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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.
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 This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current.
- For his original text on force, see: Wikisource. – via
- For his original text on dynamics, see: Wikisource. – via
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.
- (A) The law of total currents
- (B) Definition of the magnetic potential
- (E) The electric elasticity equation
- (F) Ohm's law
- (G) Gauss's law
- (H) Equation of continuity of charge
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.
Equation (D) implicitly contains the Lorentz force law and the differential form of Faraday's law of induction. For a static magnetic field, 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, 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. 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).
Maxwell's equation (G), viewed in isolation as printed in the 1864 paper, at first seems to say that ρ + ∇⋅ D = 0. However, if we trace the signs through the previous two triplets of equations, we see that what seem to be the components of D are in fact the components of −D. 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,
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.
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
If we take the curl of the curl equations we obtain
If we note the vector identity
where is any vector function of space, we recover the wave equations
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, a paper produced during his 1905 Annus Mirabilis. In it is stated:
- "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good"
- "Any ray of light moves in the "stationary" system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body."
Maxwell's equations can also be derived by extending general relativity into five physical dimensions.
- Maxwell, James Clerk (1865). "A dynamical theory of the electromagnetic field" (PDF). Philosophical Transactions of the Royal Society of London. 155: 459–512. doi:10.1098/rstl.1865.0008. (This article followed a December 8, 1864, presentation by Maxwell to the Royal Society.)
- Royal Society archives; register of papers
- Maxwell, James Clerk (1861). "On physical lines of force" (PDF). Philosophical Magazine.
- Cf. Tai, Chen-To (1972), "On the presentation of Maxwell's theory" (Invited Paper), Proceedings of the IEEE 60 (8): 936–45.
- Maxwell, James Clerk (1873). A Treatise on Electricity and Magnetism. Oxford: Clarendon Press. Vol. II, p., eq. 233 (J).
- A Dynamical Theory of the Electromagnetic Field/Part VI
- Maxwell, James C.; Torrance, Thomas F. (March 1996). A Dynamical Theory of the Electromagnetic Field. Eugene, OR: Wipf and Stock. ISBN 1-57910-015-5.
- Niven, W. D. (1952). The Scientific Papers of James Clerk Maxwell. Vol. 1. New York: Dover.
- Johnson, Kevin (May 2002). "The electromagnetic field". James Clerk Maxwell – The Great Unknown. Archived from the original on September 15, 2008. Retrieved September 7, 2009.
- Tokunaga, Kiyohisa (2002). "Part 2, Chapter V – Maxwell's Equations". Total Integral for Electromagnetic Canonical Action. Archived from the original on 2010-11-10. Retrieved Sep 7, 2009.
- Katz, Randy H. (February 22, 1997). "'Look Ma, No Wires': Marconi and the Invention of Radio". History of Communications Infrastructures. Retrieved Sep 7, 2009.
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