Maxwell's Equation

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 arethe 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.

History

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 pre-dates publications by Heaviside, Hertz and others.

The term Maxwell's equations

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, concurrently with similar work by Willard Gibbs and Heinrich Hertz. These equations were also 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 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)

The equations were called by some the Hertz-Heaviside equations, but later Einstein referred to them as the Maxwell-Hertz equations.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; 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.

The net result of Heaviside's work was the symmetrical duplex set of four equations, 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 Maxwell-Faraday 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.