View source for A Dynamical Theory of the Electromagnetic Field

{{Short description|1865 physics paper by James Maxwell}}

"'''A Dynamical Theory of the Electromagnetic Field'''" is a paper by [[James Clerk Maxwell]] on [[electromagnetism]], published in 1865.<ref name=ADTEF>{{cite journal |last=Maxwell |first=James Clerk |authorlink=James Clerk Maxwell |title=A dynamical theory of the electromagnetic field |journal=Philosophical Transactions of the Royal Society of London |volume=155 |url=https://archive.org/details/dynamicaltheoryo00maxw/ |pages=459–512 |year=1865 |doi=10.1098/rstl.1865.0008 |s2cid=186207827 |ol=25533062M}} ''(Paper read at a meeting of the Royal Society on 8 December 1864).''</ref> Physicist [[Freeman Dyson]] called the publishing of the paper the "most important event of the nineteenth century in the history of the physical sciences".<ref>{{Cite book |last=Dyson |first=F.J. |title=2nd European Conference on Antennas and Propagation (EuCAP 2007) |chapter=Why is Maxwell's theory so hard to understand? |date=2007  |url=https://digital-library.theiet.org/content/conferences/10.1049/ic.2007.1146 |language=en |publisher=Institution of Engineering and Technology |pages=619 |doi=10.1049/ic.2007.1146 |isbn=978-0-86341-842-6}}</ref>

The paper was key in establishing the classical theory of electromagnetism.<ref>{{Cite journal |last=Longair |first=Malcolm |date=2015-04-13 |title='...a paper ...I hold to be great guns': a commentary on Maxwell (1865) 'A dynamical theory of the electromagnetic field' |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |language=en |volume=373 |issue=2039 |pages=20140473 |doi=10.1098/rsta.2014.0473 |issn=1364-503X |pmc=4360095 |pmid=25750155|bibcode=2015RSPTA.37340473L }}</ref> Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and also 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 Thomson (later [[Lord Kelvin]]) on 24 December 1864.<ref>Royal Society archives; register of papers</ref> 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,<ref>[https://royalsociety.org/~/media/library/The%20library%20and%20archives%20of%20the%20royal%20society%201660-1990%20Marie%20Boas%20Hall.pdf royalsociety.org]</ref> 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.<!-- Note: the date in the PDF version of the paper is incorrect due to errors during digitization. This is due to be rectified in the future -->

==Maxwell's original equations==
In part III of the paper, which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations<ref name=ADTEF/> 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 his 1861 paper "[[On Physical Lines of Force]]".<ref name=OPLF/>

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]].<ref name=OPLF/>

===Heaviside's equations===
{{see also|Oliver Heaviside}}

Eighteen of Maxwell's twenty original equations can be [[Vectorization (mathematics)|vectorized]] into six equations, labeled [[#A|(A)]] to [[#F|(F)]] below, each of which represents a group of three original equations in [[Vector component|component form]]. The 19th and 20th of Maxwell's component equations appear as [[#G|(G)]] and [[#H|(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 1865 paper.<ref name=tai-72>{{cite journal | doi=10.1109/PROC.1972.8818 | title=On the presentation of Maxwell's theory | date=1972 | last1=Chen-To Tai | journal=Proceedings of the IEEE | volume=60 | issue=8 | pages=936–945 }}</ref>

;(A) {{anchor|A}} The law of total currents
<div class="center">
<math>\mathbf{J}_{\rm tot} = </math> [[electric current|<math>\,\mathbf{J}</math>]] <math> +\,\frac{\partial\mathbf{D}}{\partial t}</math>
</div>
;(B) {{anchor|B}} Definition of the [[magnetic vector potential|magnetic potential]]
<div class="center">
<math>\mu \mathbf{H} = \nabla \times \mathbf{A}</math>
</div>
;(C) {{anchor|C}} [[Ampère's circuital law]]
<div class="center">
<math>\nabla \times \mathbf{H} = \mathbf{J}_{\rm tot}</math>
</div>
;(D) {{anchor|D}} The [[Lorentz force]] and [[Faraday's law of induction]]
<div class="center">
<math>\mathbf{f} = \mu (\mathbf{v} \times \mathbf{H}) - \frac{\partial\mathbf{A}}{\partial t}-\nabla \phi </math>
</div>
;(E) {{anchor|E}} The electric elasticity equation
<div class="center">
<math>\mathbf{f} = \frac{1}{\varepsilon} \mathbf{D}</math>
</div>
;(F) {{anchor|F}} [[Ohm's law]]
<div class="center">
<math>\mathbf{f} = \frac{1}{\sigma} \mathbf{J}</math>
</div>
;(G) {{anchor|G}} [[Gauss's law]]
<div class="center">
<math>\nabla \cdot \mathbf{D} = \rho</math>
</div>
;(H) {{anchor|H}} Equation of [[continuity of charge]]
<div class="center">
<math>\nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\partial t}\,</math>.
</div>

;Notation
<div style="font-size:smaller;">
: <math>\mathbf{H}</math> is the [[magnetic field]], which Maxwell called the "''magnetic intensity''".
: <math>\mathbf{J}</math> is the [[electric current]] density (with <math>\mathbf{J}_{\rm tot}</math> being the total current density including [[displacement current]]).
: <math>\mathbf{D}</math> is the [[Electric displacement field|displacement field]] (called the "''electric displacement''" by Maxwell).
: <math>\rho</math> is the [[free charge]] density (called the "''quantity of free electricity''" by Maxwell).
: <math>\mathbf{A}</math> is the [[magnetic vector potential|magnetic potential]] (called the "''angular impulse''" by Maxwell).
: <math>\mathbf{f}</math> 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 [[#Clarify|below]]).
: <math>\phi</math> is the [[electric potential]] (which Maxwell also called "''electric potential''").
: <math>\sigma</math> is the [[electrical conductivity]] (Maxwell called the inverse of conductivity the "''specific resistance''", what is now called the [[resistivity]]).
: <math>\nabla</math> is the vector operator ''[[del]]''.
</div>

<div id="Clarify">'''Clarifications'''</div>

Maxwell did not consider completely general materials; his initial formulation used [[linear]], [[isotropic]], [[dispersion (optics)|nondispersive]] media with [[permittivity]] ''ϵ'' and [[Permeability (electromagnetism)|permeability]] ''μ'', although he also discussed the possibility of [[anisotropic]] materials.

[[Gauss's law for magnetism]] ({{math|∇⋅{{hsp}}'''B''' {{=}} 0}}) is not included in the above list, but follows directly from equation&nbsp;[[#B|(B)]] by taking [[divergence]]s (because the divergence of the [[curl (mathematics)|curl]] is zero).

Substituting [[#A|(A)]] into [[#C|(C)]] yields the familiar differential form of the [[Ampère's circuital law#Extending the original law: the Maxwell–Ampère equation|Maxwell-Ampère law]].

Equation [[#D|(D)]] implicitly contains the [[Lorentz force|Lorentz force law]] and the differential form of [[Faraday's law of induction]]. For a ''static'' magnetic field, <math>\partial\mathbf{A}/\partial t</math> vanishes, and the [[electric field]] {{math|'''E'''}} becomes [[conservative vector field|conservative]] and is given by {{math|−∇''ϕ''}}, so that [[#D|(D)]] reduces to
<div class="center"><math>\mathbf{f}=\mathbf{E}+\mathbf{v}\times\mathbf{B}\,</math>.</div>
This is simply the Lorentz force law on a per-unit-charge basis — although Maxwell's equation&nbsp;[[#D|(D)]] first appeared at equation ([http://upload.wikimedia.org/math/3/1/f/31fe90d9486dae0e5a1165c5637f4bbb.png 77]) in "On Physical Lines of Force" in 1861,<ref name=OPLF/> 34&nbsp;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 ''[[Lorentz force#EMF|motional emf]]{{Broken anchor|date=2025-06-09|bot=User:Cewbot/log/20201008/configuration|target_link=Lorentz force#EMF|reason= The anchor (EMF) [[Special:Diff/1283546059|has been deleted]].|diff_id=1283546059}}'' in electric generators (see also ''[[Moving magnet and conductor problem]]''). Where there is no motion through the magnetic field — e.g., in [[transformer]]s — we can drop the cross-product term, and the force per unit charge (called {{math|'''f'''}}) reduces to the electric field {{math|'''E'''}}, so that Maxwell's equation&nbsp;[[#D|(D)]] reduces to
<div class="center"><math>\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t}-\nabla\phi\,</math>.</div>
Taking curls, noting that the curl of a [[gradient]] is zero, we obtain
<div class="center"><math>\nabla\times\mathbf{E}\,=\,-\nabla\times\frac{\partial\mathbf{A}}{\partial t}\,=\,-\frac{\partial}{\partial t}\big(\nabla\times\mathbf{A}\big)\,=\,-\frac{\partial\mathbf{B}}{\partial t}\,,</math></div>
which is the [[Faraday's law of induction#Maxwell–Faraday equation|differential form of Faraday's law]]. Thus the three terms on the right side of equation&nbsp;[[#D|(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&nbsp;[[#D|(D)]], in contrast to modern textbooks which tend to work from Faraday's law (see [[#Modern equation methods|below]]).

The [[constitutive equation]]s [[#E|(E)]] and [[#F|(F)]] are now usually written in the rest frame of the medium as {{math|'''D'''{{tsp}}{{=}}{{tsp}}''ϵ'''''E'''}} and {{math|'''J'''{{tsp}}{{=}}{{tsp}}''σ'''''E'''}}.

Maxwell's equation&nbsp;[[#G|(G)]], as printed in the 1865 paper, requires his {{mvar|e}} to mean ''minus'' the charge density (if his {{mvar|f, g, h}} are the components of{{math| '''D'''}}), whereas his equation&nbsp;[[#H|(H)]] requires his {{mvar|e}} to mean ''plus'' the charge density (if his {{mvar|p, q, r}} are the components of{{math| '''J'''}}). John&nbsp;W.&nbsp;Arthur<ref name="arthur947">Arthur, J.W., "The evolution of Maxwell's equations from 1862 to the present day", ''IEEE Antennas &amp; Propagation Magazine'', vol.&nbsp;55, no.&nbsp;3 (June&nbsp;2013), incorporating "Correction" in vol.&nbsp;55, no.&nbsp;4 (Aug.&nbsp;2013), republished with permission at [https://johnwarthur.com/articles-by-john-w-arthur/evolution-of-maxwells-equations/ johnwarthur.com/articles-by-john-w-arthur/evolution-of-maxwells-equations] (cited page numbers are from this link).</ref>{{rp|7,8}} concludes that the sign of {{mvar|e}} in&nbsp;(G) is wrong, and observes{{r|arthur947|p=8}} that this sign is corrected in Maxwell's subsequent ''Treatise''.<ref>Maxwell, James Clerk (1873). ''A Treatise on Electricity and Magnetism''. Oxford: Clarendon Press. Vol.&nbsp;{{serif|II}}, [https://archive.org/stream/electricandmag02maxwrich#page/n263/mode/2up p.&nbsp;233], eq.&nbsp;({{serif|J}}).</ref> Arthur speculates that the sign confusion may have arisen from the analogy between momentum and the magnetic vector potential (Maxwell's "electromagnetic momentum"), in which positive mass corresponds to ''negative'' charge{{r|arthur947|p=4}}. Arthur{{r|arthur947|p=3}} also lists some corresponding equations from Maxwell's earlier paper of 1861-2,{{r|OPLF}} and notes that the signs do not always match the later ones. The earlier signs (1861-2) are correct if {{mvar|F, G, H}} are the components of{{math| −'''A'''}} while {{mvar|f, g, h}} are the components of{{math| −'''D'''}}.

== Maxwell – electromagnetic light wave ==

[[Image:James Clerk Maxwell.jpg|thumb|right|175px|Father of Electromagnetic Theory]] [[Image:Postcard-from-Maxwell-to-Tait.jpg|thumb|right|175px|A postcard from Maxwell to [[Peter Guthrie Tait|Peter Tait]]]]

In part VI of "A Dynamical Theory of the Electromagnetic Field",<ref name=ADTEF/>  subtitled "Electromagnetic theory of light",<ref>[[s:A Dynamical Theory of the Electromagnetic Field/Part VI|A Dynamical Theory of the Electromagnetic Field/Part VI]]</ref> Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force",<ref name=OPLF>{{cite journal |last=Maxwell |first=James Clerk |title=On physical lines of force |url=http://upload.wikimedia.org/wikipedia/commons/b/b8/On_Physical_Lines_of_Force.pdf |journal=Philosophical Magazine |year=1861}}</ref> 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,

{{blockquote|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

<div class="center">
{| style="margin:1em auto;"
|<math>\nabla \cdot \mathbf{E} = 0</math>
|-
|<math> \nabla \times \mathbf{E} = -\mu_o \frac{\partial \mathbf{H}} {\partial t}</math>
|-
|<math> \nabla \cdot \mathbf{H} = 0</math>
|-
|<math> \nabla \times \mathbf{H} =\varepsilon_o \frac{ \partial \mathbf{E}} {\partial t}</math>
|}
</div>

If we take the [[Curl (mathematics)|curl]] of the curl equations we obtain
<div class="center">
<math> \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} </math>

<math> \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}
</math>
</div>
If we note the vector identity

<div class="center">
<math>\nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V}</math>
</div>

where <math> \mathbf{V} </math> is any vector function of space, we recover the wave equations

<div class="center">
<math> {\partial^2 \mathbf{E} \over \partial t^2} \ - \  c^2 \cdot \nabla^2 \mathbf{E}  \ \ = \ \ 0</math>
</div>

<div class="center">
<math> {\partial^2 \mathbf{H} \over \partial t^2} \ - \  c^2 \cdot \nabla^2 \mathbf{H}  \ \ = \ \ 0</math>
</div>

where

<div class="center">
<math>c = { 1 \over \sqrt{ \mu_o \varepsilon_o } } = 2.99792458 \times 10^8 </math> meters per second
</div>

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."<ref>{{Cite web |last=Feynman |first=Richard |date=November 1, 2018 |others="I think that the discovery of electricity and magnetism and the electromagnetic effects which are finally worked out, the full equations were worked out by Maxwell in 1873 [is] probably the most fundamental transformation, the most remarkable thing in history, the biggest change in history" |title=Fun to Imagine |url=https://www.youtube.com/watch?v=P1ww1IXRfTA&t=2234s |access-date=May 21, 2025 |website=[[YouTube]]}}</ref>

[[Albert Einstein]] used Maxwell's equations as the starting point for his [[special theory of relativity]], presented in ''The Electrodynamics of Moving Bodies'', one of Einstein's 1905 ''[[Annus Mirabilis papers|Annus Mirabilis]]'' [[Annus Mirabilis papers|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 [[Kaluza–Klein theory|extending general relativity into five physical dimensions]].

==See also==
{{wikisource|A Dynamical Theory of the Electromagnetic Field}}
{{Wikisource|On Physical Lines of Force}}
* ''[[A Treatise on Electricity and Magnetism]]''
* [[Gauge theory]]
{{-}}

==References==
{{reflist|2}}

==Further reading==
* {{cite book |first1=James C. |last1=Maxwell |first2=Thomas F. |last2=Torrance |title=A Dynamical Theory of the Electromagnetic Field |date=March 1996 |isbn=1-57910-015-5 |publisher=Wipf and Stock |location=Eugene, OR}}
* {{cite book |last=Niven |first=W. D. |title=The Scientific Papers of James Clerk Maxwell |publisher=Dover |location=New York |year=1952 |volume=1}}
* {{cite web |last=Johnson |first=Kevin |url=http://www-gap.dcs.st-and.ac.uk/~history/Projects/Johnson/Chapters/Ch4_4.html |title=The electromagnetic field |date=May 2002 |work=James Clerk Maxwell – The Great Unknown |accessdate=Sep 7, 2009 |archive-url=https://web.archive.org/web/20080915180648/http://www-gap.dcs.st-and.ac.uk/~history/Projects/Johnson/Chapters/Ch4_4.html |archive-date=September 15, 2008 |url-status=dead |df=mdy-all }}
* Darrigol, Olivier (2000). ''Electromagnetism from Ampère to Einstein.'' Oxford University Press. ISBN 978-0198505945
* {{cite web |last=Katz |first=Randy H. |url=http://www.cs.berkeley.edu/~randy/Courses/CS39C.S97/radio/radio.html |title='Look Ma, No Wires': Marconi and the Invention of Radio |work=History of Communications Infrastructures |date=February 22, 1997 |accessdate=Sep 7, 2009}}

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