The symbol was used by André-Marie Ampère, after whom the unit of electric current is named, in formulating the eponymous Ampère's force law which he discovered in 1820.

Using this approach to justify the electromotive force equation ( the precursor of the Lorentz force equation ), he derived a wave equation from a set of eight equations which appeared in the paper and which included the electromative force equation and Ampère's circuital law.

Further, Ampère derived both Ampère's force law describing the force between two currents and Ampère's law which, like the Biot – Savart law, correctly described the magnetic field generated by a steady current.

Later, Franz Ernst Neumann proved that, for a moving conductor in a magnetic field, induction is a consequence of Ampère's force law.

Although implicit in Ampère's force law the force due to a magnetic field on a moving electric charge was not correctly and explicitly stated until 1892 by Hendrik Lorentz who theoretically derived it from Maxwell's equations.

The ampere is defined using Ampère's force law ; the definition relies in part on the mass of the international prototype kilogram, a metal cylinder housed in France.

A magnetic field is generated by a feedback loop: current loops generate magnetic fields ( Ampère's circuital law ); a changing magnetic field generates an electric field ( Faraday's law ); and the electric and magnetic fields exert a force on the charges that are flowing in currents ( the Lorentz force ).

Another school of thought invokes Ampère's force law and asserts that it acts along the length of the rails ( which is their strongest axis ).

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 ".

** Ampère's force law relating the current through two nearby conductors to the force between them.

The magnetomotive force can often be quickly calculated using Ampère's law.

In order to establish the numerical value of ε < sub > 0 </ sub >, one makes use of the fact that if one uses the rationalized forms of Coulomb's law and Ampère's force law ( and other ideas ) to develop Maxwell's equations, then the relationship stated above is found to exist between ε < sub > 0 </ sub >, μ < sub > 0 </ sub > and c < sub > 0 </ sub >.

Pairs of parallel Birkeland currents will also interact due to Ampère's force law: parallel Birkeland currents moving in the same direction will attract each other with an electromagnetic force inversely proportional to their distance apart whilst parallel Birkeland currents moving in opposite directions will repel each other.

Ampère's force law --

* Of the four Maxwell's equations, two — Faraday's law and Ampère's law — can be compactly expressed using curl.

Faraday's law states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and rate of change of the electric field.

According to Ampère's circuital law | Ampère's law, an electric current produces a magnetic field.

In the electromagnetic cgs system, electric current is a fundamental quantity defined via Ampère's law and takes the permeability as a dimensionless quantity ( relative permeability ) whose value in a vacuum is unity.

Although Maxwell did not explicitly mention the sea of molecular vortices, his derivation of Ampère's circuital law was carried over from the 1861 paper and he used a dynamical approach involving rotational motion within the electromagnetic field which he likened to the action of flywheels.

) The other two equations describe how the fields ' circulate ' around their respective sources ; the magnetic field ' circulates ' around electric currents and time varying electric field in Ampère's law with Maxwell's correction, while the electric field ' circulates ' around time varying magnetic fields in Faraday's law.

An Wang's magnetic core memory ( 1954 ) is an application of Ampère's law.

Ampère's law with Maxwell's correction states that magnetic fields can be generated in two ways: by electrical current ( this was the original " Ampère's law ") and by changing electric fields ( this was " Maxwell's correction ").

Maxwell's correction to Ampère's law is particularly important: it shows that not only does a changing magnetic field induce an electric field, but also a changing electric field induces a magnetic field.

File: Andre-marie-ampere2. jpg | André-Marie Ampère ( 1777-1836 ): main founder of electrodynamics, showed how an electric current produces a magnetic field, stated that the mutual action of two lengths of current-carrying wire is proportional to their lengths and to the intensities of their currents ( Ampère's law ), namesake of the unit of electric current ( the ampere )

However, Ampère's law in its original form states:

Substituting this form for J into Ampère's law, and assuming there is no bound or free current density contributing to J:

Assuming each wire carries current I, and there are N wires per unit length, the magnetic field can be derived using Ampère's law:

A corollary of Faraday's Law, together with Ampère's law and Ohm's law is Lenz's law: The EMF induced in an electric circuit always acts in such a direction that the current it drives around the circuit opposes the change in magnetic flux which produces the EMF.

The term is the English version of A. M. Ampère's cinématique,

It is one of the four Maxwell's equations which form the basis of classical electrodynamics, the other three being Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction.

Gauss's law is something of an electrical analogue of Ampère's law, which deals with magnetism.

The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism.

By Ampère's law, we know that the line integral of B ( the magnetic flux density vector ) around this loop is zero, since it encloses no electrical currents ( it can be also assumed that the circuital electric field passing through the loop is constant under such conditions: a constant or constantly changing current through the solenoid ).

Now we can consider the imaginary loop b. Take the line integral of B around the loop of height l. The horizontal components vanish, and the field outside is practically zero, so Ampère's Law gives us:

Ampère's circuital law is now known to be a correct law of physics in a magnetostatic situation: The system is static except possibly for continuous steady currents within closed loops.

In SI units ( cgs units are later ), the " integral form " of the original Ampère's circuital law is a line integral of the magnetic field around some closed curve C ( arbitrary but must be closed ).

The result is that the more microscopic Ampère's law, expressed in terms of B and the microscopic current ( which includes free, magnetization and polarization currents ), is sometimes put into the equivalent form below in terms of H and the free current only.

They influenced French physicist André-Marie Ampère's developments of a single mathematical form to represent the magnetic forces between current-carrying conductors.

The explanation is that a displacement current I < sub > D </ sub > flows in the vacuum, and this current produces the magnetic field in the region between the plates according to Ampère's law:

Maxwell's derivation is unrelated to the modern day derivation for displacement current in the vacuum, which is based on consistency between Ampère's law for the magnetic field and the continuity equation for electric charge.

There are two important issues regarding Ampère's law that require closer scrutiny.

This version of the rule is used in two complementary applications of Ampère's circuital law: