## Ampère's circuital law |

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In **Ampère's circuital law** (not to be confused with ^{[1]}) relates the ^{[2]} and it is now one of the

- maxwell's original circuital law
- free current versus bound current
- shortcomings of the original formulation of the circuital law
- extending the original law: the maxwell–ampère equation
- ampère's circuital law in cgs units
- see also
- notes
- further reading
- external links

The original form of Maxwell's circuital law, which he derived in his 1855 paper "On Faraday's Lines of Force"^{[3]} based on an analogy to hydrodynamics, relates

The original circuital law is only a correct law of physics in a

The original circuital law can be written in several different forms, which are all ultimately equivalent:

- An "integral form" and a "differential form". The forms are exactly equivalent, and related by the
Kelvin–Stokes theorem .(see the "proof" section below) - Forms using
SI units , and those usingcgs units . Other units are possible, but rare. This section will use SI units, with cgs units discussed later. - Forms using either
. These two forms use the total current density and free current density, respectively. The**H**magnetic fields**B**and**H**fields are related by theconstitutive equation :**B**=*μ*_{0}**H**where*μ*_{0}is themagnetic constant .

The integral form of the original circuital law is a ^{[4]}^{[5]}

In terms of total **B**-field*I*_{enc} passing through a surface S (enclosed by C). In terms of free current, the line integral of the **H**-field^{−1}) around closed curve C equals the free current *I*_{f,enc} through a surface S.

Forms of the original circuital law written in SI units Integral formDifferential formUsing **B**-field and total currentUsing **H**-field and free current

**J**is the totalcurrent density (inamperes per squaremetre , A·m^{−2}),**J**_{f}is the free current density only,- ∮
_{C}is the closedline integral around the closed curve C, - ∬
_{S}denotes a 2-Dsurface integral over S enclosed by C, - · is the vector
dot product , - d
is an**l**infinitesimal element (adifferential ) of the curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C) - d
**S**is thevector area of aninfinitesimal element of surface S (that is, a vector with magnitude equal to the area of the infinitesimal surface element, and direction normal to surface S. The direction of the normal must correspond with the orientation of C by the right hand rule), see below for further explanation of the curve C and surface S. - ∇ × is the
curl operator.

There are a number of ambiguities in the above definitions that require clarification and a choice of convention.

- First, three of these terms are associated with sign ambiguities: the line integral ∮
_{C}could go around the loop in either direction (clockwise or counterclockwise); the vector area d**S**could point in either of the two directionsnormal to the surface; and*I*_{enc}is the net current passing through the surface S, meaning the current passing through in one direction, minus the current in the other direction—but either direction could be chosen as positive. These ambiguities are resolved by theright-hand rule : With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area d**S**. Also the current passing in the same direction as d**S**must be counted as positive. Theright hand grip rule can also be used to determine the signs. - Second, there are infinitely many possible surfaces S that have the curve C as their border. (Imagine a soap film on a wire loop, which can be deformed by moving the wire). Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter; it can be proven that any surface with boundary C can be chosen.

Other Languages

العربية: قانون أمبير

asturianu: Llei d'Ampère

বাংলা: অম্পেয়্যারের বর্তনী সূত্র

беларуская: Тэарэма аб цыркуляцыі магнітнага поля

български: Закон на Ампер

català: Llei d'Ampère

čeština: Ampérův zákon

Deutsch: Ampèresches Gesetz

eesti: Ampère'i seadus

Ελληνικά: Νόμος του Αμπέρ

español: Ley de Ampère

Esperanto: Ampera cirkvita leĝo

فارسی: قانون آمپر

français: Théorème d'Ampère

한국어: 앙페르 회로 법칙

हिन्दी: एम्पीयर का नियम

hrvatski: Ampèreov zakon

íslenska: Lögmál Amperes

italiano: Legge di Ampère

עברית: חוק אמפר

ქართული: ამპერის კანონი

қазақша: Ампер заңы

latviešu: Magnētiskās indukcijas cirkulācija

lietuvių: Ampero dėsnis

magyar: Ampère-törvény

македонски: Амперов закон

Nederlands: Wet van Ampère

नेपाली: एम्पीयरको नियम

日本語: アンペールの法則

norsk: Ampères sirkulasjonslov

ភាសាខ្មែរ: ច្បាប់អំពែរ

polski: Prawo Ampère’a

português: Lei de Ampère

русский: Теорема о циркуляции магнитного поля

Scots: Ampère's circuital law

shqip: Ligji i Amperit

Simple English: Ampère's circuital law

slovenčina: Ampérov zákon

српски / srpski: Амперов закон

srpskohrvatski / српскохрватски: Ampèreov zakon

suomi: Ampèren laki

svenska: Ampères lag

Tagalog: Batas na pang-sirkito ni Ampère

தமிழ்: ஆம்ப்பியர் விதி

Türkçe: Ampère yasası

українська: Закон Ампера для циркуляції магнітного поля

اردو: قانون ایمپیئر

Tiếng Việt: Định luật Ampère

吴语: 安培定律

中文: 安培定律