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

Maxwell's four equations express, respectively, how electric charges produce electric fields (Gauss' law), the experimental absence of magnetic charges, how currents produce magnetic fields (Ampere's law), and how changing magnetic fields produce electric fields (Faraday's law of induction). Maxwell, in 1864, was the first to put all four equations together and to notice that a correction was required to Ampere's law: changing electric fields act like currents, likewise producing magnetic fields. (This additional term is called the displacement current)

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

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

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

Historical developments of Maxwell's equations and relativity

Maxwell's 1865 formulation was in terms of 20 equations in 20 variables, which included several equations now considered to be auxiliary to what are now called "Maxwell's equations" — the corrected Ampere's law (three component equations), Gauss' law for charge (one equation), the relationship between total and displacement current densities (three component equations), the relationship between magnetic field and the vector potential (three component equations, which imply the absence of magnetic charge), the relationship between electric field and the scalar and vector potentials (three component equations, which imply Faraday's law), the relationship between the electric and displacement fields (three component equations), Ohm's law relating current density and electric field (three component equations), and the continuity equation relating current density and charge density (one equation).

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

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

in the equations, Maxwell's equations were only thought to express electromagnetism in the rest frame of the luminiferous aether (the postulated medium for light, whose interpretation was considerably debated). When the Michelson-Morley experiment, conducted by Edward Morley and Albert Abraham Michelson, produced a null result for the change of the velocity of light due to the Earth's hypothesized motion through the aether, however, alternative explanations were sought by Lorentz and others. This culminated in Einstein's theory of special relativity, which postulated the absence of any absolute rest frame (or aether) and the invariance of Maxwell's equations in all frames of reference.

The electromagnetic field equations have an intimate link with special relativity: the magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities. (In relativity, the equations are written in an even more compact, "manifestly covariant" form, in terms of the rank-2 antisymmetric field-strength 4-tensor that unifies the electric and magnetic fields into a single object.)

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

where:

ρ is the free electric charge density (SI unit: coulomb per cubic meter), not including dipole charges bound in a material

is the magnetic flux density (SI unit: tesla, volt × second per square meter), also called the magnetic induction.

is the electric displacement field (SI unit: coulomb per square meter).

is the electric field (SI unit: volt per meter),

is the magnetic field strength (SI unit: ampere per meter)

is the current density (SI unit: ampere per square meter)

is the divergence operator (SI unit: 1 per meter),

is the curl operator (SI unit: 1 per meter).

• For full treatment of "Maxwell's equations" click here.

References

• 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.)
• James Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vols. 1-2 (1891) (reprinted: Dover, New York NY, 1954.
• John David Jackson, Classical Electrodynamics (Wiley, New York, 1998).
• Edward M. Purcell, Electricity and Magnetism (McGraw-Hill, New York, 1985).
• Banesh Hoffman, Relativity and Its Roots (Freeman, New York, 1983).
• Charles F. Stevens, The Six Core Theories of Modern Physics, (MIT Press, 1995).
• Landau, L. D., The Classical Theory of Fields (Course of Theoretical Physics: Volume 2), (Butterworth-Heinemann: Oxford, 1987).
• Fitzpatrick, Richard, "Lecture series: Relativity and electromagnetism (http://farside.ph.utexas.edu/~rfitzp/teaching/jk1/lectures/node6.html)". Advanced Classical Electromagnetism, PHY387K. University of Texas at Austin, Fall 1996.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; . (Provides a treatment of Maxwell's equations in terms of differential forms.)