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| {{Infobox book
| | Title of the writer is almost certainly [http://Pinterest.com/search/pins/?q=Adrianne+Quesada Adrianne Quesada]. Her [http://Www.Encyclopedia.com/searchresults.aspx?q=husband+doesn%27t husband doesn't] like the idea the way she has been doing but what she really likes doing is that will bake but she's bearing in mind on starting something newbie. After being out of your man's job for years he became an order sales person. Vermont will where her house could be described as. Her husband and her always maintain a website. You might need to check it out: http://circuspartypanama.com<br><br> |
| | name = A dynamical theory of the electromagnetic field
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| | title_orig =
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| | translator =
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| | image = <!-- include the [[File:]] and the image size -->
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| | caption =
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| | author = [[James Clerk Maxwell]]
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| | country =
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| | language = [[English language|English]]
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| | series =
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| | subject = [[Classical electromagnetism]]
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| | genre = [[Scientific paper]]
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| | publisher = [[Philosophical Transactions of the Royal Society]]
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| | pub_date = 1865
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| }}
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| "'''A Dynamical Theory of the Electromagnetic Field'''" is the third of [[James Clerk Maxwell]]'s papers regarding [[electromagnetism]], published in 1865.<ref name=ADTEF>{{cite journal |doi=10.1098/rstl.1865.0008 |last=Maxwell |first=James Clerk |authorlink=James Clerk Maxwell |title=A dynamical theory of the electromagnetic field | url=http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf |format=PDF |journal=Philosophical Transactions of the Royal Society of London |volume=155 |pages=459–512 |year=1865}} (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)</ref> 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]].<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 |format=PDF |journal=Philosophical Magazine |year=1861}}</ref>
| | Feel free to surf to my page ... [http://circuspartypanama.com clash of clans hack] |
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| ==Maxwell's original equations==
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| In part III of "[http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf A Dynamical Theory of the Electromagnetic Field]", 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 "On physical lines of force".<ref name=OPLF/>
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| 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/>
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| :''For his original text on force, see'': {{cite wikisource|title=On Physical Lines of Force|last=|first=|year=|publisher=|page=|wspage=|scan=}}
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| :''For his original text on dynamics, see'': {{cite wikisource|title=A Dynamical Theory of the Electromagnetic Field|last=|first=|year=|publisher=|page=|wspage=|scan=}}
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| ===Heaviside's equations===
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| {{see also|Oliver Heaviside}}
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| Eighteen of the twenty original Maxwell's equations can be [[Vectorization (mathematics)|vectorized]] into 6 equations. Each vectorized equation represents 3 original equations in [[Vector component|component form]]. Including the other two equations, in modern vector notation, they can form a set of eight equations. They are listed below:
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| ;(A) The law of total currents
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| <center>
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| <math>\mathbf{J}_{tot} = </math>[[electric current|<math>\mathbf{J}</math>]]<math> + \frac{\partial\mathbf{D}}{\partial t}</math>
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| </center>
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| ;(B) Definition of the [[magnetic potential]]
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| <center>
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| <math>\mu \mathbf{H} = \nabla \times \mathbf{A}</math>
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| </center>
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| ;(C) [[Ampère's circuital law]]
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| <center>
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| <math>\nabla \times \mathbf{H} = \mathbf{J}_{tot}</math>
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| </center>
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| <div id="D">
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| ;(D) The [[Lorentz force]]</div>
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| <center>
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| <math>\mathbf{f} = \mu (\mathbf{v} \times \mathbf{H}) - \frac{\partial\mathbf{A}}{\partial t}-\nabla \phi </math>
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| </center>
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| This ''Maxwellian electromotive force'' represents the effect of [[electric field]]s created by [[convection]], induction,<ref>See: [[Electromagnetic induction]] and [[Electrostatic induction]]</ref> and by [[electric charge]]s.
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| ;(E) The electric elasticity equation
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| <center>
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| <math>\mathbf{f} = \frac{1}{\epsilon} \mathbf{D}</math>
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| </center>
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| ;(F) [[Ohm's law]]
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| <center>
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| <math>\mathbf{f} = \frac{1}{\sigma} \mathbf{J}</math>
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| </center>
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| ;(G) [[Gauss's law]]
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| <center>
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| <math>\nabla \cdot \mathbf{D} = - \rho</math>
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| </center>
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| ;(H) Equation of [[continuity of charge]]
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| <center>
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| <math>\nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\partial t}</math>
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| </center>
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| ;Notation
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| <small>
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| : <math>\mathbf{H}</math> is the [[magnetic field]], which Maxwell called the "''magnetic intensity''".
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| : <math>\mathbf{J}</math> is the [[electric current]] density (with <math>\mathbf{J}_{tot}</math> being the total current including [[displacement current]]).
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| : <math>\mathbf{D}</math> is the [[Electric displacement field|displacement field]] (called the "''electric displacement''" by Maxwell).
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| : <math>\rho</math> is the [[free charge]] density (called the "''quantity of free electricity''" by Maxwell).
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| : <math>\mathbf{A}</math> is the [[magnetic potential]] (called the "''angular impulse''" by Maxwell).
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| : <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]]; [[#Clarify|see below]]).
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| : <math>\phi</math> is the [[electric potential]] (which Maxwell also called "''electric potential''").
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| : <math>\sigma</math> is the [[electrical conductivity]] (Maxwell called the inverse of conductivity the "''specific resistance''", what is now called the [[resistivity]]).</small>
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| <div id="Clarify">'''Clarifications'''</div>
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| Maxwell did not consider completely [[materials|general materials]]; his initial formulation used [[linear]], [[isotropic]], [[Dispersive and nondispersive media|nondispersive]] [[permittivity]] ''ε'' and [[Permeability (electromagnetism)|permeability]] ''μ'', although he also discussed the possibility of [[anisotropic]] materials.
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| It is of particular interest to note that Maxwell includes a <math>\mu (\mathbf{v} \times \mathbf{H})</math> term in his expression for the "electromotive force" at equation "[[#D|D]]", which [[Correspondence (mathematics)|mathematically corresponds]]{{elucidate|date=August 2013}} to the [[Electromotive force#Notation and units of measurement|magnetic force per unit charge]] on a [[Moving magnet and conductor problem|moving conductor]] with [[velocity]] <math>\mathbf{v}</math>. This means that equation "[[#D|D]]" is effectively the [[Lorentz force]]. This equation first appeared at equation ([http://upload.wikimedia.org/math/3/1/f/31fe90d9486dae0e5a1165c5637f4bbb.png 77]) in "''[[s:On Physical Lines of Force|On Physical Lines of Force]]''" quite some time before Lorentz thought of it.<ref name=OPLF/> 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.
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| When Maxwell derives the [[electromagnetic wave equation]] in his 1864 paper, he uses equation "[[#D|D]]" as opposed to using Faraday's law of electromagnetic induction as in modern textbooks. Maxwell however drops the <math>\mu (\mathbf{v} \times \mathbf{H})</math> term from equation "[[#D|D]]" when he is deriving the electromagnetic wave equation, and he considers the situation only from the rest frame.
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| == Maxwell – electromagnetic light wave ==
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| [[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]].]]
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| 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".<ref name=ADTEF/> In part VI of his 1864 paper "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><ref>[http://www.ams.org/journals/bull/1915-21-06/S0002-9904-1915-02631-5/S0002-9904-1915-02631-5.pdf The Structure Of The Ether]. By Dr. H. Bateman. 1915-21-06.</ref><ref>[http://joseheras.com/documents/Electrodynamics/Bromberg.pdf Maxwell's Displacement Current and His Theory of Light]. Joan Bromberg. Communicated by S. G. Brush.</ref> 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,
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| {{quote|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.}}
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| 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.
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| ===Modern equation methods===
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| 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
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| <center>
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| {{Equation box 1|indent =:|equation =
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| <math>\nabla \cdot \mathbf{E} = 0</math>
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| <math> \nabla \times \mathbf{E} = -\mu_o \frac{\partial \mathbf{H}} {\partial t}</math>
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| <math> \nabla \cdot \mathbf{H} = 0</math>
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| <math> \nabla \times \mathbf{H} =\varepsilon_o \frac{ \partial \mathbf{E}} {\partial t}</math>
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| |cellpadding= 6|border = 0|border colour = black|background colour=white}}
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| </center>
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| If we take the [[Curl (mathematics)|curl]] of the curl equations we obtain
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| <center>
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| <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>
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| <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}
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| </math>
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| </center>
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| If we note the vector identity
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| <center>
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| <math>\nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V}</math>
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| </center>
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| where <math> \mathbf{V} </math> is any vector function of space, we recover the wave equations
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| <center>
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| <math> {\partial^2 \mathbf{E} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{E} \ \ = \ \ 0</math>
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| </center>
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| <center>
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| <math> {\partial^2 \mathbf{H} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{H} \ \ = \ \ 0</math>
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| </center>
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| where
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| <center>
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| <math>c = { 1 \over \sqrt{ \mu_o \varepsilon_o } } = 2.99792458 \times 10^8 </math> meters per second
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| </center>
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| is the speed of light in free space.
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| ==See also==
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| {{Wikipedia books|Maxwell's equations}}
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| * ''[[A Treatise on Electricity and Magnetism]]''
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| * ''[[On Physical Lines of Force]]''
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| ==References and notes==
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| ;General
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| {{Wikisource}}
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| ;Citations
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| <references/>
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| ==Further reading==
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| * {{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}}
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| * {{cite book |last=Niven |first=W. D. |title=The Scientific Papers of James Clerk Maxwell'' |publisher=Dover |location=New York |year=1952 |volume=Vol. 1}}
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| * {{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}}
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| *{{cite web |last=Tokunaga |first=Kiyohisa |url=http://www.d3.dion.ne.jp/~kiyohisa/tieca/251.htm |title=Part 2, Chapter V – Maxwell's Equations |work=Total Integral for Electromagnetic Canonical Action |year=2002 |accessdate=Sep 7, 2009}}
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| *{{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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| {{DEFAULTSORT:Dynamical Theory Of The Electromagnetic Field}}
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| [[Category:1860s in science]]
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| [[Category:Electromagnetism]]
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| [[Category:Physics papers]]
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| [[Category:Works by James Clerk Maxwell]]
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| [[Category:Maxwell's equations]]
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| [[Category:1865 works]]
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| [[Category:Works originally published in Philosophical Transactions of the Royal Society]]
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