square meter  
net free electric charge within the threedimensional volume V (not including bound charge)  coulombs  
net electric charge within the threedimensional volume V (including both free and bound charge)  coulombs  
line integral of the electric field along the boundary ∂S of a surface S (∂S is always a closed curve).  joules per coulomb  
line integral of the magnetic field over the closed boundary ∂S of the surface S  teslameters  
the electric flux (surface integral of the electric field) through the (closed) surface (the boundary of the volume V)  joulemeter per coulomb  
the magnetic flux (surface integral of the magnetic Bfield) through the (closed) surface (the boundary of the volume V)  tesla meterssquared or webers  
magnetic flux through any surface S, not necessarily closed  webers or equivalently, voltseconds  
electric flux through any surface S, not necessarily closed  joulemeters per coulomb  
flux of electric displacement field through any surface S, not necessarily closed  coulombs  
net free electrical current passing through the surface S (not including bound current)  amperes  
net electrical current passing through the surface S (including both free and bound current)  amperes 
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. At microscopic level, Maxwell's equations, ignoring quantum effects, describe fields, charges and currents in free space—but at this level of detail one must include all charges, even those at an atomic level, generally an intractable problem.
Although James Clerk Maxwell is said by some not to be the originator of these equations, he nevertheless derived them independently in conjunction with his molecular vortex model of Faraday's "lines of force". In doing so, he made an important addition to Ampère's circuital law.
All four of what are now described as Maxwell's equations can be found in recognizable form (albeit without any trace of a vector notation, let alone ∇) in his 1861 paper On Physical Lines of Force, in his 1865 paper A Dynamical Theory of the Electromagnetic Field, and also in vol. 2 of Maxwell's "A Treatise on Electricity & Magnetism", published in 1873, in Chapter IX, entitled "General Equations of the Electromagnetic Field". This book by Maxwell predates publications by Heaviside, Hertz and others.
The term Maxwell's equations originally applied to a set of eight equations published by Maxwell in 1865, but nowadays applies to modified versions of four of these equations that were grouped together in 1884 by Oliver Heaviside,^{[6]} concurrently with similar work by Willard Gibbs and Heinrich Hertz.^{[7]} These equations were also known variously as the HertzHeaviside equations and the MaxwellHertz equations,^{[6]} and are sometimes still known as the Maxwell–Heaviside equations.^{[8]}
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 concept of fields was introduced by, among others, Faraday. Albert Einstein wrote:
The precise formulation of the timespace 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 fellowworkers—(Science, May 24, 1940)
The equations were called by some the HertzHeaviside equations, but later Einstein referred to them as the MaxwellHertz equations.^{[6]} However, in 1940 Einstein referred to the equations as Maxwell's equations in "The Fundamentals of Theoretical Physics" published in the Washington periodical Science, May 24, 1940.
Heaviside worked to eliminate the potentials (electrostatic potential and vector potential) that Maxwell had used as the central concepts in his equations;^{[6]} this effort was somewhat controversial,^{[9]} though it was understood by 1884 that the potentials must propagate at the speed of light like the fields, unlike the concept of instantaneous actionatadistance like the then conception of gravitational potential.^{[7]} 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.
The net result of Heaviside's work was the symmetrical duplex set of four equations,^{[6]} all of which originated in Maxwell's previous publications, in particular Maxwell's 1861 paper On Physical Lines of Force, the 1865 paper A Dynamical Theory of the Electromagnetic Field and the Treatise. The fourth was a partial time derivative version of Faraday's law of induction that doesn't include motionally induced EMF; this version is often termed the MaxwellFaraday equation or Faraday's law in differential form to keep clear the distinction from Faraday's law of induction, though it expresses the same law.^{[10]}^{[11]}
The four modern day Maxwell's equations appeared throughout Maxwell's 1861 paper On Physical Lines of Force:
The difference between the and the 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.
It is later clarified in his concept of a sea of molecular vortices that appears in his 1861 paper On Physical Lines of Force  1861. Within that context, represented pure vorticity (spin), whereas 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
was essentially a rotational analogy to the linear electric current relationship,
(2) Electric convection current
where ρ is electric charge density. was seen as a kind of magnetic current of vortices aligned in their axial planes, with being the circumferential velocity of the vortices. With µ representing vortex density, it follows that the product of µ with vorticity leads to the magnetic field denoted as .
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 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 is to , and where is to ρ, then it necessarily follows from Gauss's law and from the equation of continuity of charge that is to . i.e. parallels with , whereas parallels with .
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" is exacerbated because it is also sometimes 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,"^{[12]} a confusion 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:
It is interesting to note the 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 term from equation D when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame.
In A Treatise on Electricity and Magnetism, an 1873 textbook on electromagnetism written by James Clerk Maxwell, the equations are compiled into two sets.
The first set is
The second set is
If an electric field is applied to a dielectric material, each of the molecules responds by forming a microscopic electric dipole—its atomic nucleus will move a tiny distance in the direction of the field, while its electrons will move a tiny distance in the opposite direction. This is called polarization of the material. In an idealized situation like that shown in the figure, the distribution of charge that results from these tiny movements turns out to be identical (outside the material) to having a layer of positive charge on one side of the material, and a layer of negative charge on the other side (a macroscopic separation of charge) even though all of the charges involved are bound to individual molecules. The volume polarization P is a result of bound charge. (Mathematically, once physical approximation has established the electric dipole density P based upon the underlying behavior of atoms, the surface charge that is equivalent to the material with its internal polarization is provided by the divergence theorem applied to a region straddling the interface between the material and the surrounding vacuum.)^{[13]}^{[14]}
Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the angular momentum of the atoms' components, 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. The bound currents can be described using M. (Mathematically, once physical approximation has established the magnetic dipole density based upon the underlying behavior of atoms, the surface current that is equivalent to the material with its internal magnetization is provided by Stokes' theorem applied to a path straddling the interface between the material and the surrounding vacuum.)^{[15]}^{[16]}
These ideas suggest that for some situations the microscopic details of the atomic and electronic behavior can be treated in a simplified fashion that ignores many details on a fine scale that may be unimportant to understanding matters on a grosser scale. That notion underlies the bound/free partition of behavior.
In this section, a simple proof is outlined which shows that the two alternate general formulations of Maxwell's equations given in Section 1 are mathematically equivalent.
The relation between polarization, magnetization, bound charge, and bound current is as follows:
where P and M are polarization and magnetization, and ρ_{b} and J_{b} are bound charge and current, respectively. Plugging in these relations, it can be easily demonstrated that the two formulations of Maxwell's equations given in Section 2 are precisely equivalent.
In order to apply Maxwell's equations (the formulation in terms of free/bound charge and current using D and H), it is necessary to specify the relations between D and E, and H and B.
Finding relations between these fields is another way to say that to solve Maxwell's equations by employing the free/bound partition of charges and currents, one needs the properties of the materials relating the response of bound currents and bound charges to the fields applied to these materials.^{[note 3]} 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. These material properties specifying the response of bound charge and current to the field are called constitutive relations, and correspond physically to how much polarization and magnetization a material acquires in the presence of electromagnetic fields.
Once the responses of bound currents and charges are related to the fields, Maxwell's equations can be fully formulated in terms of the E and Bfields alone, with only the free charges and currents appearing explicitly in the equations.
In the absence of magnetic or dielectric materials, the relations are simple:
where ε_{0} and μ_{0} are two universal constants, called the permittivity of free space and permeability of free space, respectively.
In a linear, isotropic, nondispersive, uniform material, the relations are also straightforward:
where ε and μ are constants (which depend on the material), called the permittivity and permeability, respectively, of the material.
For realworld materials, the constitutive relations are not simple proportionalities, except approximately. The relations can usually still be written:
but ε and μ are not, in general, simple constants, but rather functions. For example, ε and μ can depend upon:
If further there are dependencies on:
then the constitutive relations take a more complicated form:^{[17]}^{[18]}
in which the permittivity and permeability functions are replaced by integrals over the more general electric and magnetic susceptibilities.
It may be noted that manmade materials can be designed to have customized permittivity and permeability, such as metamaterials and photonic crystals.
Substituting in the constitutive relations above, Maxwell's equations in linear, dispersionless, timeinvariant materials (differential form only) are:
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 (see also displacement field, electric susceptibility and polarization density), the permeability of free space was replaced with the permeability of the material (see also magnetization, magnetic susceptibility and magnetic field), 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 lefthand sides.
The fields in Maxwell's equations are generated by charges and currents. Conversely, the charges and currents are affected by the fields through the Lorentz force equation:
where q is the charge on the particle and v is the particle velocity. (It also should be remembered that the Lorentz force is not the only force exerted upon charged bodies, which also may be subject to gravitational, nuclear, etc. forces.) Therefore, in both classical and quantum physics, the precise dynamics of a system form a set of coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of statistical mechanics.^{[note 4]} This remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents, which enter Maxwell's equations through the constitutive equations, as described next.
Commonly, real materials are approximated as continuous media with bulk properties such as the refractive index, permittivity, permeability, conductivity, and/or various susceptibilities. These lead to the macroscopic Maxwell's equations, which are written (as given above) in terms of free charge/current densities and D, H, E, and B ( rather than E and B alone ) along with the constitutive equations relating these fields. For example, although a real material consists of atoms whose electronic charge densities can be individually polarized by an applied field, for most purposes behavior at the atomic scale is not relevant and the material is approximated by an overall polarization density related to the applied field by an electric susceptibility.
Continuum approximations of atomicscale inhomogeneities cannot be determined from Maxwell's equations alone, but require some type of quantum mechanical analysis such as quantum field theory as applied to condensed matter physics. See, for example, density functional theory, GreenKubo relations and Green's function (manybody theory). Various approximate transport equations have evolved, for example, the Boltzmann equation or the FokkerPlanck equation or the NavierStokes 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^{[19]}^{[20]} (valid for excitations with wavelengths much larger than the scale of the inhomogeneity).^{[21]}^{[22]}^{[23]}^{[24]}
Theoretical results have their place, but often require fitting to experiment. Continuumapproximation properties of many real materials rely upon measurement,^{[25]} for example, ellipsometry measurements.
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 where frequency is limited to a narrow bandwidth; material absorption can be neglected for wavelengths where 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).
And, of course, some situations demand that Maxwell's equations and the Lorentz force be combined with other forces that are not electromagnetic. An obvious example is gravity. A more subtle example, which applies where electrical forces are weakened due to charge balance in a solid or a molecule, is the Casimir force from quantum electrodynamics.^{[26]}
The connection of Maxwell's equations to the rest of the physical world is via the fundamental charges and currents. These charges and currents are a response of their sources to electric and magnetic fields and to other forces. The determination of these responses involves the properties of physical materials.
Starting with the equations appropriate in the case without dielectric or magnetic materials, and assuming that there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:^{[note 5]}
One set of solutions to these equations is in the form of traveling 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, traveling at the speed^{[note 6]}
In fact, Maxwell's equations explains specifically how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. That electric field, in turn, creates a changing magnetic field through Maxwell's correction to Ampère's law. This perpetual cycle allows these waves, known as electromagnetic radiation, to move through space, always at velocity c.
Maxwell knew from an 1856 leyden jar experiment by Wilhelm Eduard Weber and Rudolf Kohlrausch, that c was very close to the measured speed of light in vacuum (known from early experiments), and concluded (correctly) that light is a form of electromagnetic radiation.
Maxwell's equations of electromagnetism relate the electric and magnetic fields to the motions of electric charges. The standard form of the equations provide for an electric charge, but posit no magnetic charge. There is no known magnetic analog of an electron, however recently scientists have described behavior in a crystalline state of matter known as spinice which have macroscopic behavior like magnetic monopoles.^{[28]}^{[29]} (in accordance with the fact that magnetic charge has never been seen and may not exist). Except for this, the equations are symmetric under interchange of electric and magnetic field. In fact, symmetric equations can be written when all charges are zero, and this is how the wave equation is derived (see immediately above).
Fully symmetric equations can also be written if one allows for the possibility of magnetic charges.^{[30]} With the inclusion of a variable for these magnetic charges, say , there will also be a "magnetic current" variable in the equations, . The extended Maxwell's equations, simplified by nondimensionalization via Planck units, are as follows:
Name  Without magnetic monopoles  With magnetic monopoles (hypothetical)  

Gauss's law:  
Gauss's law for magnetism:  
Maxwell–Faraday equation (Faraday's law of induction): 

Ampère's law (with Maxwell's extension): 

Note: the Bivector notation embodies the sign swap, and these four equations can be written as only one equation. 
If magnetic charges do not exist, or if they exist but where they are not present in a region, then the new variables are zero, and the symmetric equations reduce to the conventional equations of electromagnetism such as .
Although Maxwell's equations apply throughout space and time, practical problems are finite and solutions to Maxwell's equations inside the solution region are joined to the remainder of the universe through boundary conditions^{[31]}^{[32]}^{[33]} and started in time using initial conditions.^{[34]}
In particular, in a region without any free currents or free charges, the electromagnetic fields in the region originate elsewhere, and are introduced via boundary and/or initial conditions. An example of this type is a an electromagnetic scattering problem, where an electromagnetic wave originating outside the scattering region is scattered by a target, and the scattered electromagnetic wave is analyzed for the information it contains about the target by virtue of the interaction with the target during scattering.^{[35]}
In some cases, like waveguides or cavity resonators, the solution region is largely isolated from the universe, for example, by metallic walls, and boundary conditions at the walls define the fields with influence of the outside world confined to the input/output ends of the structure.^{[36]} In other cases, the universe at large sometimes is approximated by an artificial absorbing boundary,^{[37]}^{[38]}^{[39]} or, for example for radiating antennas or communication satellites, these boundary conditions can take the form of asymptotic limits imposed upon the solution.^{[40]} In addition, for example in an optical fiber or thinfilm optics, the solution region often is broken up into subregions with their own simplified properties, and the solutions in each subregion must be joined to each other across the subregion interfaces using boundary conditions.^{[41]}^{[42]}^{[43]} A particular example of this use of boundary conditions is the replacement of a material with a volume polarization by a charged surface layer, or of a material with a volume magnetization by a surface current, as described in the section Bound charge and current.
Following are some links of a general nature concerning boundary value problems: Examples of boundary value problems, SturmLiouville theory, Dirichlet boundary condition, Neumann boundary condition, mixed boundary condition, Cauchy boundary condition, Sommerfeld radiation condition. Needless to say, one must choose the boundary conditions appropriate to the problem being solved. See also Kempel^{[44]} and the extraordinary book by Friedman.^{[45]}
The above equations are given in the International System of Units, or SI for short. In a related unit system, called CGS (short for centimetergramsecond), there are several ways of defining the unit of electric current in terms of the CGS base units (cm, g, s). In one of those variants, called Gaussian units, the equations take the following form:^{[46]}
where c is the speed of light in a vacuum. For the electromagnetic field in a vacuum, assuming that there is no current or electric charge present in the vacuum, the equations become:
In this system of units the relation between displacement field, electric field and polarization density is:
And likewise the relation between magnetic induction, magnetic field and total magnetization is:
In the linear approximation, the electric susceptibility and magnetic susceptibility can be defined so that:
(Note that although the susceptibilities are dimensionless numbers in both cgs and SI, they have different values in the two unit systems, by a factor of 4π.) The permittivity and permeability are:
so that
In vacuum, one has the simple relations , 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:
where is the charge on the particle and is the particle velocity. This is slightly different from the SIunit expression above. For example, here the magnetic field has the same units as the electric field .
Some equations in the article are given in Gaussian units but not SI or viceversa. Fortunately, there are general rules to convert from one to the other; see the article Gaussian units for details.
Maxwell's equations have a close relation to special relativity: Not only were Maxwell's equations a crucial part of the historical development of special relativity, but also, special relativity has motivated a compact mathematical formulation of Maxwell's equations, in terms of covariant tensors.
Maxwell's electromagnetic wave equation only applied in what he believed to be the rest frame of the luminiferous medium because he didn't use the v×B term of his equation (D) when he derived it. Maxwell's idea of the luminiferous medium was that it consisted of aethereal vortices aligned solenoidally along their rotation axes.
The American scientist A.A. Michelson set out to determine the velocity of the earth through the luminiferous medium aether using a light wave interferometer that he had invented. When the MichelsonMorley experiment was conducted by Edward Morley and Albert Abraham Michelson in 1887, it produced a null result for the change of the velocity of light due to the Earth's motion through the hypothesized aether. This null result was in line with the theory that was proposed in 1845 by George Stokes which suggested that the aether was entrained with the Earth's orbital motion.
Hendrik Lorentz objected to Stokes' aether drag model and along with George FitzGerald and Joseph Larmor, he suggested another approach. 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 mathematically the group property of the Lorentz transformation (Poincaré 1905).
This culminated in Albert Einstein's theory of special relativity, which postulated the absence of any absolute rest frame, dismissed the aether as unnecessary (a bold idea, which did not come to Lorentz nor to Poincaré), and established the invariance of Maxwell's equations in all inertial frames of reference, in contrast to the famous Newtonian equations for classical mechanics. But the transformations between two different inertial frames had to correspond to Lorentz' equations and not  as formerly believed  to those of Galileo (called Galilean transformations).^{[47]} Indeed, Maxwell's equations played a key role in Einstein's famous paper on special relativity; for example, in the opening paragraph of the paper, he motivated his 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 force is calculated in the rest frame of the magnet or that of the conductor.^{[48]}
General relativity has also had a close relationship with Maxwell's equations. For example, 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 continues to be an active area of research in particle physics.
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, Maxwell's equations are written in terms of fourvectors and tensors in the "manifestly covariant" form. The purely spatial components of the following are in SI units.
One ingredient in this formulation is the electromagnetic tensor, a rank2 covariant antisymmetric tensor combining the electric and magnetic fields:
and the result of raising its indices
The other ingredient is the fourcurrent: where ρ is the charge density and J is the current density.
With these ingredients, Maxwell's equations can be written:
and
The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss's law and Ampere'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. The second equation is equivalent to
where is the contravariant version of the LeviCivita symbol, and
is the 4gradient. In the tensor equations above, repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations. Upper and lower components of a vector, v^{α} and v_{α} respectively, are interchanged with the fundamental tensor g, e.g., g=η=diag(1,+1,+1,+1).
Alternative covariant presentations of Maxwell's equations also exist, for example in terms of the fourpotential; see Covariant formulation of classical electromagnetism for details.
Maxwell's equations can be written in an alternative form, involving the electric potential (also called scalar potential) and magnetic potential (also called vector potential), as follows.^{[18]} (The following equations are valid in the absence of dielectric and magnetic materials; or if such materials are present, they are valid as long as bound charge and bound current are included in the total charge and current densities.)
First, Gauss's law for magnetism states:
By Helmholtz's theorem, B can be written in terms of a vector field A, called the magnetic potential:
Second, plugging this into Faraday's law, we get:
By Helmholtz's theorem, the quantity in parentheses can be written in terms of a scalar function , called the electric potential:
Combining these with the remaining two Maxwell's equations yields the four relations:
These equations, taken together, are as powerful and complete as Maxwell's equations. Moreover, if we work only with the potentials and ignore the fields, the problem has been reduced somewhat, as the electric and magnetic fields each have three components which need to be solved for (six components altogether), while the electric and magnetic potentials have only four components altogether.
Many different choices of A and are consistent with a given E and B, making these choices physically equivalent – a flexibility known as gauge freedom. Suitable choice of A and can simplify these equations, or can adapt them to suit a particular situation. For more information, see the article gauge freedom.
The two equations that represent the potentials can be reduced to one manifestly Lorentz invariant equation, using fourvectors: the fourcurrent defined by
formed from the current density j and charge density ρ, and the electromagnetic fourpotential defined by
formed from the vector potential A and the scalar potential . The resulting single equation, due to Arnold Sommerfeld, a generalization of an equation due to Bernhard Riemann and known as the RiemannSommerfeld equation^{[49]} or the covariant form of the MaxwellLorentz equations,^{[50]} is:
where is the d'Alembertian operator, or fourLaplacian, , sometimes written , or , where is the fourgradient.
In free space, where ε = ε_{0} and μ = μ_{0} are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. In what follows, cgsGaussian units, not SI units are used. (To convert to SI, see here.) The electric and magnetic fields are now jointly described by a 2form F in a 4dimensional spacetime manifold. Maxwell's equations then reduce to the Bianchi identity
where d denotes the exterior derivative — a natural coordinate and metric independent differential operator acting on forms — and the source equation
where the (dual) Hodge star operator * is a linear transformation from the space of 2forms to the space of (42)forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric), and the fields are in natural units where 1 / 4πε_{0} = 1. Here, the 3form J is called the electric current form or current 3form satisfying the continuity equation
The current 3form can be integrated over a 3dimensional spacetime 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 4dimensional 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 2forms. We call
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:
where the current 3form J still satisfies the continuity equation dJ= 0.
When the fields are expressed as linear combinations (of exterior products) of basis forms ,
the constitutive relation takes the form
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
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 4dimensional 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.
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 nonphysical degrees of freedom which can be removed by gauge transformation A→A' = Adα: see also gauge fixing and FadeevPopov ghosts.
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 which is a twoform that automatically satisfies and can be interpreted as a fieldstrength. If the line bundle is trivial with flat reference connection d we can write and F = dA with A the 1form 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 AharonovBohm 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 crosssection of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the spacetime 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 noncontractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantummechanically with a doubleslit 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. (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.)
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 will also generate 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 sourcefree equations become (cgsGaussian units):
and
Here,
is a Christoffel symbol that characterizes the curvature of spacetime and D_{γ} is the covariant derivative.
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 1forms dx^{α} in every point of the open set where the coordinates are defined. Using this basis and cgsGaussian units we define
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 torsionfreeness of the Levi Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:
The developments before relativity
see

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In 1868, James Clerk Maxwell found four equations to describe all the things relating to electromagnetism:
Contents 
Name  Differential form  Integral form 

Gauss' law:  $\backslash nabla\; \backslash cdot\; \backslash mathbf\{D\}\; =\; \backslash rho$  $\backslash oint\_S\; \backslash mathbf\{D\}\; \backslash cdot\; d\backslash mathbf\{A\}\; =\; \backslash int\_V\; \backslash rho\; \backslash cdot\; dV$ 
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:  $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$  $\backslash oint\_C\; \backslash mathbf\{E\}\; \backslash cdot\; d\backslash mathbf\{l\}\; \; \backslash oint\_C\; \backslash mathbf\; \{B\}\; \backslash times\; \backslash mathbf\{v\}\; \backslash cdot\; d\{\backslash mathbf\; \{l\}\}\; =\; \; \backslash \; \{\; d\; \backslash over\; dt\; \}\; \backslash int\_S\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}$ 
Ampère's law (with Maxwell's extension):  $\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{D\}\}\; \{\backslash partial\; t\}$  $\backslash oint\_C\; \backslash mathbf\{H\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; \backslash int\_S\; \backslash mathbf\{J\}\; \backslash cdot\; d\; \backslash mathbf\{A\}\; +$ \int_S\frac{\partial \mathbf{D} }{\partial t}\cdot d \mathbf{A} 
where the following table provides the meaning of each symbol and the SI unit of measure:
Symbol  Meaning  SI Unit of Measure 

$\backslash mathbf\{E\}$  electric field  volt per metre 
$\backslash mathbf\{H\}$  magnetic field strength  ampere per metre 
$\backslash mathbf\{D\}$  electric displacement field  coulomb per square metre 
$\backslash mathbf\{B\}$  magnetic flux density also called the magnetic induction.  tesla, or equivalently, weber per square metre 
$\backslash \; \backslash rho\; \backslash $  free electric charge density, not counting the dipole charges bound in a material.  coulomb per cubic metre 
$\backslash mathbf\{J\}$  free current density, not counting polarization or magnetization currents bound in a material.  ampere per square metre 
$d\backslash mathbf\{A\}$  differential vector element of surface area A, with very small magnitude and direction normal to surface S  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 c  meters 
$\backslash mathbf\{v\}$  instantaneous velocity of the line element $d\backslash mathbf\{l\}$ defined above (for moving circuits).  meters per second 
and
where $\{\backslash rho\}$ is the free electric charge density (in units of C/m^{3}), not counting the dipole charges bound in a material, and $\backslash mathbf\{D\}$ is the electric displacement field (in units of C/m^{2}). This equation is like Coulomb's law for nonmoving charges in vacuum.
The next integral form (by the divergence theorem), also known as Gauss' law, says the same thing:
$d\backslash mathbf\{A\}$ is the area of a differential square on the closed surface A. The surface normal pointing out is the direction, and $Q\_\backslash mbox\{enclosed\}$ is the free charge that is inside the surface.
In a linear material, $\backslash mathbf\{D\}$ is directly related to the electric field $\backslash mathbf\{E\}$ with a constant called the permittivity, $\backslash epsilon$ (This constant is different for different materials):
You can pretend a material is linear, if the electric field is not very strong.
The permittivity of free space is called $\backslash epsilon\_0$, and is used in this equation:
Here $\backslash mathbf\{E\}$ is the electric field again (in units of V/m), $\backslash rho\_t$ is the total charge density (including the bound charges), and $\backslash epsilon\_0$ (approximately 8.854 pF/m) is the permittivity of free space. You can also write $\backslash epsilon$ as $\backslash varepsilon\_0\; \backslash cdot\; \backslash varepsilon\_r$. Here $\backslash epsilon\_r$ is the permittivity of the material when you compare it to the permittivity of free space. This is called the relative permittivity or dielectric constant.
See also Poisson's equation.
$\backslash mathbf\{B\}$ is the magnetic flux density (in units of teslas, T), also called the magnetic induction.
This next integral form says the same thing:
$d\backslash mathbf\{A\}$ is the area of a differential square on the surface $A$. The surface normal pointing out is the direction.
This equation only works if the integral is done over a closed surface. This equation says, that in every volume the sum of the magnetic field lines that go in equals the sum of the magnetical field lines that go out. This means that the magnetic field lines must be closed loops. Another way of saying this is that the field lines cannot start from somewhere. This is the mathematical way of saying: "There are no magnetic monopoles".
This next integral form says the same thing:
Here $\backslash Phi\_\{\backslash mathbf\{B\}\}\; =\; \backslash int\_\{A\}\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}$
This is what the symbols mean:
Φ_{B} is the magnetic flux that goes through the area A that the second equation describes,
E is the electric field that the magnetic flux causes,
s is a closed path in which current is induced, for example a wire,
v is the instantaneous velocity of the line element (for moving circuits).
The electromotive force is equal to the value of this integral. Sometimes this symbol is used for the electromotive force: $\backslash mathcal\{E\}$, do not confuse it with the symbol for permittivity that was used before.
This law is like Faraday's law of electromagnetic induction.
Some textbooks show the right hand sign of the integral form with an N (N is 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's left out here.
The negative sign is needed for conservation of energy. It is so important that it even has its own name, Lenz's law.
This equation shows how the electric and magnetic fields have to do with each other. For example, this equation explains how electric motors and electric generators work. In a motor or generator, the field circuit has a fixed electric field that causes a magnetic field. This is called fixed excitation. The varying voltage is measured across the armature circuit. Maxwell's equations are used in a righthanded coordinate system. To use them in a lefthanded system, without having to change the equations, the polarity of magnetic fields has to made opposite (this is not wrong, but it is confusing because it is not usually done like this).
H is the magnetic field strength (in units of A/m), which you can get by multiplying the magnetic flux B by a constant called the permeability, μ (B = μH), and J is the current density, defined by:
J = ∫ρ_{q}vdV
v is a vector field called the drift velocity. It describes the speeds of the charge carriers that have a density described by the scalar function ρ_{q}.
In free space, the permeability μ is the permeability of free space, μ_{0}, which is exactly 4π×10^{7} W/A·m, by definition. Also, the permittivity is the permittivity of free space ε_{0}. So, in free space, the equation is:
The next integral form says the same thing:
s is the edge of the open surface A (any surface with the curve s as its edge is okay here), and I_{encircled} is the current encircled by the curve s (the current through any surface is defined by the equation: I_{through A} = ∫_{A}J·dA).
If the electric flux density does not change very fast, the second term on the right hand side (the displacement flux) is very small and can be left out, and then the equation is the same as Ampere's law.
There are only two covariant Maxwell Equations, because the covariant field vector includes the electrical and the magnetical field.
Mathematical note: In this section the abstract index notation will be used.
In special relativity, Maxwell's equations for the vacuum are written in terms of fourvectors and tensors in the "manifestly covariant" form. This has been done to show more clearly the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system. This is the "manifestly covariant" form:
and
The second equation is the same as:
Here $\backslash ,\; J^a$ is the 4current, $\backslash ,\; F^\{ab\}$ is the field strength tensor (written as a 4 × 4 matrix), $\backslash ,\; \backslash epsilon\_\{abcd\}$ is the LeviCivita symbol, and $\backslash partial\_a\; =\; (\backslash partial/\backslash partial\; ct,\; \backslash nabla)$ is the 4gradient (so that $\backslash partial\_a\; \backslash partial^a$ is the d'Alembertian operator). (The $a$ in the first equation is implicitly summed over, according to Einstein notation.) The first tensor equation says the same thing as the two inhomogeneous Maxwell's equations: Gauss' law and Ampere's law with Maxwell's correction. The second equation say the same thing as the other two equations, the homogeneous equations: Faraday's law of induction and the absence of magnetic monopoles.
$\backslash ,\; J^a$ can also be described more explicitly by this equation: $J^a\; =\; \backslash ,\; (c\; \backslash rho,\; \backslash vec\; J)$ (as a contravariant vector), where you get $\backslash ,\; J^a$ from the charge density ρ and the current density $\backslash vec\; J$. The 4current is a solution to the continuity equation:
$J^a\{\}\_\{,a\}\; \backslash ,\; =\; 0$
In terms of the 4potential (as a contravariant vector) $A^\{a\}\; =\; \backslash left(\backslash phi,\; \backslash vec\; A\; c\; \backslash right)$, where φ is the electric potential and $\backslash vec\; A$ is the magnetic vector potential in the Lorentz gauge $\backslash left\; (\; \backslash partial\_a\; A^a\; =\; 0\; \backslash right\; )$, F can be written as:
which leads to the 4 × 4 matrix rank2 tensor:
\begin{matrix} 0 & \frac {E_x}{c} & \frac {E_y}{c} & \frac {E_z}{c} \\ \frac{E_x}{c} & 0 & B_z & B_y \\ \frac{E_y}{c} & B_z & 0 & B_x \\ \frac{E_z}{c} & B_y & B_x & 0 \end{matrix} \right) .
The fact that both electric and magnetic fields are combined into a single tensor shows the fact that, according to relativity, both of these are different parts of the same thing—by changing frames of reference, what looks like an electric field in one frame can look like a magnetic field in another frame, and the other way around.
Using the tensor form of Maxwell's equations, the first equation implies
$\backslash Box\; F^\{ab\}\; =\; 0$ (See Electromagnetic fourpotential for the relationship between the d'Alembertian of the fourpotential and the fourcurrent, expressed in terms of the older vector operator notation).
Different authors sometimes use different sign conventions for these tensors and 4vectors (but this does not change what they mean).
$\backslash ,\; F^\{ab\}$ and $\backslash ,\; F\_\{ab\}$ are not the same: they are related by the Minkowski metric tensor $\backslash eta$: $F\_\{ab\}\; =\backslash ,\; \backslash eta\_\{ac\}\; \backslash eta\_\{bd\}\; F^\{cd\}$. This changes the sign of some of F's components; more complex metric dualities can be seen in general relativity.
