Maxwell's equations

Maxwell's equations


Four equations, formulated by James Clerk Maxwell, that together form a complete description of the production and interrelation of electric and magnetic fields. The statements of these four equations are (1) electric field diverges from electric charge, (2) there are no isolated magnetic poles, (3) electric fields are produced by changing magnetic fields, and (4) circulating magnetic fields are produced by changing electric fields and by electric currents. Maxwell based his description of electromagnetic fields on these four statements.

In electromagnetism, Maxwell's equations are a set of four equations that were first presented as a distinct group in 1884 by Oliver Heaviside in conjunction with Willard Gibbs. These equations had appeared throughout James Clerk Maxwell's 1861 paper entitled On Physical Lines of Force.

Those equations describe the interrelationship between electric field, magnetic field, electric charge, and electric current. Although Maxwell himself was the originator of only one of these equations (by virtue of modifying an already existing equation), he derived them all again independently in conjunction with his molecular vortex model of Faraday's "lines of force".

Although Maxwell's equations were known before special relativity, they can be derived from Coulomb's law and special relativity if one assumes invariance of electric charge.[1][2] For more information, see links to relativity section.

History

Maxwell's equations are a set of four equations originally appearing separately in Maxwell's 1861 paper On Physical Lines of Force as equation (54) Faraday's law, equation (56) div B = 0, equation (112) Ampère's law with Maxwell's correction, and equation (113) Gauss's law. They express respectively how changing magnetic fields produce electric fields, the experimental absence of magnetic monopoles, how electric currents and changing electric fields produce magnetic fields (Ampère's circuital law with Maxwell's correction), and how electric charges produce electric fields.

Maxwell introduced an extra term to Ampère's circuital law which is the time derivative of electric field and known as Maxwell's displacement current. This modification is the most significant aspect of Maxwell's work in electromagnetism.

In Maxwell's 1865 paper, A Dynamical Theory of the Electromagnetic Field Maxwell's modified version of Ampère's circuital law enabled him to derive the electromagnetic wave equation, hence demonstrating that light is an electromagnetic wave.

Apart from Maxwell's amendment to Ampère's circuital law, none of these equations were original. Maxwell however uniquely re-derived them hydrodynamically and mechanically using his vortex model of Faraday's lines of force.

In 1884 Oliver Heaviside, in conjunction with Willard Gibbs, grouped these equations together and restated them in modern vector notation. It is important however to note that in doing so, Heaviside used partial time derivative notation as opposed to the total time derivative notation used by Maxwell at equation (54). The consequence of this is that we lose the vXB term that appeared in Maxwell's follow up equation (77). Nowadays, the vXB term sits beside the group known as Maxwell's equations and bears the name Lorentz Force.

This whole matter is confused because the term Maxwell's equations is also used for a set of eight equations in Maxwell's 1865 paper, A Dynamical Theory of the Electromagnetic Field, and this confusion is yet further confused by virtue of the fact that six of those eight equations are each written as three separate equations for the x, y, and z, axes, hence allowing even Maxwell to refer to them as twenty equations in twenty unknowns.

The two sets of Maxwell's equations are nearly physically equivalent, although the vXB term at equation (D) of the original eight is absent from the modern Heaviside four. The Maxwell-Ampère equation in Heaviside's restatement is an amalgamation of two equations in the set of eight that Maxwell published in his 1865 paper.

Summary of the modern Heaviside versions

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


The equations are given here 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. For example, the electric field and the magnetic field have the same unit (gauss) in the Gaussian system. 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[3]), Lorentz-Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics).