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By the first half of the 19th century, the understanding ofelectromagnetics had improved through many experiments and theoretical work. In the 1780s,Charles-Augustin de Coulomb established hislaw ofelectrostatics. In 1825,André-Marie Ampère published hisforce law. In 1831,Michael Faraday discoveredelectromagnetic induction through his experiments, and proposedlines of forces to describe it. In 1834,Emil Lenz solved the problem of the direction of the induction, andFranz Ernst Neumann 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 byJames Clerk Maxwell through a series of papers published from the 1850s to the 1870s. In the 1850s, Maxwell was working at theUniversity of Cambridge where he was impressed by Faraday'slines of forces concept. Faraday created this concept by impression ofRoger Boscovich, a physicist that impacted Maxwell's work as well.^[1] In 1856, he published his first paper in electromagnetism:On Faraday's Lines of Force.^[2] He tried to use the analogy of incompressible fluid flow to model the magnetic lines of forces. Later, Maxwell moved toKing's College London 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 ofOn Physical Lines of Force.^{[3][4][5][6][7]} 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 thespeed of lightc 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 1865A Dynamical Theory of the Electromagnetic Field,^[8] in which the theory is formulated in strictly mathematical form. In 1873, Maxwell publishedA Treatise on Electricity and Magnetism 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.^[9]

Maxwell built a simple flywheel model of electromagnetism, andBoltzmann built an elaborate mechanical model ("Bicykel") based on Maxwell's flywheel model, which he used for lecture demonstrations.^[10] Figures are at the end of Boltzmann's 1891 book.^[11]

Later,Oliver Heaviside studied Maxwell'sA Treatise on Electricity and Magnetism and employedvector calculus to synthesize Maxwell's over 20 equations into the four recognizable ones which modern physicists use. Maxwell's equations also inspiredAlbert Einstein in developing thetheory of special relativity.^[13]

The experimental proof of Maxwell's equations was demonstrated byHeinrich Hertz in a series of experiments in the 1890s.^[14] After that, Maxwell's equations were fully accepted by scientists.

The relationships amongst electricity, magnetism, and the speed of light can be summarized by the modern equation:

${\displaystyle c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}.}$

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, whenWilhelm Eduard Weber andRudolf 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${\displaystyle 1/\sqrt{{\mu }_0{\epsilon }_0}}$), and determined that it should have units of velocity. They then measured this ratio by an experiment which involved charging and discharging aLeyden jar and measuring the magnetic force from the discharge current, and found a value3.107×10⁸ m/s,^[15] remarkably close to the speed of light, which had recently been measured at3.14×10⁸ m/s byHippolyte Fizeau in 1848 and at2.98×10⁸ m/s byLéon Foucault in 1850.^[15] However, Weber and Kohlrausch did not make the connection to the speed of light.^[15] Towards the end of 1861 while working on Part III of his paperOn 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.^[16]

The four modern Maxwell's equations can be found individually throughout his 1861 paper, derived theoretically using a molecular vortex model ofMichael Faraday's "lines of force" and in conjunction with the experimental result of Weber and Kohlrausch. But it was not until 1884 thatOliver Heaviside, concurrently with similar work byJosiah Willard Gibbs andHeinrich Hertz, grouped the twenty equations together into a set of only four, viavector notation.^[17] This group of four equations was known variously as the Hertz–Heaviside equations and the Maxwell–Hertz equations, but are now universally known asMaxwell's equations.^[18] 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.^[19]

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${\displaystyle F=ma}$ is attributed to Newton, but Newton in fact wrote his second law as${\displaystyle F=\dot{p}}$. This is clearly visible in a glass case in theWren Library ofTrinity College, Cambridge, where Newton's manuscript is open to the relevant page, showing the equation${\displaystyle F=\dot{p}}$, where${\displaystyle \dot{p}}$ is thetime derivative of the momentum${\displaystyle p}$. This seems a trivial enough fact until you realize that${\displaystyle F=\dot{p}}$ remains true inspecial relativity, without modification.

Maxwell's contribution to science in producing these equations lies in the correction he made toAmpère's circuital law in his 1861 paperOn Physical Lines of Force. He added thedisplacement current term to Ampère's circuital law and this enabled him to derive theelectromagnetic wave equation in his later 1865 paperA Dynamical Theory of the Electromagnetic Field and to demonstrate the fact that light is anelectromagnetic wave. This fact was later confirmed experimentally by Heinrich Hertz in 1887. The physicistRichard Feynman 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."^[20]

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 polarized 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.

Heaviside worked to eliminate the potentials (electric potential andmagnetic potential) that Maxwell had used as the central concepts in his equations;^[21] this effort was somewhat controversial,^[22] 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.^[23]

The four equations we use today appeared separately in Maxwell's 1861 paper,On Physical Lines of Force:

1. Equation (56) in Maxwell's 1861 paper isGauss's law for magnetism,∇ •B = 0.
2. Equation (112) isAmpère's circuital law, with Maxwell's addition ofdisplacement current. This may be the most remarkable contribution of Maxwell's work, enabling him to derive theelectromagnetic wave equation 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 thetelegrapher's equations in 1857 without using displacement current, but he did usePoisson's equation 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).
3. Equation (115) isGauss's law.
4. 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.^[24][25] Maxwell deals with the motion-related aspect of electromagnetic induction,v ×B, 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,F =q(E +v ×B), which sits adjacent to Maxwell's equations and bears the nameLorentz force, even though Maxwell derived it when Lorentz was still a young boy.

The difference between theB and theH vectors can be traced back to Maxwell's 1855 paper entitledOn Faraday's Lines of Force which was read to theCambridge 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 ofdifferential equations.

It is later clarified in his concept of a sea of molecular vortices that appears in his 1861 paperOn Physical Lines of Force. Within that context,H represented pure vorticity (spin), whereasB was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell consideredmagnetic permeabilityµ to be a measure of the density of the vortex sea. Hence the relationship,

1. Magnetic induction current causes a magnetic current densityB = μH was essentially a rotational analogy to the linear electric current relationship,
2. Electric convection currentJ =ρv whereρ is electric charge density.B was seen as a kind of magnetic current of vortices aligned in their axial planes, withH being the circumferential velocity of the vortices. Withµ representing vortex density, it follows that the product ofµ with vorticityH leads to themagnetic field denoted asB.

The electric current equation can be viewed as a convective current ofelectric 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 theB vector. The magnetic inductive current represents lines of force. In particular, it represents lines ofinverse-square law force.

The extension of the above considerations confirms that whereB is toH, and whereJ is toρ, then it necessarily follows from Gauss's law and from the equation of continuity of charge thatE is toD i.e.B parallels withE, whereasH parallels withD.

In 1865 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 1865 paper "A dynamical theory of the electromagnetic field", entitled "General equations of the electromagnetic field",^[26] 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.^[a]

The eight original Maxwell's equations can be written in the modern form of Heaviside's vector notation as follows:

[A] The law of total currents	${\displaystyle {\mathbf{J}}_{\mathrm{t}\mathrm{o}\mathrm{t}}=\mathbf{J}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}}$
[B] The equation of magnetic force	${\displaystyle \mu \mathbf{H}=\mathrm{\nabla }\times \mathbf{A}}$
[C] Ampère's circuital law	${\displaystyle \mathrm{\nabla }\times \mathbf{H}={\mathbf{J}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}$
[D] Electromotive force created by convection, induction, and by static electricity. (This is in effect theLorentz force)	${\displaystyle \mathbf{E}=\mu \mathbf{v}\times \mathbf{H}-\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}-\mathrm{\nabla }\varphi }$
[E] The electric elasticity equation	${\displaystyle \mathbf{E}=\frac{1}{\epsilon }\mathbf{D}}$
[F] Ohm's law	${\displaystyle \mathbf{E}=\frac{1}{\sigma }\mathbf{J}}$
[G] Gauss's law	${\displaystyle \mathrm{\nabla }\cdot \mathbf{D}=\rho }$
[H] Equation of continuity	${\displaystyle \mathrm{\nabla }\cdot \mathbf{J}=-\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}}$ or ${\displaystyle \mathrm{\nabla }\cdot {\mathbf{J}}_{\mathrm{t}\mathrm{o}\mathrm{t}}=0}$

Notation

H is themagnetizing field, which Maxwell called themagnetic intensity.

J is thecurrent density (withJ_tot being the total current including displacement current).^[b]

D is thedisplacement field (called theelectric displacement by Maxwell).

ρ is thefree charge density (called thequantity of free electricity by Maxwell).

A is themagnetic potential (called theangular impulse by Maxwell).

E is called theelectromotive force by Maxwell. The termelectromotive force is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern termelectric field.

ϕ is theelectric potential (which Maxwell also calledelectric potential).

σ is theelectrical conductivity (Maxwell called the inverse of conductivity thespecific resistance, what is now called theresistivity).

Equation [D], with theμv ×H 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 [D] to cater for electromagnetic induction rather thanFaraday's law of induction which is used in modern textbooks. (Faraday's law itself does not appear among his equations.) However, Maxwell drops theμv ×H term from equation [D] when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame.

EnglishWikisource has original text related to this article:

"A treatise on electricity and magnetism"

InA Treatise on Electricity and Magnetism, an 1873treatise onelectromagnetism written byJames Clerk Maxwell, twelve general equations of the electromagnetic field are listed and these include the eight that are listed in the 1865 paper.^[27] 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[A] and[B] relates the electric scalar potential and magnetic vector potential to the electric and magnetic fields. The third equation[C] relates the electromagnetic field to electromagnetic force. The rest of the equations[D] to[L] 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 originalFraktur typeface. For comparison Maxwell's equations in their originalquaternion form and their vector form have been given. The${\displaystyle S.}$ and${\displaystyle V.}$ notations are used to denote thescalar and vector parts of quaternion product.

Name	Quaternion Form	Vector Form
[A] Magnetic induction	${\displaystyle \mathbf{B}=V.\mathrm{\nabla }\mathbf{A}}$; ${\displaystyle S.\mathrm{\nabla }\mathbf{A}=0}$	${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A}}$; ${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}=0}$
[B] Electromotive force	${\displaystyle \mathbf{E}=V.\mathbf{G}\mathbf{B}-\dot{\mathbf{A}}-\mathrm{\nabla }\mathrm{\Psi }}$	${\displaystyle \mathbf{E}=\mathbf{G}\times \mathbf{B}-\dot{\mathbf{A}}-\mathrm{\nabla }\mathrm{\Psi }}$
[C] Mechanical force	${\displaystyle \mathbf{F}=V.\mathbf{C}\mathbf{B}+e\mathbf{E}-m\mathrm{\nabla }\mathrm{\Omega }}$	${\displaystyle \mathbf{F}=\mathbf{C}\times \mathbf{B}+e\mathbf{E}-m\mathrm{\nabla }\mathrm{\Omega }}$
[D] Magnetization	${\displaystyle \mathbf{B}=\mathbf{H}+4\pi \mathbf{J}}$	${\displaystyle \mathbf{B}=\mathbf{H}+4\pi \mathbf{J}}$
[E] Electric currents	${\displaystyle 4\pi \mathbf{C}=V.\mathrm{\nabla }\mathbf{H}}$	${\displaystyle 4\pi \mathbf{C}=\mathrm{\nabla }\times \mathbf{H}}$
[F] Ohm's law	${\displaystyle \mathbf{K}=C\mathbf{E}}$	${\displaystyle \mathbf{K}=C\mathbf{E}}$
[G] Electric displacement	${\displaystyle \mathbf{D}=\frac{1}{4\pi }K\mathbf{E}}$	${\displaystyle \mathbf{D}=\frac{1}{4\pi }K\mathbf{E}}$
[H] Total current	${\displaystyle \mathbf{C}=\mathbf{K}+\dot{\mathbf{D}}}$	${\displaystyle \mathbf{C}=\mathbf{K}+\dot{\mathbf{D}}}$
[I] When magnetization arises from magnetic induction	${\displaystyle \mathbf{B}=\mu \mathbf{H}}$	${\displaystyle \mathbf{B}=\mu \mathbf{H}}$
[J] Electric volume density	${\displaystyle e=S.\mathrm{\nabla }\mathbf{D}}$	${\displaystyle e=\mathrm{\nabla }\cdot \mathbf{D}}$
[K] Magnetic volume density	${\displaystyle m=S.\mathrm{\nabla }\mathbf{J}}$	${\displaystyle m=\mathrm{\nabla }\cdot \mathbf{J}}$
[L] When magnetic force can be derived from a potential	${\displaystyle \mathbf{H}=-\mathrm{\nabla }\mathrm{\Omega }}$	${\displaystyle \mathbf{H}=-\mathrm{\nabla }\mathrm{\Omega }}$

Unfamiliar notation

G is the velocity of a point.

C is total current.

J is the intensity of magnetization.

K is the current of conduction.

Ψ is the electric potential.

Ω is the magnetic potential.

K is the dielectric constant.

C is electrical conductivity.

e is electric charge density.

m is magnetic charge density.

In the same chapter, Maxwell points out that the consequence of equation [A] is (in vector notation)${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$. Similarly, taking divergence of equation [E] gives conservation of electric charge,${\displaystyle \mathrm{\nabla }\cdot \mathbf{C}=0}$, which, Maxwell points out, is true only if the total current includes the variation of electric displacement. Lastly, combining equation [A] and equation [E], the formula${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}=4\pi \mu \mathbf{C}}$ 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${\displaystyle {\mathrm{\nabla }}^2\mathrm{\Psi }=-\frac{4\pi }{K}e}$ 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.

Maxwell's equations were an essential inspiration for the development of special relativity. Possibly the most important aspect was their denial ofinstantaneous action at a distance. Rather, according to them, forces are propagated at the velocity of light through the electromagnetic field.^[28]: 189

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.^[c]

A more theoretical approach was suggested byHendrik Lorentz along withGeorge FitzGerald andJoseph Larmor. Both Larmor (1897) and Lorentz (1899, 1904) ignored aether motion and derived theLorentz transformation (so named byHenri 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 themathematical 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 onspecial relativity. For example, in the opening paragraph of his paper, he began his theory by noting that a description of anelectric conductor moving with respect to a magnet must generate a consistent set of fields regardless of whether the force is calculated in therest frame of the magnet or that of the conductor.^[29]

Thegeneral theory of relativity has also had a close relationship with Maxwell's equations. For example,Theodor Kaluza andOskar Kleinin the 1920s showed that Maxwell's equations could be derived by extendinggeneral relativity into five physicaldimensions. This strategy of using additional dimensions to unify different forces remains an active area of research inphysics.

• Classical electromagnetism and special relativity
• History of electromagnetic theory
• The Maxwellians

• Turnbull, Graham, ed. (29 October 2019)."Maxwell's Equations".Engineering and Technology History Wiki.

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• History of physics
• Maxwell's equations

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