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. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.
Publication.
Following standard procedure for the time, the paper was first read to the Royal Society on 8 December 1864, having been sent by Maxwell to the Society on 27 October. It then underwent peer review, being sent to William Thompson (later Lord Kelvin) on 24 December 1864. 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, 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.
Maxwell's original equations.
In part III of the paper, which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations 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 "On Physical Lines of Force".
Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that they are written in modern vector notation. They actually only contain one of the original eight—equation "G" (Gauss's Law). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation "A") with Ampère's circuital law (equation "C"). This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current.
Heaviside's equations.
Eighteen of Maxwell's twenty original equations can be vectorized into six equations, labeled (A) to (F) below, each of which represents a group of three original equations in component form. The 19th and 20th of Maxwell's component equations appear as (G) and (H) below, making a total of eight vector equations. These are listed below in Maxwell's original order, designated by the letters that Maxwell assigned to them in his 1864 paper.
formula_1 formula_2 formula_3
formula_4
formula_5
formula_6
formula_7
formula_8
formula_9
formula_10.
Clarifications
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 () 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 .
Equation (D) implicitly contains the Lorentz force law and the differential form of Faraday's law of induction. For a "static" magnetic field, formula_11 vanishes, and the electric field becomes conservative and is given by , so that (D) reduces to
formula_12.
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, 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 ) reduces to the electric field , so that Maxwell's equation (D) reduces to
formula_13.
Taking curls, noting that the curl of a gradient is zero, we obtain
formula_14
which is the differential form of Faraday's law. Thus the three terms on the right side of equation (D) may be described, from left to right, as the motional term, the transformer term, and the conservative term.
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. But he still works from equation (D), in contrast to modern textbooks which tend to work from Faraday's law (see below).
The constitutive equations (E) and (F) are now usually written in the rest frame of the medium as and .
Maxwell's equation (G), viewed in isolation as printed in the 1864 paper, at first seems to say that .  However, if we trace the signs through the previous two triplets of equations, we see that what seem to be the components of are in fact the components of . The notation used in Maxwell's later "Treatise on Electricity and Magnetism" is different, and avoids the misleading first impression.
Maxwell – electromagnetic light wave.
In part VI of "A Dynamical Theory of the Electromagnetic Field", subtitled "Electromagnetic theory of light", Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force", which is defined as displacement current, to derive the electromagnetic wave equation.
He obtained a wave equation with a speed in close agreement to experimental determinations of the speed of light. He commented,
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.
Modern equation methods.
To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. Using (SI units) in a vacuum, these equations are
 {\partial t}</math>
formula_15
formula_16
If we take the curl of the curl equations we obtain
formula_17
formula_18
If we note the vector identity
formula_19
where formula_20 is any vector function of space, we recover the wave equations
formula_21
formula_22
where
formula_23 meters per second
is the speed of light in free space.
Legacy and impact.
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.