"A Dynamical Theory of the Electromagnetic Field" is a paper by
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
on
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, published in 1865.
[ ''(Paper read at a meeting of the Royal Society on 8 December 1864).''] 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
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
on 8 December 1864, having been sent by Maxwell to the Society on 27 October. It then underwent
peer review
Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work (peers). It functions as a form of self-regulation by qualified members of a profession within the relevant field. Peer review ...
, being sent to William Thompson (later
Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
) on 24 December 1864. It was then sent to
George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish migration to Great Britain, Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University ...
, the Society's Physical Sciences Secretary, on 23 March 1865. It was approved for publication in the ''
Philosophical Transactions of the Royal Society
''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journa ...
'' 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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
, 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
"On Physical Lines of Force" is a four-part paper written by James Clerk Maxwell published in 1861. In it, Maxwell derived the equations of electromagnetism in conjunction with a "sea" of "molecular vortices" which he used to model Faraday's li ...
".[
Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that they are written in modern ]vector notation
In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space.
For representing a vector, the common typographic convention is l ...
. They actually only contain one of the original eight—equation "G" (Gauss's Law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
). 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
In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
.[
]
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.[Cf. Tai, Chen-To (1972)]
"On the presentation of Maxwell's theory"
(Invited Paper), ''Proceedings of the IEEE'' 60(8): 936–45.
;(A) The law of total currents
;(B) Definition of the magnetic potential Magnetic potential may refer to:
* Magnetic vector potential, the vector whose curl is equal to the magnetic B field
* Magnetic scalar potential
Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electr ...
;(C) Ampère's circuital law
;(D) The Lorentz force and Faraday's law of induction
;(E) The electric elasticity equation
;(F) Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
;(G) Gauss's law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
;(H) Equation of continuity of charge
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
.
;Notation
:
is the
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, which Maxwell called the "''magnetic intensity''".
:
is the
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
density (with
being the total current density including
displacement current
In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
).
:
is the
displacement field (called the "''electric displacement''" by Maxwell).
:
is the
free charge
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
density (called the "''quantity of free electricity''" by Maxwell).
:
is the
magnetic potential Magnetic potential may refer to:
* Magnetic vector potential, the vector whose curl is equal to the magnetic B field
* Magnetic scalar potential
Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electr ...
(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
In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...
; see
below).
:
is the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
(which Maxwell also called "''electric potential''").
:
is the
electrical conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allow ...
(Maxwell called the inverse of conductivity the "''specific resistance''", what is now called the
resistivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
).
:
is the vector operator ''
del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
''.
Clarifications
Maxwell did not consider completely general materials; his initial formulation used linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
, isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
, nondispersive media with permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
''ϵ'' and permeability ''μ'', although he also discussed the possibility of anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
materials.
Gauss's law for magnetism
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
() is not included in the above list, but follows directly from equation (B) by taking divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
s (because the divergence of the curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
is zero).
Substituting (A) into (C) yields the familiar differential form of the Maxwell-Ampère law.
Equation (D) implicitly contains the Lorentz force law
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboar ...
and the differential form of Faraday's law of induction. For a ''static'' magnetic field, vanishes, and the electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
becomes conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
and is given by , 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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
". 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
The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant vel ...
''). Where there is no motion through the magnetic field — e.g., in transformer
A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
s — 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
.
Taking curls, noting that the curl of a gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
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 In special relativity, the rest frame of a particle is the frame of reference (a coordinate system attached to physical markers) in which the particle is at rest.
The rest frame of compound objects (such as a fluid, or a solid made of many vibratin ...
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 equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
s (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",][ A Dynamical Theory of the Electromagnetic Field/Part VI] 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
In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
, 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
If we take the curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of the curl equations we obtain
If we note the vector identity
where is any vector function of space, we recover the wave equations
where
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
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
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
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
used Maxwell's equations as the starting point for his special theory of relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
, presented in ''The Electrodynamics of Moving Bodies'', one of Einstein's 1905 ''Annus Mirabilis
''Annus mirabilis'' (pl. ''anni mirabiles'') is a Latin phrase that means "marvelous year", "wonderful year", "miraculous year", or "amazing year". This term has been used to refer to several years during which events of major importance are re ...
'' papers. 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.
See also
* '' A Treatise on Electricity and Magnetism''
* Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
References
Further reading
*
*
*
*Darrigol, Olivier (2000). ''Electromagnetism from Ampère to Einstein.'' Oxford University Press. ISBN 978-0198505945
*
{{DEFAULTSORT:Dynamical Theory Of The Electromagnetic Field
1860s in science
Electromagnetism
Physics papers
Works by James Clerk Maxwell
Maxwell's equations
1865 documents
Works originally published in Philosophical Transactions of the Royal Society