Ampère's circuital law
"Ampère's law" redirects here. For the law describing forces between current-
carrying wires, see Ampère's force law.
In classical electromagnetism, Ampère's circuital law (not to be confused
with Ampère's force law) relates the integrated magnetic field around a closed
[1]
loop to the electric current passing through the loop. James Clerk
Maxwell (not Ampère) derived it using hydrodynamics in his 1861 published
paper "On Physical Lines of Force" In 1865 he generalized the equation to
[2]
apply to time-varying currents by adding the displacement current term,
resulting in the modern form of the law, sometimes called the Ampère–Maxwell
law, which is one of Maxwell's equations which form the basis
[3][4][5]
of classical electromagnetism.
Contents
Maxwell's original circuital law[edit]
In 1820 Danish physicist Hans Christian Ørsted discovered that an electric
current creates a magnetic field around it, when he noticed that the needle of
a compass next to a wire carrying current turned so that the needle was
perpendicular to the wire. He investigated and discovered the rules which
[6][7]
govern the field around a straight current-carrying wire: [8]
The magnetic field lines encircle the current-carrying wire.
The magnetic field lines lie in a plane perpendicular to the wire.
If the direction of the current is reversed, the direction of the magnetic field
reverses.
The strength of the field is directly proportional to the magnitude of the
current.
The strength of the field at any point is inversely proportional to the distance
of the point from the wire.
This sparked a great deal of research into the relation between electricity and
magnetism. André-Marie Ampère investigated the magnetic force between two
current-carrying wires, discovering Ampère's force law. In the 1850s Scottish
mathematical physicist James Clerk Maxwell generalized these results and
others into a single mathematical law. The original form of Maxwell's circuital
law, which he derived as early as 1855 in his paper "On Faraday's Lines of
Force" based on an analogy to hydrodynamics, relates magnetic
[9]
fields to electric currents that produce them. It determines the magnetic field
, associated with a given current, or the current associated with a given
magnetic field.
The original circuital law only applies to a magnetostatic situation, to
continuous steady currents flowing in a closed circuit. For systems with
electric fields that change over time, the original law (as given in this section)
must be modified to include a term known as Maxwell's correction (see below).
Equivalent forms[edit]
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 using cgs units. Other units are possible, but
rare. This section will use SI units, with cgs units discussed later.
Forms using either B or H magnetic fields. These two forms use the total
current density and free current density, respectively. The B and H fields are
related by the constitutive equation: B = μ H in non-magnetic materials 0
where μ is the magnetic constant.
0
Explanation[edit]
The integral form of the original circuital law is a line integral of the magnetic
field around some closed curve C (arbitrary but must be closed). The curve C in
turn bounds both a surface S which the electric current passes through (again
arbitrary but not closed—since no three-dimensional volume is enclosed by S),
and encloses the current. The mathematical statement of the law is a relation
between the total amount of magnetic field around some path (line integral)
due to the current which passes through that enclosed path (surface integral).
[10][11]
In terms of total current, (which is the sum of both free current and bound
current) the line integral of the magnetic B-field (in teslas, T) around closed
curve C is proportional to the total current I passing through a enc
surface S (enclosed by C). In terms of free current, the line integral of
the magnetic H-field (in amperes per metre, A·m ) around closed −1
curve C equals the free current I through a surface S. f,enc
[clarification needed]
Forms of the original circuital law written in SI units
Integral form Differential form
Using B-field and total current
Using H-field and free current
"Ampère's law" redirects here. For the law describing forces between current-
carrying wires, see Ampère's force law.
In classical electromagnetism, Ampère's circuital law (not to be confused
with Ampère's force law) relates the integrated magnetic field around a closed
[1]
loop to the electric current passing through the loop. James Clerk
Maxwell (not Ampère) derived it using hydrodynamics in his 1861 published
paper "On Physical Lines of Force" In 1865 he generalized the equation to
[2]
apply to time-varying currents by adding the displacement current term,
resulting in the modern form of the law, sometimes called the Ampère–Maxwell
law, which is one of Maxwell's equations which form the basis
[3][4][5]
of classical electromagnetism.
Contents
Maxwell's original circuital law[edit]
In 1820 Danish physicist Hans Christian Ørsted discovered that an electric
current creates a magnetic field around it, when he noticed that the needle of
a compass next to a wire carrying current turned so that the needle was
perpendicular to the wire. He investigated and discovered the rules which
[6][7]
govern the field around a straight current-carrying wire: [8]
The magnetic field lines encircle the current-carrying wire.
The magnetic field lines lie in a plane perpendicular to the wire.
If the direction of the current is reversed, the direction of the magnetic field
reverses.
The strength of the field is directly proportional to the magnitude of the
current.
The strength of the field at any point is inversely proportional to the distance
of the point from the wire.
This sparked a great deal of research into the relation between electricity and
magnetism. André-Marie Ampère investigated the magnetic force between two
current-carrying wires, discovering Ampère's force law. In the 1850s Scottish
mathematical physicist James Clerk Maxwell generalized these results and
others into a single mathematical law. The original form of Maxwell's circuital
law, which he derived as early as 1855 in his paper "On Faraday's Lines of
Force" based on an analogy to hydrodynamics, relates magnetic
[9]
fields to electric currents that produce them. It determines the magnetic field
, associated with a given current, or the current associated with a given
magnetic field.
The original circuital law only applies to a magnetostatic situation, to
continuous steady currents flowing in a closed circuit. For systems with
electric fields that change over time, the original law (as given in this section)
must be modified to include a term known as Maxwell's correction (see below).
Equivalent forms[edit]
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 using cgs units. Other units are possible, but
rare. This section will use SI units, with cgs units discussed later.
Forms using either B or H magnetic fields. These two forms use the total
current density and free current density, respectively. The B and H fields are
related by the constitutive equation: B = μ H in non-magnetic materials 0
where μ is the magnetic constant.
0
Explanation[edit]
The integral form of the original circuital law is a line integral of the magnetic
field around some closed curve C (arbitrary but must be closed). The curve C in
turn bounds both a surface S which the electric current passes through (again
arbitrary but not closed—since no three-dimensional volume is enclosed by S),
and encloses the current. The mathematical statement of the law is a relation
between the total amount of magnetic field around some path (line integral)
due to the current which passes through that enclosed path (surface integral).
[10][11]
In terms of total current, (which is the sum of both free current and bound
current) the line integral of the magnetic B-field (in teslas, T) around closed
curve C is proportional to the total current I passing through a enc
surface S (enclosed by C). In terms of free current, the line integral of
the magnetic H-field (in amperes per metre, A·m ) around closed −1
curve C equals the free current I through a surface S. f,enc
[clarification needed]
Forms of the original circuital law written in SI units
Integral form Differential form
Using B-field and total current
Using H-field and free current
