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{{electromagnetism}}

In [[electromagnetism]], one of the [[fundamental interaction|fundamental]] [[field (physics)|field]]s of [[physics]], the introduction of [[Maxwell's equations]] (mainly in "''[[A Dynamical Theory of the Electromagnetic Field]]''") was one of the most important aggregation of [[Empirical evidence|empirical fact]]s in the [[history of physics]] that took place in the nineteenth century, starting from basic experimental observations to the formulations of numerous mathematical equations, notably by [[Charles-Augustin de Coulomb]], [[Hans Christian Ørsted]], [[Carl Freidrich Gauss]], [[Jean-Baptiste Biot]], [[Félix Savart]], [[André-Marie Ampère]], and [[Michael Faraday]]. The apparently separate laws and phenomena of electricity and magnetism culminated by [[James Clerk Maxwell]], who published an early form of the equations completing [[Ampère's circuital law]] by introducing the [[displacement current]] term, and showed these equations predict [[light]] to propagate as [[electromagnetic wave]]s. They were rewritten in by [[Oliver Heaviside]] in the more modern and compact [[vector calculus]] formalism he independently developed. Increasingly more powerful [[mathematical descriptions of the electromagnetic field]] were developed into the twentieth century, enabling the equations to take simpler forms using more advanced mathematics.

==Relation between electricity, magnetism, and the speed of light==

The relation between electricity, magnetism, and the speed of light can be summarized by the modern equation:
:<math>c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \ .</math>
The left-hand side is the speed of light, and the right-hand side is a quantity related to 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]] and [[Rudolf Kohlrausch]] 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 <math>1/\sqrt{\mu_0 \varepsilon_0}</math>), 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]] and measuring the magnetic force from the discharge current, and found a value {{val|3.107|e=8|u=m/s}},<ref name=Keithley>[http://books.google.com/books?id=uwgNAtqSHuQC&pg=PA115 The story of electrical and magnetic measurements: from 500 B.C. to the 1940s, by Joseph F. Keithley, p115]</ref> remarkably close to the speed of light, which had recently been measured at {{val|3.14|e=8|u=m/s}} by [[Hippolyte Fizeau]] in 1848 and at {{val|2.98|e=8|ul=m/s}} by [[Léon Foucault]] in 1850.<ref name=Keithley/> However, Weber and Kohlrausch did not make the connection to the speed of light.<ref name=Keithley/> Towards the end of 1861 while working on part III of his paper ''[[On Physical Lines of Force]]'', 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.<ref>"The Dictionary of Scientific Biography", by Charles Coulston Gillispie</ref>

==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]]'s "lines of force" and in conjunction with the experimental result of Weber and Kohlrausch. But it wasn't until 1884 that [[Oliver Heaviside]],<ref name=nahin/> concurrently with similar work by [[Willard Gibbs]] and [[Heinrich Hertz]],<ref name=buchwald/> grouped the four together into a distinct set. This group of four equations was known variously as the Hertz–Heaviside equations and the Maxwell–Hertz equations,<ref name=nahin>but are now universally known as ''Maxwell's 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.

{{cite book
| title = Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age
| author = Paul J. Nahin
| publisher = JHU Press
| isbn = 978-0-8018-6909-9
| pages = 108–112
| url = http://books.google.com/?id=e9wEntQmA0IC&pg=PA111&dq=nahin+hertz-heaviside+maxwell-hertz
| date = 2002-10-09
}}</ref> and are sometimes still known as the Maxwell–Heaviside equations.<ref>
{{cite book
| title = Modern nonlinear optics
| author = Myron Evans
| publisher = John Wiley and Sons
| isbn = 978-0-471-38931-6
| page = 240
| url = http://books.google.com/?id=9p0kK6IG94gC&pg=PA240&dq=maxwell-heaviside+equations
| date = 2001-10-05
}}</ref>

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 physicist [[Richard Feynman]] predicted that, "The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade."<ref>Crease, Robert. ''[http://books.google.com/books?id=IU04tZsVjXkC&lpg=PA133&dq=%22Civil%20War%20will%20pale%20into%20provincial%20insignificance%22&pg=PA133#v=onepage&q=%22Civil%20War%20will%20pale%20into%20provincial%20insignificance%22&f=false The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg]'', page 133 (2008).</ref>

The concept of fields was introduced by, among others, Faraday. [[Albert Einstein]] wrote:
{{quote|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)}}

Heaviside worked to eliminate the potentials ([[electric potential]] and [[magnetic potential]]) that Maxwell had used as the central concepts in his equations;<ref name=nahin/> this effort was somewhat controversial,<ref>
{{cite journal
| journal = Electrical Engineer
| volume = 7
| author = Oliver J. Lodge
| title = Sketch of the Electrical Papers in Section A, at the Recent Bath Meeting of the British Association
| date = November 1888
| page = 535
}}</ref> 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.<ref name=buchwald>
{{cite book
| title = The creation of scientific effects: Heinrich Hertz and electric waves
| author = Jed Z. Buchwald
| publisher = University of Chicago Press
| isbn = 978-0-226-07888-5
| page = 194
| url = http://books.google.com/?id=2bDEvvGT1EYC&pg=PA194&dq=maxwell+faraday+time-derivative+vector-potential
| year = 1994
}}</ref>

==''On Physical Lines of Force''==
{{Main|On Physical Lines of Force}}
The four modern day Maxwell's equations appeared throughout Maxwell's 1861 paper ''On Physical Lines of Force'':

#Equation (56) in Maxwell's 1861 paper is [[Gauss's law for magnetism|∇ • '''B''' = 0]].
#Equation (112) is [[Ampère's circuital law]] with Maxwell's displacement current added. It is the addition of [[displacement current]] that is the most significant aspect of Maxwell's work in [[electromagnetism]], as it enabled him to later derive the [[electromagnetic wave equation]] in his 1865 paper [[A Dynamical Theory of the Electromagnetic Field]], and hence show that light is an electromagnetic wave. It is therefore this aspect of Maxwell's work which gives the equations their full significance. (Interestingly, Kirchhoff derived the [[telegrapher's equations]] in 1857 without using [[displacement current]]. But he did use Poisson's equation and the equation of continuity which are the mathematical ingredients of the [[displacement current]]. Nevertheless, Kirchhoff believed his equations to be applicable only inside an electric wire and so he is not credited with having discovered that light is an electromagnetic wave).
#Equation (115) is [[Gauss's law]].
#Equation (54) is an equation that [[Oliver Heaviside]] referred to as 'Faraday's law'. This equation caters for the time varying aspect of electromagnetic induction, but not for the motionally induced aspect, whereas Faraday's original flux law caters for both aspects.<ref>
{{cite book
| title = Optical spectroscopies of electronic absorption
| author = J. R. Lalanne, F. Carmona, and L. Servant
| publisher = World Scientific
| isbn = 978-981-02-3861-2
| page = 8
| url = http://books.google.com/?id=7rWD-TdxKkMC&pg=PA8&dq=maxwell-faraday+derivative
| date = November 1999
}}</ref><ref>
{{cite book
| title = Introduction to Electromagnetic Engineering
| author = Roger F. Harrington
| publisher = Courier Dover Publications
| isbn = 978-0-486-43241-0
| pages = 49–56
| url = http://books.google.com/?id=ZlC2EV8zvX8C&pg=PR7&dq=maxwell-faraday-equation+law-of-induction
| date = 2003-10-17
}}</ref> Maxwell deals with the motionally dependent aspect of electromagnetic induction, '''v''' × '''B''', at equation (77). Equation (77) which is the same as equation (D) in the original eight Maxwell's equations listed below, corresponds to all intents and purposes to the modern day force law '''F ''' = ''q''( '''E''' + '''v''' × '''B''' ) which sits adjacent to Maxwell's equations and bears the name [[Lorentz force]], even though Maxwell derived it when Lorentz was still a young boy.

The difference between the '''B''' and the '''H''' 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 equation]]s.

[[File:Molecular Vortex Model.svg|right|thumb|300px|Figure of Maxwell's molecular vortex model. For a uniform magnetic field, the field lines point outward from the display screen, as can be observed from the black dots in the middle of the hexagons. The vortex of each hexagonal molecule rotates counter-clockwise. The small green circles are clockwise rotating particles sandwiching between the molecular vortices.]]

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, '''H''' represented pure vorticity (spin), whereas '''B''' 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,

#'''Magnetic induction current''' causes a magnetic current density '''B''' = μ '''H''' was essentially a rotational analogy to the linear electric current relationship,
#'''Electric convection current''' '''J''' = ρ '''v''' where ρ is electric charge density. '''B''' was seen as a kind of magnetic current of vortices aligned in their axial planes, with '''H''' being the circumferential velocity of the vortices. With ''µ'' representing vortex density, it follows that the product of ''µ'' with vorticity '''H''' leads to the [[magnetic field]] denoted as '''B'''.

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 '''B''' 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 '''B''' is to '''H''', and where '''J''' is to ρ, then it necessarily follows from Gauss's law and from the equation of continuity of charge that '''E''' is to '''D'''. i.e. '''B''' parallels with '''E''', whereas '''H''' parallels with '''D'''.

==''A Dynamical Theory of the Electromagnetic Field''==
{{Main|A Dynamical Theory of the Electromagnetic Field}}

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" sometimes arises because it has been 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",<ref>[http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf page 480.]</ref> 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. (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:

:{|class="wikitable" style="text-align: center;"
|-
!scope="col" width="250"|(A) The law of total currents
|scope="col" width="250"|<math>\mathbf{J}_\mathrm{tot} = \mathbf{J} + \frac{\partial\mathbf{D}}{\partial t}</math>
|-
!(B) The equation of magnetic force
|<math>\mu \mathbf{H} = \nabla \times \mathbf{A}</math>
|-
!(C) Ampère's circuital law
|<math>\nabla \times \mathbf{H} = \mathbf{J}_\mathrm{tot}</math>
|-
!(D) Electromotive force created by convection, induction, and by static electricity. (This is in effect the [[Lorentz force]])
|<math>\mathbf{E} = \mu \mathbf{v} \times \mathbf{H} - \frac{\partial\mathbf{A}}{\partial t}-\nabla \phi </math>
|-
!(E) The electric elasticity equation
|<math>\mathbf{E} = \frac{1}{\varepsilon} \mathbf{D}</math>
|-
!(F) Ohm's law
|<math>\mathbf{E} = \frac{1}{\sigma} \mathbf{J}</math>
|-
!(G) Gauss's law
|<math>\nabla \cdot \mathbf{D} = \rho</math>
|-
!(H) Equation of continuity
|<math>\nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\partial t}</math>

or

<math>\nabla \cdot \mathbf{J}_\mathrm{tot} = 0</math>
|-
|}

;Notation

: '''H''' is the [[Effective magnetic field|magnetizing field]], which Maxwell called the ''magnetic intensity''.
:'''J''' is the [[current density]] (with'''J'''<sub>tot</sub> being the total current including displacement current).<ref group="note">Here it is noted that a quite different quantity, the ''magnetic polarization'', μ<sub>0</sub>'''M''' by decision of an international [[IUPAP]] commission has been given the same name '''J'''. So for the electric current density, a name with small letters, '''j''' would be better. But even then the mathematicians would still use the large-letter name '''J''' for the corresponding current two-form (see below).</ref>
: '''D''' is the [[electric displacement field|displacement field]] (called the ''electric displacement'' by Maxwell).
: ρ is the [[free charge]] density (called the ''quantity of free electricity'' by Maxwell).
: '''A''' is the [[magnetic potential]] (called the ''angular impulse'' by Maxwell).
: '''E''' is called the ''electromotive force'' by Maxwell. The term [[electromotive force]] is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern term [[electric field]].
: φ is the [[electric potential]] (which Maxwell also called ''electric potential'').
: σ is the [[electrical conductivity]] (Maxwell called the inverse of conductivity the ''specific resistance'', what is now called the [[resistivity]]).

It is interesting to note the μ'''v''' × '''H''' 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 μ'''v''' × '''H''' term from equation D when he is deriving the [[electromagnetic wave equation]], as he considers the situation only from the rest frame.

==''A Treatise on Electricity and Magnetism''==
{{Main|A Treatise on Electricity and Magnetism}}
{{Wikisourcelang|en|A Treatise on Electricity and Magnetism|''A Treatise on Electricity and Magnetism''}}

In ''[[A Treatise on Electricity and Magnetism]]'', an 1873 [[treatise]] on [[electromagnetism]] written by [[James Clerk Maxwell]], eleven general equations of the electromagnetic field are listed and these include the eight that are listed in the 1865 paper.<ref>http://www.mathematik.tu-darmstadt.de/~bruhn/Original-MAXWELL.htm</ref>

==Relativity==
{{main|Maxwell's equations in curved spacetime}}
{{see also|History of special relativity}}

Maxwell's original equations are based on the idea that light travels through a sea of molecular vortices known as the "[[luminiferous aether]]", 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.<ref group="note">Experiments like the [[Michelson–Morley experiment]] in 1887 showed that the "aether" moved at the same speed as Earth. While other experiments, such as measurements of the [[aberration of light]] from the [[star]]s, showed that the ether is moving relative to the Earth.</ref>

A more theoretical approach was suggested by [[Hendrik Lorentz]] along with [[George FitzGerald]] and [[Joseph Larmor]]. 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 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]] dismissed the notion of the aether as an unnecessary one, and he concluded that Maxwell's equations predicted the existence of a fixed speed of light, ''independent'' of the velocity of the observer. Hence, he used the Maxwell's equations as the starting point for his [[Special Theory of Relativity]]. In doing so, he established that the FitzGerald–Lorentz transformation is valid for all matter and space, and not just Maxwell's equations. Maxwell's equations played a key role in Einstein's groundbreaking scientific paper on [[special relativity]] (1905). For example, in the opening paragraph of his paper, he began his theory by noting that a description of an [[moving magnet and conductor problem|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]] of the magnet or that of the conductor.<ref>
{{cite web
|url=http://www.fourmilab.ch/etexts/einstein/specrel/www/
|title=On the Electrodynamics of Moving Bodies
|publisher=Fourmilab.ch
|date=
|accessdate=2008-10-19
}}</ref>

The [[general theory of relativity]] has also had a close relationship with Maxwell's equations. For example, [[Theodor Kaluza]] and [[Oskar Klein]] [[Kaluza–Klein theory|in the 1920s showed]] that Maxwell's equations could be derived by extending [[general relativity]] into five physical [[dimension]]s. This strategy of using additional dimensions to unify different forces remains an active area of research in [[physics]].

==See also==

* [[Classical electromagnetism and special relativity]]
* [[History of electromagnetic theory]]

==Notes==
{{Reflist|group="note"|1}}

==References==
{{Reflist|2}}

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