A Dynamical Theory of the Electromagnetic Field
[1] Physicist Freeman Dyson called the publishing of the paper the "most important event of the nineteenth century in the history of the physical sciences.
"[2] The paper was key in establishing the classical theory of electromagnetism.
[3] Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and also deduces that light is an electromagnetic wave.
It then underwent peer review, being sent to William Thomson (later Lord Kelvin) on 24 December 1864.
[4] It was then sent to George Gabriel Stokes, the Society's physical sciences secretary, on 23 March 1865.
It was approved for publication in the Philosophical Transactions of the Royal Society on 15 June 1865, by the Committee of Papers (essentially the society's governing council) and sent to the printer the following day (16 June).
During this period, Philosophical Transactions was only published as a bound volume once a year,[5] and would have been prepared for the society's anniversary day on 30 November (the exact date is not recorded).
However, the printer would have prepared and delivered to Maxwell offprints, for the author to distribute as he wished, soon after 16 June.
In part III of the paper, which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations[1] which were to become known as Maxwell's equations, until this term became applied instead to a vectorized set of four equations selected in 1884, which had all appeared in his 1861 paper "On Physical Lines of Force".
[6] Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that they are written in modern vector notation.
This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current.
Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive media with permittivity ϵ and permeability μ, although he also discussed the possibility of anisotropic materials.
Gauss's law for magnetism (∇⋅ B = 0) is not included in the above list, but follows directly from equation (B) by taking divergences (because the divergence of the curl is zero).
Substituting (A) into (C) yields the familiar differential form of the Maxwell-Ampère law.
vanishes, and the electric field E becomes conservative and is given by −∇ϕ, so that (D) reduces to This is simply the Lorentz force law on a per-unit-charge basis — although Maxwell's equation (D) first appeared at equation (77) in "On Physical Lines of Force" in 1861,[6] 34 years before Lorentz derived his force law, which is now usually presented as a supplement to the four "Maxwell's equations".
The cross-product term in the Lorentz force law is the source of the so-called motional emf in electric generators (see also Moving magnet and conductor problem).
Where there is no motion through the magnetic field — e.g., in transformers — we can drop the cross-product term, and the force per unit charge (called f) reduces to the electric field E, so that Maxwell's equation (D) reduces to Taking curls, noting that the curl of a gradient is zero, we obtain which is the differential form of Faraday's law.
In deriving the electromagnetic wave equation, Maxwell considers the situation only from the rest frame of the medium, and accordingly drops the cross-product term.
The constitutive equations (E) and (F) are now usually written in the rest frame of the medium as D = ϵE and J = σE.
Maxwell's equation (G), as printed in the 1865 paper, requires his e to mean minus the charge density (if his f, g, h are the components of D), whereas his equation (H) requires his e to mean plus the charge density (if his p, q, r are the components of J).
[9] Arthur speculates that the sign confusion may have arisen from the analogy between momentum and the magnetic vector potential (Maxwell's "electromagnetic momentum"), in which positive mass corresponds to negative charge[8]: 4 .
Arthur[8]: 3 also lists some corresponding equations from Maxwell's earlier paper of 1861-2,[6] and notes that the signs do not always match the later ones.
In part VI of "A Dynamical Theory of the Electromagnetic Field",[1] subtitled "Electromagnetic theory of light",[10] Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force",[6] which is defined as displacement current, to derive the electromagnetic wave equation.
He commented, The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics by a much less cumbersome method which combines the corrected version of Ampère's Circuital Law with Faraday's law of electromagnetic induction.
To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations.
is any vector function of space, we recover the wave equations
meters per second is the speed of light in free space.
Of this paper and Maxwell's related works, fellow physicist Richard Feynman said: "From the long view of this history of mankind – seen from, say, 10,000 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 electromagnetism."
Albert Einstein used Maxwell's equations as the starting point for his special theory of relativity, presented in The Electrodynamics of Moving Bodies, one of Einstein's 1905 Annus Mirabilis papers.
In it is stated: and Maxwell's equations can also be derived by extending general relativity into five physical dimensions.
