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By the first half of the 19th century, the understanding of
electromagnetics In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
had improved through many experiments and theoretical work. In the 1780s,
Charles-Augustin de Coulomb Charles-Augustin de Coulomb ( ; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of att ...
established his
law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a science and as the ar ...
of
electrostatics Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
. In 1825,
André-Marie Ampère André-Marie Ampère (, ; ; 20 January 177510 June 1836) was a French physicist and mathematician who was one of the founders of the science of classical electromagnetism, which he referred to as ''electrodynamics''. He is also the inventor of ...
published his force law. In 1831,
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English chemist and physicist who contributed to the study of electrochemistry and electromagnetism. His main discoveries include the principles underlying electromagnetic inducti ...
discovered
electromagnetic induction Electromagnetic or magnetic induction is the production of an electromotive force, electromotive force (emf) across an electrical conductor in a changing magnetic field. Michael Faraday is generally credited with the discovery of induction in 1 ...
through his experiments, and proposed '' lines of forces'' to describe it. In 1834, Emil Lenz solved the problem of the direction of the induction, and
Franz Ernst Neumann Franz Ernst Neumann (11 September 1798 – 23 May 1895) was a German mineralogist and physicist. He devised the first formulas to calculate inductance. He also formulated Neumann's law for molecular heat. In electromagnetism, he is credited for ...
wrote down the equation to calculate the induced force by change of magnetic flux. However, these experimental results and rules were not well organized and sometimes confusing to scientists. A comprehensive summary of the electrodynamic principles was needed. This work was done by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
through a series of papers published from the 1850s to the 1870s. In the 1850s, Maxwell was working at the
University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...
where he was impressed by Faraday's ''lines of forces'' concept. Faraday created this concept by impression of Roger Boscovich, a physicist that impacted Maxwell's work as well. In 1856, he published his first paper in electromagnetism: ''On Faraday's Lines of Force''. He tried to use the analogy of incompressible fluid flow to model the magnetic lines of forces. Later, Maxwell moved to
King's College London King's College London (informally King's or KCL) is a public university, public research university in London, England. King's was established by royal charter in 1829 under the patronage of George IV of the United Kingdom, King George IV ...
where he actually came into regular contact with Faraday, and became life-long friends. From 1861 to 1862, Maxwell published a series of four papers under the title of ''
On Physical Lines of Force "On Physical Lines of Force" is a four-part paper written by James Clerk Maxwell, published in 1861. In it, Maxwell derived the Maxwell's equations, equations of electromagnetism in conjunction with a "sea" of "molecule, molecular vortex, vortice ...
''. In these papers, he used mechanical models, such as rotating vortex tubes, to model the electromagnetic field. He also modeled the vacuum as a kind of insulating elastic medium to account for the stress of the magnetic lines of force given by Faraday. These works had already laid the basis of the formulation of the Maxwell's equations. Moreover, the 1862 paper already derived the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
from the expression of the velocity of the electromagnetic wave in relation to the vacuum constants. The final form of Maxwell's equations was published in 1865 ''
A Dynamical Theory of the Electromagnetic Field "A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. ''(Paper read at a meeting of the Royal Society on 8 December 1864).'' Physicist Freeman Dyson called the publishing of the ...
'', in which the theory is formulated in strictly mathematical form. In 1873, Maxwell published ''
A Treatise on Electricity and Magnetism ''A Treatise on Electricity and Magnetism'' is a two-volume treatise on electromagnetism written by James Clerk Maxwell in 1873. Maxwell was revising the ''Treatise'' for a second edition when he died in 1879. The revision was completed by Wil ...
'' as a summary of his work on electromagnetism. In summary, Maxwell's equations successfully unified theories of light and electromagnetism, which is one of the great unifications in physics. Maxwell built a simple flywheel model of electromagnetism, and
Boltzmann Ludwig Eduard Boltzmann ( ; ; 20 February 1844 – 5 September 1906) was an Austrian mathematician and theoretical physicist. His greatest achievements were the development of statistical mechanics and the statistical explanation of the seco ...
built an elaborate mechanical model ("''Bicykel''") based on Maxwell's flywheel model, which he used for lecture demonstrations. Figures are at the end of Boltzmann's 1891 book. Later,
Oliver Heaviside Oliver Heaviside ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, an ...
studied Maxwell's ''
A Treatise on Electricity and Magnetism ''A Treatise on Electricity and Magnetism'' is a two-volume treatise on electromagnetism written by James Clerk Maxwell in 1873. Maxwell was revising the ''Treatise'' for a second edition when he died in 1879. The revision was completed by Wil ...
'' and employed
vector calculus Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...
to synthesize Maxwell's over 20 equations into the four recognizable ones which modern physicists use. Maxwell's equations also inspired
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
in developing the
theory of special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, "On the Electrodynamics of Moving Bodies", the theory is present ...
. The experimental proof of Maxwell's equations was demonstrated by
Heinrich Hertz Heinrich Rudolf Hertz (; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. Biography Heinri ...
in a series of experiments in the 1890s. After that, Maxwell's equations were fully accepted by scientists.


Relationships among electricity, magnetism, and the speed of light

The relationships amongst electricity, magnetism, and the speed of light can be summarized by the modern equation: : c = \frac \ . The left-hand side is the speed of light and the right-hand side is a quantity related to the constants that appear in 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 Wilhelm Eduard Weber ( ; ; 24 October 1804 – 23 June 1891) was a German physicist and, together with Carl Friedrich Gauss, inventor of the first electromagnetic telegraph. Biography Early years Weber was born in Schlossstrasse in Witte ...
and
Rudolf Kohlrausch Rudolf Hermann Arndt Kohlrausch (November 6, 1809 in Göttingen – March 8, 1858 in Erlangen) was a German physicist. Biography He was a native of Göttingen, the son of the Royal Hanovarian director general of schools Friedrich Kohlrausch. He ...
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), 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 A Leyden jar (or Leiden jar, or archaically, Kleistian jar) is an electrical component that stores a high-voltage electric charge (from an external source) between electrical conductors on the inside and outside of a glass jar. It typically co ...
and measuring the magnetic force from the discharge current, and found a value , remarkably close to the speed of light, which had recently been measured at by
Hippolyte Fizeau Armand Hippolyte Louis Fizeau (; 23 September 1819 – 18 September 1896) was a French physicist who, in 1849, measured the speed of light to within 5% accuracy. In 1851, he measured the speed of light in moving water in an experiment known as t ...
in 1848 and at by
Léon Foucault Jean Bernard Léon Foucault (, ; ; 18 September 1819 – 11 February 1868) was a French physicist best known for his demonstration of the Foucault pendulum, a device demonstrating the effect of Earth's rotation. He also made an early measuremen ...
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 "On Physical Lines of Force" is a four-part paper written by James Clerk Maxwell, published in 1861. In it, Maxwell derived the Maxwell's equations, equations of electromagnetism in conjunction with a "sea" of "molecule, molecular vortex, vortice ...
'', 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 Michael Faraday (; 22 September 1791 – 25 August 1867) was an English chemist and physicist who contributed to the study of electrochemistry and electromagnetism. His main discoveries include the principles underlying electromagnetic inducti ...
's "lines of force" and in conjunction with the experimental result of Weber and Kohlrausch. But it was not until 1884 that
Oliver Heaviside Oliver Heaviside ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, an ...
, concurrently with similar work by
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American mechanical engineer and scientist who made fundamental theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynami ...
and
Heinrich Hertz Heinrich Rudolf Hertz (; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. Biography Heinri ...
, grouped the twenty equations together into a set of only four, via
vector notation In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space. For denoting a vector, the common typographic convention is lower ...
. This group of four equations was known variously as the Hertz–Heaviside equations and the Maxwell–Hertz equations, but are now universally known as ''Maxwell's equations''. Heaviside's equations, which are taught in textbooks and universities as Maxwell's equations are not exactly the same as the ones due to Maxwell, and, in fact, the latter are more easily made to conform to quantum physics. This very subtle and paradoxical sounding situation can perhaps be most easily understood in terms of the similar situation that exists with respect to Newton's second law of motion: In textbooks and in classrooms the law F=ma is attributed to Newton, but Newton in fact wrote his second law as F=\dot. This is clearly visible in a glass case in the
Wren Library The Wren Library is the library of Trinity College in Cambridge. It was designed by Christopher Wren in 1676 and completed in 1695. Description The library is a single large room built over an open colonnade on the ground floor of Nevi ...
of
Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any ...
, where Newton's manuscript is open to the relevant page, showing the equation F=\dot, where \dot is the
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
of the momentum p. This seems a trivial enough fact until you realize that F=\dot remains true in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
, without modification. Maxwell's contribution to science in producing these equations lies in the correction he made to
Ampère's circuital law In classical electromagnetism, Ampère's circuital law, often simply called Ampère's law, and sometimes Oersted's law, relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop. James ...
in his 1861 paper ''On Physical Lines of Force''. He added the
displacement current In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
term to Ampère's circuital law and this enabled him to derive the
electromagnetic wave equation The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous for ...
in his later 1865 paper ''
A Dynamical Theory of the Electromagnetic Field "A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. ''(Paper read at a meeting of the Royal Society on 8 December 1864).'' Physicist Freeman Dyson called the publishing of the ...
'' and to demonstrate the fact that light is an
electromagnetic wave In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ...
. This fact was later confirmed experimentally by Heinrich Hertz in 1887. The physicist
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
predicted that, "From a long view of the history of mankind, seen from, say, ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. 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: Heaviside worked to eliminate the potentials (
electric potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
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.


''On Physical Lines of Force''

The four equations we use today appeared separately in Maxwell's 1861 paper, ''On Physical Lines of Force'': #Equation (56) in Maxwell's 1861 paper is
Gauss's law for magnetism In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
, . #Equation (112) is
Ampère's circuital law In classical electromagnetism, Ampère's circuital law, often simply called Ampère's law, and sometimes Oersted's law, relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop. James ...
, with Maxwell's addition of
displacement current In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
. This may be the most remarkable contribution of Maxwell's work, enabling him to derive the
electromagnetic wave equation The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous for ...
in his 1865 paper A Dynamical Theory of the Electromagnetic Field, showing that light is an electromagnetic wave. This lent the equations their full significance with respect to understanding the nature of the phenomena he elucidated. (Kirchhoff derived the
telegrapher's equations The telegrapher's equations (or telegraph equations) are a set of two coupled, linear partial differential equations that model voltage and current along a linear electrical transmission line. The equations are important because they allow trans ...
in 1857 without using displacement current, but he did use
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
and the equation of continuity, which are the mathematical ingredients of the displacement current. Nevertheless, believing his equations to be applicable only inside an electric wire, he cannot be credited with the discovery that light is an electromagnetic wave). #Equation (115) is Gauss's law. #Equation (54) expresses what Oliver Heaviside referred to as 'Faraday's law', which addresses the time-variant aspect of electromagnetic induction, but not the one induced by motion; Faraday's original flux law accounted for both. Maxwell deals with the motion-related aspect of electromagnetic induction, , in equation (77), which is the same as equation (D) in Maxwell's original equations as listed below. It is expressed today as the force law equation, , which sits adjacent to Maxwell's equations and bears the name
Lorentz force In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...
, even though Maxwell derived it when Lorentz was still a young boy. 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 Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ...
. 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''. 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 In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. It is the ratio of the magnetic ...
to be a measure of the density of the vortex sea. Hence the relationship, # Magnetic induction current causes a magnetic current density was essentially a rotational analogy to the linear electric current relationship, # 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 representing vortex density, it follows that the product of with vorticity leads to the
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
denoted as . The electric current equation can be viewed as a convective current of
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
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 In science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental ca ...
force. The extension of the above considerations confirms that where is to , and where is to , 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 .


''A Dynamical Theory of the Electromagnetic Field''

In 1865 Maxwell published "
A dynamical theory of the electromagnetic field "A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. ''(Paper read at a meeting of the Royal Society on 8 December 1864).'' Physicist Freeman Dyson called the publishing of the ...
" 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 1865 paper "
A dynamical theory of the electromagnetic field "A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. ''(Paper read at a meeting of the Royal Society on 8 December 1864).'' Physicist Freeman Dyson called the publishing of the ...
", 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. The eight original Maxwell's equations can be written in the modern form of Heaviside's vector notation as follows: : ; Notation : ; : is the
magnetizing field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
, which Maxwell called the ''magnetic intensity''. ; : is the
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ...
(with being the total current including displacement current). ; : is the displacement field (called the ''electric displacement'' by Maxwell). ; : is the
free charge Free may refer to: Concept * Freedom, the ability to act or change without constraint or restriction * Emancipate, attaining civil and political rights or equality * Free (''gratis''), free of charge * Gratis versus libre, the difference betw ...
density (called the ''quantity of free electricity'' by Maxwell). ; : is the magnetic potential (called the ''angular impulse'' by Maxwell). ; : is called the ''electromotive force'' by Maxwell. The term
electromotive force In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transducer ...
is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern term
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
. ; : is the
electric potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
(which Maxwell also called ''electric potential''). ; : is the
electrical conductivity Electrical resistivity (also called volume resistivity or specific electrical resistance) is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity in ...
(Maxwell called the inverse of conductivity the ''specific resistance'', what is now called the
resistivity Electrical resistivity (also called volume resistivity or specific electrical resistance) is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity i ...
). Equation [], with the term, is 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 [] 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 [] when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame.


''A Treatise on Electricity and Magnetism''

In ''A Treatise on Electricity and Magnetism'', an 1873
treatise A treatise is a Formality, formal and systematic written discourse on some subject concerned with investigating or exposing the main principles of the subject and its conclusions."mwod:treatise, Treatise." Merriam-Webster Online Dictionary. Acc ...
on
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
written by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
, twelve general equations of the electromagnetic field are listed and these include the eight that are listed in the 1865 paper. His theoretical investigations of the electromagnetic field was guided by the notions of work, energy, potential, the principle of conservation of energy, and Lagrangian dynamics. All the principal equations concerning Maxwell's electromagnetic theory are recapitulated in Chapter IX of Part IV. At the end of this chapter, all the equations are listed and set in quaternion form. The first two equations [] and [] relates the electric scalar potential and magnetic vector potential to the electric and magnetic fields. The third equation [] relates the electromagnetic field to electromagnetic force. The rest of the equations [] to [] relates the electromagnetic field to material data: the current and charge densities as well as the material medium. Here the twelve Maxwell's equations have been given, respecting the original notations used by Maxwell. The only difference is that the vectors have been denoted using bold typeface instead of the original
Fraktur Fraktur () is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand. It is designed such that the beginnings and ends of the individual strokes that make up each letter will be clearly vis ...
typeface. For comparison Maxwell's equations in their original
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
form and their vector form have been given. The S. and V. notations are used to denote the scalar and vector parts of quaternion product. : ; Unfamiliar notation : ; : is the velocity of a point. ; : is total current. ; : is the intensity of magnetization. ; : is the current of conduction. ; : is the electric potential. ; : is the magnetic potential. ; : is the dielectric constant. ; : is electrical conductivity. ; : is electric charge density. ; : is magnetic charge density. In the same chapter, Maxwell points out that the consequence of equation [] is (in vector notation) \nabla \cdot \mathbf = 0. Similarly, taking divergence of equation [] gives conservation of electric charge, \nabla \cdot \mathbf = 0, which, Maxwell points out, is true only if the total current includes the variation of electric displacement. Lastly, combining equation [] and equation [], the formula \nabla^2 \mathbf = 4 \pi \mu \mathbf is obtained which relates magnetic potential with current. Elsewhere in the Part I of the book, the electric potential is related to charge density as \nabla^2 \Psi = - \frac e in the absence of motion. Presciently, Maxwell also mentions that although some of the equations could be combined to eliminate some quantities, the objective of his list was to express every relation of which there was any knowledge of, rather than to obtain compactness of mathematical formulae.


Relativity

Maxwell's equations were an essential inspiration for the development of special relativity. Possibly the most important aspect was their denial of instantaneous action at a distance. Rather, according to them, forces are propagated at the velocity of light through the electromagnetic field. Maxwell's original equations are based on the idea that light travels through a sea of molecular vortices known as the "
luminiferous aether Luminiferous aether or ether (''luminiferous'' meaning 'light-bearing') was the postulated Transmission medium, medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empt ...
", 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 Hendrik Antoon Lorentz ( ; ; 18 July 1853 – 4 February 1928) was a Dutch theoretical physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for their discovery and theoretical explanation of the Zeeman effect. He derive ...
along with George FitzGerald and
Joseph Larmor Sir Joseph Larmor (; 11 July 1857 – 19 May 1942) was an Irish mathematician and physicist who made breakthroughs in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influential work was ...
. Both Larmor (1897) and Lorentz (1899, 1904) ignored aether motion and derived the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
(so named by
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
) 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 also dismissed the notion of the aether, and relied on Lorentz's conclusion about the fixed speed of light, ''independent'' of the velocity of the observer. He applied the FitzGerald–Lorentz transformation to kinematics, and not just Maxwell's equations. Maxwell's equations played a key role in Einstein's groundbreaking 1905 scientific paper on
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
. 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 In special relativity, the rest frame of a particle is the frame of reference (a coordinate system attached to physical markers) in which the particle is at rest. The rest frame of compound objects (such as a fluid, or a solid made of many vibrati ...
of the magnet or that of the conductor. The
general theory of relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physi ...
has also had a close relationship with Maxwell's equations. For example,
Theodor Kaluza Theodor Franz Eduard Kaluza (; 9 November 1885 – 19 January 1954) was a German mathematician and physicist known for the Kaluza–Klein theory, involving field equations in five-dimensional space-time. His idea that fundamental forces can b ...
and
Oskar Klein Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physics, theoretical physicist. Oskar Klein is known for his work on Kaluza–Klein theory, which is partially named after him. Biography Klein was born ...
in the 1920s showed that Maxwell's equations could be derived by extending
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
into five physical
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
s. This strategy of using additional dimensions to unify different forces remains an active area of research in
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
.


See also

* Classical electromagnetism and special relativity * History of electromagnetic theory *
The Maxwellians ''The Maxwellians'' is a book by Bruce J. Hunt, published in 1991 by Cornell University Press; a paperback edition appeared in 1994, and the book was reissued in 2005. It chronicles the development of electromagnetic theory in the years after the ...


Notes


References

* {{DEFAULTSORT:Maxwell's Equations Electrodynamics History of physics
History History is the systematic study of the past, focusing primarily on the Human history, human past. As an academic discipline, it analyses and interprets evidence to construct narratives about what happened and explain why it happened. Some t ...