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Ampère's circuital law
From Wikipedia, the free encyclopedia
Ampère's law redirects here. For the law describing forces between current-carrying wires, see Ampère's force law.
Electromagnetism
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Ampère's circuital law
From Wikipedia, the free encyclopedia
Ampère's law redirects here. For the law describing forces between current-carrying wires,see Ampère's force law .
Electromagnetism
Electricity
·
Magnetism
[show]
Electrostatics
[hide]
Magnetostatics
Ampère's law
·
Electric current
·
Magnetic
field
·
Magnetization
·
Magnetic flux
·
Biot
–
Savart law
·
Magnetic dipole
moment
·
Gauss's law for magnetism
[show]
Electrodynamics
[show]
Electrical Network
[show]
Covariant formulation
[show]
Scientists
v
·
d
·
e
In classical electromagnetism,
Ampère's circuital law
, discovered by André-MarieAmpère in 1826,
[1]
relates the integrated magnetic field around a closed loop to the electric
current passing through the loop. James Clerk Maxwell derived it again
using hydrodynamics in his 1861 paper
On Physical Lines of Force
and it is now one ofthe Maxwell equations,which form the basis of classical electromagnetism.
Contents
[hide]
1 Original Ampère's circuital law
o
1.1 Integral form
o
1.2 Differential form
2 Note on free current versus bound current 3 Shortcomings of the srcinal formulation of Ampère's circuital law
o
3.1 Displacement current 4 Extending the srcinal law: the Maxwell
–
Ampère equation
o
4.1 Proof of equivalence 5 Ampère's law in cgs units 6 See also 7 Notes 8 Further reading 9 External links
[edit]
Original Ampère's circuital law
An electric current produces a magnetic field.
It relates magnetic fields to electric currents that produce them. Using Ampere's law, youcan determine the magnetic field associated with a given current or current associated witha given magnetic field, providing there is no time changing electric field present. In itshistorically srcinal form, Ampère's Circuital Law relates the magnetic field to its electriccurrent source. The law can be written in two forms, the integral form and the differentialform . The forms are equivalent, and related by the Kelvin
–
Stokes theorem.It can also bewritten in terms of either the
B
or
H
magnetic fields.Again, the two forms are equivalent(see the proof section below).
Ampère's circuital law is now known to be a correct law of physics ina magnetostatic situation: The system is static except possibly for continuous steady
currents within closed loops. In all other cases the law is incorrect unless Maxwell'scorrection is included (see below).
[edit]
Integral form
In SI units (the version in cgs units is in a later section), the integral form of the srcinal
Ampère's circuital law is:
[2][3]
or equivalently,whereis the closed line integral around the closed curve
C
;
B
is the magnetic B-field in teslas;
H
is the magnetic H-field in ampere per metre;
· is the vector dot product;
d
ℓ
is an infinitesimal element (a differential)of the curve
C
(i.e. a vector withmagnitude equal to the length of the infinitesimal line element, and direction given bythe tangent to the curve
C
, see below);denotes an integral over the surface
S
enclosed by the curve
C
(see below; thedouble integral sign is meant simply to denote that the integral is two-dimensional innature);
μ
0
is the magnetic constant;
J
f
is the free current density through the surface
S
enclosed by the curve
C
(seebelow);
J
is the total current density through the surface
S
enclosed by the curve
C
,including both free and bound current (see below);d
S
is the vector area of an infinitesimal element of surface
S
(that is, a vector withmagnitude equal to the area of the infinitesimal surface element, and direction
normal to surface
S
. The direction of the normal must correspond with the orientationof
C
by the right hand rule, see below for further discussion);
I
f,enc
is the net free current that penetrates through the surface
S
(see below);
I
enc
is the total net current that penetrates through the surface
S
, including both freeand bound current (see below).There are anumber ofambiguities inthe abovedefinitions thatwarrantelaboration.First, three ofthese terms areassociated withsignambiguities: theline integralcould goaround the loopin eitherdirection(clockwise orcounterclockwise); the vectorarea d
S
couldpoint in either ofthe twodirections normal to thesurface;and
I
enc
is thenet currentpassing throughthe surface
S
,

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