# 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.[1] 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.[2] 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,[3] 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[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 "On physical lines of force".[4]

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 This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current.[4]

For his original text on force, see: On Physical Lines of Force  – via Wikisource.
For his original text on dynamics, see: A Dynamical Theory of the Electromagnetic Field  – via Wikisource.

### 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.[5]

(A) The law of total currents

${\displaystyle \mathbf {J} _{tot}=}$ ${\displaystyle \,\mathbf {J} }$ ${\displaystyle +\,{\frac {\partial \mathbf {D} }{\partial t}}}$

(B) Definition of the magnetic potential

${\displaystyle \mu \mathbf {H} =\nabla \times \mathbf {A} }$

(C) Ampère's circuital law

${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{tot}}$

(D) The Lorentz force and Faraday's law of induction

${\displaystyle \mathbf {f} =\mu (\mathbf {v} \times \mathbf {H} )-{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi }$

(E) The electric elasticity equation

${\displaystyle \mathbf {f} ={\frac {1}{\epsilon }}\mathbf {D} }$

(F) Ohm's law

${\displaystyle \mathbf {f} ={\frac {1}{\sigma }}\mathbf {J} }$

(G) Gauss's law

${\displaystyle \nabla \cdot \mathbf {D} =\rho }$

(H) Equation of continuity of charge

${\displaystyle \nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}\,}$.

Notation

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.

Equation (D) implicitly contains the Lorentz force law and the differential form of Faraday's law of induction. For a static magnetic field, ${\displaystyle \partial \mathbf {A} /\partial t}$ 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,[4] 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).

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), 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.[6]

## Maxwell – electromagnetic light wave

Father of Electromagnetic Theory
A postcard from Maxwell to Peter Tait.

In part VI of "A Dynamical Theory of the Electromagnetic Field",[1] subtitled "Electromagnetic theory of light",[7] Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force",[4] 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

 ${\displaystyle \nabla \cdot \mathbf {E} =0}$ ${\displaystyle \nabla \times \mathbf {E} =-\mu _{o}{\frac {\partial \mathbf {H} }{\partial t}}}$ ${\displaystyle \nabla \cdot \mathbf {H} =0}$ ${\displaystyle \nabla \times \mathbf {H} =\varepsilon _{o}{\frac {\partial \mathbf {E} }{\partial t}}}$

If we take the curl of the curl equations we obtain

${\displaystyle \nabla \times \nabla \times \mathbf {E} =-\mu _{o}{\frac {\partial }{\partial t}}\nabla \times \mathbf {H} =-\mu _{o}\varepsilon _{o}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}$

${\displaystyle \nabla \times \nabla \times \mathbf {H} =\varepsilon _{o}{\frac {\partial }{\partial t}}\nabla \times \mathbf {E} =-\mu _{o}\varepsilon _{o}{\frac {\partial ^{2}\mathbf {H} }{\partial t^{2}}}}$

If we note the vector identity

${\displaystyle \nabla \times \left(\nabla \times \mathbf {V} \right)=\nabla \left(\nabla \cdot \mathbf {V} \right)-\nabla ^{2}\mathbf {V} }$

where ${\displaystyle \mathbf {V} }$ is any vector function of space, we recover the wave equations

${\displaystyle {\partial ^{2}\mathbf {E} \over \partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {E} \ \ =\ \ 0}$

${\displaystyle {\partial ^{2}\mathbf {H} \over \partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {H} \ \ =\ \ 0}$

where

${\displaystyle c={1 \over {\sqrt {\mu _{o}\varepsilon _{o}}}}=2.99792458\times 10^{8}}$ 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, 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"

and

"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.

## References

1. ^ a b c 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.)
2. ^ Royal Society archives; register of papers
3. ^ royalsociety.org
4. ^ a b c d Maxwell, James Clerk (1861). "On physical lines of force" (PDF). Philosophical Magazine.
5. ^ Cf.  Tai, Chen-To (1972), "On the presentation of Maxwell's theory" (Invited Paper), Proceedings of the IEEE  60 (8): 936–45.
6. ^ Maxwell, James Clerk (1873). A Treatise on Electricity and Magnetism. Oxford: Clarendon Press. Vol.II, p.233, eq.(J).
7. ^ 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. Retrieved Sep 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.
1864

1864 (MDCCCLXIV)

was a leap year starting on Friday of the Gregorian calendar and a leap year starting on Wednesday of the Julian calendar, the 1864th year of the Common Era (CE) and Anno Domini (AD) designations, the 864th year of the 2nd millennium, the 64th year of the 19th century, and the 5th year of the 1860s decade. As of the start of 1864, the Gregorian calendar was

12 days ahead of the Julian calendar, which remained in localized use until 1923.

1864 in Scotland

Events from the year 1864 in Scotland.

1864 in science

The year 1864 in science and technology included many events, some of which are listed here.

1864 in the United Kingdom

Events from the year 1864 in the United Kingdom.

1865 in Scotland

Events from the year 1865 in Scotland.

1865 in science

The year 1865 in science and technology involved some significant events, listed below.

1865 in the United Kingdom

Events from the year 1865 in the United Kingdom.

Carl Neumann

Carl Gottfried Neumann (also Karl; 7 May 1832 – 27 March 1925) was a German mathematician.

Displacement current

In electromagnetism, displacement current density is the quantity ∂D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials (as opposed to vacuum), there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

The idea was conceived by James Clerk Maxwell in his 1861 paper On Physical Lines of Force, Part III in connection with the displacement of electric particles in a dielectric medium. Maxwell added displacement current to the electric current term in Ampère's Circuital Law. In his 1865 paper A Dynamical Theory of the Electromagnetic Field Maxwell used this amended version of Ampère's Circuital Law to derive the electromagnetic wave equation. This derivation is now generally accepted as a historical landmark in physics by virtue of uniting electricity, magnetism and optics into one single unified theory. The displacement current term is now seen as a crucial addition that completed Maxwell's equations and is necessary to explain many phenomena, most particularly the existence of electromagnetic waves.

Electric displacement field

In physics, the electric displacement field, denoted by D, is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter squared (C⋅m−2).

Electromagnetic field

An electromagnetic field (also EMF or EM field) is a physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction. It is one of the four fundamental forces of nature (the others are gravitation, weak interaction and strong interaction).

The field can be viewed as the combination of an electric field and a magnetic field. The electric field is produced by stationary charges, and the magnetic field by moving charges (currents); these two are often described as the sources of the field. The way in which charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz force law. The force created by the electric field is much stronger than the force created by the magnetic field.From a classical perspective in the history of electromagnetism, the electromagnetic field can be regarded as a smooth, continuous field, propagated in a wavelike manner; whereas from the perspective of quantum field theory, the field is seen as quantized, being composed of individual particles.

Electromagnetic wave equation

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

{\displaystyle {\begin{aligned}\left(v_{ph}^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {E} &=\mathbf {0} \\\left(v_{ph}^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {B} &=\mathbf {0} \end{aligned}}}

where

${\displaystyle v_{ph}={\frac {1}{\sqrt {\mu \varepsilon }}}}$

is the speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and 2 is the Laplace operator. In a vacuum, vph = c0 = 299,792,458 meters per second, a fundamental physical constant. The electromagnetic wave equation derives from Maxwell's equations. In most older literature, B is called the magnetic flux density or magnetic induction.

History of Maxwell's equations

In electromagnetism, one of the fundamental fields of physics, the introduction of Maxwell's equations (mainly in "A Dynamical Theory of the Electromagnetic Field") was one of the most important aggregations of empirical facts in the history of physics. It took place in the nineteenth century, starting from basic experimental observations, and leading to the formulations of numerous mathematical equations, notably by Charles-Augustin de Coulomb, Hans Christian Ørsted, Carl Friedrich Gauss, Jean-Baptiste Biot, Félix Savart, André-Marie Ampère, and Michael Faraday. The apparently disparate laws and phenomena of electricity and magnetism were integrated by James Clerk Maxwell, who published an early form of the equations, which modify Ampère's circuital law by introducing a displacement current term. He showed that these equations imply that light propagates as electromagnetic waves. His laws were reformulated by Oliver Heaviside in the more modern and compact vector calculus formalism he independently developed. Increasingly powerful mathematical descriptions of the electromagnetic field were developed, continuing into the twentieth century, enabling the equations to take on simpler forms by advancing more sophisticated mathematics.

Index of physics articles (A)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

James Clerk Maxwell

James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist in the field of mathematical physics. His most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as different manifestations of the same phenomenon. Maxwell's equations for electromagnetism have been called the "second great unification in physics" after the first one realised by Isaac Newton.

With the publication of "A Dynamical Theory of the Electromagnetic Field" in 1865, Maxwell demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light. Maxwell proposed that light is an undulation in the same medium that is the cause of electric and magnetic phenomena. The unification of light and electrical phenomena led to the prediction of the existence of radio waves.

Maxwell helped develop the Maxwell–Boltzmann distribution, a statistical means of describing aspects of the kinetic theory of gases. He is also known for presenting the first durable colour photograph in 1861 and for his foundational work on analysing the rigidity of rod-and-joint frameworks (trusses) like those in many bridges.

His discoveries helped usher in the era of modern physics, laying the foundation for such fields as special relativity and quantum mechanics. Many physicists regard Maxwell as the 19th-century scientist having the greatest influence on 20th-century physics. His contributions to the science are considered by many to be of the same magnitude as those of Isaac Newton and Albert Einstein. In the millennium poll—a survey of the 100 most prominent physicists—Maxwell was voted the third greatest physicist of all time, behind only Newton and Einstein. On the centenary of Maxwell's birthday, Einstein described Maxwell's work as the "most profound and the most fruitful that physics has experienced since the time of Newton".

James Clerk Maxwell Foundation

The James Clerk Maxwell Foundation is a registered Scottish charity set up in 1977. By supporting physics and mathematics, it honours one of the greatest of physicists, James Clerk Maxwell (1831–1879), and works to increase the public awareness of science. It maintains a small museum in Maxwell's birthplace which is in the ownership of the Foundation.

Line of force

A line of force in Faraday's extended sense is synonymous with Maxwell's line of induction. According to J.J. Thomson, Faraday usually discusses lines of force as chains of polarized particles in a dielectric, yet sometimes Faraday discusses them as having an existence all their own as in stretching across a vacuum. In addition to lines of force, J.J. Thomson—similar to Maxwell—also calls them tubes of electrostatic inductance, or simply Faraday tubes. From the 20th century perspective, lines of force are energy linkages embedded in a 19th-century unified field theory that led to more mathematically and experimentally sophisticated concepts and theories, including Maxwell's equations, electromagnetic waves, and Einstein's relativity.

Lines of force originated with Michael Faraday, whose theory holds that all of reality is made up of force itself. His theory predicts that electricity, light, and gravity have finite propagation delays. The theories and experimental data of later scientific figures such as Maxwell, Hertz, Einstein, and others are in agreement with the ramifications of Faraday's theory. Nevertheless, Faraday's theory remains distinct. Unlike Faraday, Maxwell and others (e.g., J.J. Thomson) thought that light and electricity must propagate through an ether. In Einstein's relativity, there is no ether, yet the physical reality of force is much weaker than in the theories of Faraday.Historian Nancy J. Nersessian in her paper "Faraday's Field Concept" distinguishes between the ideas of Maxwell and Faraday:

The specific features of Faraday's field concept, in its 'favourite' and most complete form, are that force is a substance, that it is the only substance and that all forces are interconvertible through various motions of the lines of force. These features of Faraday's 'favourite notion' were not carried on. Maxwell, in his approach to the problem of finding a mathematical representation for the continuous transmission of electric and magnetic forces, considered these to be states of stress and strain in a mechanical aether. This was part of the quite different network of beliefs and problems with which Maxwell was working.

On Physical Lines of Force

"On Physical Lines of Force" is a famous four-part paper written by James Clerk Maxwell published between 1861 and 1862. In it, Maxwell derived the equations of electromagnetism in conjunction with a "sea" of "molecular vortices" which he used to model Faraday's lines of force. Maxwell had studied and commented on the field of electricity and magnetism as early as 1855/6 when "On Faraday's Lines of Force" was read to the Cambridge Philosophical Society. Maxwell made an analogy between the density of this medium and the magnetic permeability, as well as an analogy between the transverse elasticity and the dielectric constant, and using the results of a prior experiment by Wilhelm Eduard Weber and Rudolf Kohlrausch performed in 1856, he established a connection between the speed of light and the speed of propagation of waves in this medium.

The paper ushered in a new era of classical electrodynamics and catalyzed further progress in the mathematical field of vector calculus. Because of this, it is considered one of the most historically significant publications in the field of physics and of science in general, comparable with Einstein's Annus Mirabilis papers and Newton's Principia Mathematica.

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