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A Dynamical Theory of the Electromagnetic Field
From Wikipedia, the free encyclopedia
"A Dynamical Theory of the Electromagnetic Field" is
the third of James Clerk Maxwell's papers regarding
electromagnetism, published in 1865.[1] It is the paper in
which the original set of four Maxwell's equations first
appeared. The concept of displacement current, which he
had introduced in his 1861 paper "On Physical Lines of
Force", was utilized for the first time, to derive the
electromagnetic wave equation.[2]
Contents
A dynamical theory of the
electromagnetic field
Author(s)
James Clerk Maxwell
Language
English
Subject(s)
Classical electromagnetism
Genre(s)
Scientific paper
Publisher
Philosophical Transactions
of the Royal Society
Publication date 1865
■ 1 Maxwell's original equations
■ 2 Maxwell – First to propose that light is an
electromagnetic wave
■ 3 See also
■ 4 References
■ 5 Further reading
Maxwell's original equations
In part III of "A Dynamical Theory of the Electromagnetic Field", 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 set of four vectorized equations selected
in 1884 by Oliver Heaviside, which had all appeared in "On physical lines of force".[2]
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had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies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.[2]
Eighteen of the twenty original Maxwell's equations can be vectorized into 6 equations. Each vectorized
equation represents 3 original equations in component form. Including the other two equations, in modern
vector notation, they can form a set of eight equations. They are listed below:
(A) The law of total currents
(B) Definition of the magnetic potential
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(C) Ampère's circuital law
(D) The Lorentz force
This force represents the effect of electric fields created by convection, induction, and by charges.
(E) The electric elasticity equation
(F) Ohm's law
(G) Gauss's law
(H) Equation of continuity of charge
Notation
is the magnetic field, which Maxwell called the "magnetic intensity".
is the electric current density (with
being the total current including displacement current).
is the displacement field (called the "electric displacement" by Maxwell).
ρ is the free charge density (called the "quantity of free electricity" by Maxwell).
is the magnetic potential (called the "angular impulse" by Maxwell).
is the electric field (called the "electromotive force" by Maxwell, not to be confused with the
scalar quantity that is now called electromotive force).
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).
Maxwell did not consider completely general materials; his initial formulation used linear, isotropic,
nondispersive permittivity ε and permeability µ, although he also discussed the possibility of anisotropic
materials.
It is of particular interest to note that Maxwell includes a
term in his expression for the
"electromotive force" at equation "D" , which corresponds to the magnetic force per unit charge on a
moving conductor with velocity . This means that equation "D" is effectively the Lorentz force. This
equation first appeared at equation (77) in "On Physical Lines of Force" quite some time before Lorentz
thought of it.[2] Nowadays, the Lorentz force sits alongside Maxwell's equations as an additional
electromagnetic equation that is not included as part of the set.
When Maxwell derives the electromagnetic wave equation in his 1864 paper, he uses equation "D" as
opposed to using Faraday's law of electromagnetic induction as in modern textbooks. Maxwell however
drops the
term from equation "D" when he is deriving the electromagnetic wave equation, and
he considers the situation only from the rest frame.
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Maxwell – First to propose that light is an electromagnetic wave
In "A dynamical theory of the electromagnetic field", Maxwell utilized
the correction to Ampère's Circuital Law that he had made in part III of
"On physical lines of force".[1] In part VI of his 1864 paper
"Electromagnetic theory of light",[citation needed] Maxwell combined
displacement current with some of the other equations of
electromagnetism and obtained a wave equation with a speed equal to 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.
Father of Electromagnetic
Theory
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
A postcard from Maxwell to
Peter Tait.
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
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where
meters per second
is the speed of light in free space.
See also
■ A Treatise on Electricity and Magnetism
■ On Physical Lines of Force
References
1. ^ a b c Maxwell, James Clerk (1865). "A dynamical theory of the electromagnetic
field" (http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pd
(PDF). Philosophical Transactions of the Royal Society of London 155: 459–512. doi:10.1098/rstl.1865.0008
(http://dx.doi.org/10.1098%2Frstl.1865.0008) .
http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf
(This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
2. ^ a b c d Maxwell, James Clerk (1861). "On physical lines of
force" (http://upload.wikimedia.org/wikipedia/commons/b/b8/On_Physical_Lines_of_Force.pdf) (PDF).
Philosophical Magazine.
http://upload.wikimedia.org/wikipedia/commons/b/b8/On_Physical_Lines_of_Force.pdf.
Further reading
■ 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" (http://www-gap.dcs.stand.ac.uk/~history/Projects/Johnson/Chapters/Ch4_4.html) . James Clerk Maxwell – The Great
Unknown. http://www-gap.dcs.st-and.ac.uk/~history/Projects/Johnson/Chapters/Ch4_4.html.
Retrieved Sept. 7, 2009.
■ Tokunaga, Kiyohisa (2002). "Part 2, Chapter V – Maxwell's
Equations" (http://www.d3.dion.ne.jp/~kiyohisa/tieca/251.htm) . Total Integral for
Electromagnetic Canonical Action. http://www.d3.dion.ne.jp/~kiyohisa/tieca/251.htm. Retrieved
Sept. 7, 2009.
■ Katz, Randy H. (February 22, 1997). "'Look Ma, No Wires': Marconi and the Invention of
Radio" (http://www.cs.berkeley.edu/~randy/Courses/CS39C.S97/radio/radio.html) . History of
Communications Infrastructures.
http://www.cs.berkeley.edu/~randy/Courses/CS39C.S97/radio/radio.html. Retrieved Sept. 7, 2009.
Retrieved from "http://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field"
Categories: 1860s in science | Electromagnetism | Physics papers | Works by James Clerk Maxwell |
1865 works | Works originally published in Philosophical Transactions of the Royal Society
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