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{{electromagnetism}}
In physics, '''Ampère's Circuital law''', discovered by [[André-Marie Ampère]], relates the circulating [[magnetic field]] in a closed loop to the [[electric current]] passing through the loop. It is the magnetic equivalent of [[Faraday's law of induction]].
==Original Ampère's Circuital law==
[[Image:Electromagnetism.svg|thumb|left|An electric current produces a magnetic field.]]
In its original form, Ampère's Circuital law relates the magnetic field <math>\vec{H}</math> to its source, the current density <math>\vec{J}</math>:
:<math>\oint_C \vec{H} \cdot \mathrm{d}\vec{l} = \int\!\!\!\!\int_S \vec{J} \cdot \mathrm{d}\vec{S} = I_{\mathrm{enc}} </math>
where
:<math>\oint_C </math> is the closed [[line integral]] around contour (closed curve) ''C''.
:<math>\vec{H} </math> is the [[magnetic field]] in [[amperes]] per metre.
:<math>\mathrm{d}\vec{l} </math> is an [[infinitesimal]] element ([[differential (mathematics)|differential]]) of the contour ''C'',
:<math>\vec{J} </math> is the [[current density]] (in amperes per square meter) through the surface ''S'' enclosed by contour ''C''
:<math> \mathrm{d}\vec{S} \!\ </math> is a differential [[vector area]] element of surface ''S'', with infinitesimally small magnitude and direction normal to surface ''S'',
:<math>I_{\mathrm{enc}} \!\ </math> is the current enclosed by the curve ''C'', or strictly, the current that penetrates surface ''S''.
Equivalently, the original equation in differential form is
:<math>\vec{\nabla} \times \vec{H} = \vec{J} </math>
where
:<math>\vec{\nabla} \times \!\ </math> is the [[curl]] operator.
The [[magnetic field]] <math>\vec{H}</math> in linear media, is related to the [[magnetic flux density]] <math>\vec{B}</math> (in [[Tesla (unit)|tesla]]s) by
:<math> \vec{B} \ = \ \mu \vec{H} </math>
where <math>\mu \!\ </math> is the [[Permeability (electromagnetism)|permeability]] of the medium (in [[henries]] per meter), which by definition is <math>4 \pi \times 10^{-7}</math> in free space. In non-linear media, <math>\mu \!\ </math> is a rank-2 tensor.
==Corrected Ampère's Circuital law: the Ampère-Maxwell equation==
[[James Clerk Maxwell]] conceived of [[displacement current]] as a polarization current in the dielectric vortex sea which he used to model the magnetic field hydrodynamically and mechanically. He added this [[displacement current]] to Ampère's Circuital law at equation (112) in his 1861 paper [http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf On Physical Lines of Force].
The generalized law, as corrected by Maxwell, takes the following integral form:
:<math>\oint_C \vec{H} \cdot \mathrm{d}\vec{l} = \iint_S \vec{J} \cdot \mathrm{d} \vec{A} +
{\mathrm{d} \over \mathrm{d}t} \iint_S \vec{D} \cdot \mathrm{d} \vec{A}</math>
where in linear media
:<math> \vec{D} \ = \ \varepsilon \vec{E}</math>
is the displacement flux density (in coulombs per square meter).
This ''Ampère-Maxwell law'' can also be stated in differential form:
:<math>\vec{\nabla} \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}</math>
where the second term arises from the displacement current.
With the addition of the displacement current, Maxwell was able to postulate (correctly) that light was a form of [[electromagnetic wave]]. See [[Electromagnetic wave equation]] for a discussion on this important discovery.
==See also==
* [[Maxwell's equations]]
* [[Biot-Savart law]]
* [[Faraday's law of induction]]
* [[Gauss's law]]
* [[Electric current]]
* [[Vector calculus]]
* [[Stokes theorem|Stokes' theorem]]
==References==
* {{cite book | author=Griffiths, David J.|title=Introduction to Electrodynamics (3rd ed.)| publisher=Prentice Hall |year=1998 |id=ISBN 013805326X}}
* {{cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.) | publisher=W. H. Freeman | year=2004 | id=ISBN 0716708108}}
==External links==
* [http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html#Section11.3 a section on Ampere's law from an online textbook]
* [http://35.9.69.219/home/modules/pdf_modules/m138.pdf <small>MISN-0-138</small> ''Ampere's Law''] ([[Portable Document Format|PDF file]]) by Kirby Morgan for [http://www.physnet.net Project PHYSNET].
* [http://35.9.69.219/home/modules/pdf_modules/m145.pdf <small>MISN-0-145</small> ''The Ampere-Maxwell Equation; Displacement Current''] (PDF file) by J.S. Kovacs for Project PHYSNET.
* [http://www.haverford.edu/physics-astro/songs/ampere.PDF ''The Ampère's Law Song''] (PDF file) by Walter Fox Smith; [http://www.haverford.edu/physics-astro/songs/ Main page], with recordings of the song.
[[Category:Electrostatics|Ampere's law]]
[[Category:Magnetostatics|Ampere's law]]
[[Category:Introductory physics|Ampere's law]]
[[Category:Fundamental physics concepts]]
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