A Dynamical Theory of the Electromagnetic Field
|Author||James Clerk Maxwell|
|Publisher||Philosophical Transactions of the Royal Society|
"A Dynamical Theory of the Electromagnetic Field" is the third of James Clerk Maxwell's papers regarding electromagnetism, published in 1865. 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.
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 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 This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current.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: On Physical Lines of Force. Wikisource.
- For his original text on dynamics, see: A Dynamical Theory of the Electromagnetic Field. Wikisource.
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
- (D) The Lorentz force
- (E) The electric elasticity equation
- (F) Ohm's law
- (G) Gauss's law
- (H) Equation of continuity of charge
- 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).
- 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 force per unit charge (called the "electromotive force" by Maxwell, not to be confused with the scalar quantity that is now called electromotive force; see below).
- is the electric potential (which Maxwell also called "electric potential").
- is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).
- ∇ is the vector operator del.
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 mathematically corresponds[further explanation needed] 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. In the modern age, the force equations described by Hendrik Lorentz are placed 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.
Maxwell – electromagnetic light 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". In part VI of his 1864 paper "Electromagnetic theory of light", 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.
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
meters per second
is the speed of light in free space.
References and notes
|Wikisource has original text related to this article:|
- 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 accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
- Maxwell, James Clerk (1861). "On physical lines of force" (PDF). Philosophical Magazine.
- See: Electromagnetic induction and Electrostatic induction
- A Dynamical Theory of the Electromagnetic Field/Part VI
- The Structure Of The Ether. By Dr. H. Bateman. 1915-21-06.
- Maxwell's Displacement Current and His Theory of Light. Joan Bromberg. Communicated by S. G. Brush.
- 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. Retrieved Sep 7, 2009.
- Tokunaga, Kiyohisa (2002). "Part 2, Chapter V – Maxwell's Equations". Total Integral for Electromagnetic Canonical Action. 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.