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In physics, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is the magnetic analogue of Gauss's law, and one of the four Maxwell's equations that form the basis of classical electromagnetism.

Contents 1 Original Ampère's circuital law 1.1 Integral form 1.2 Differential form 2 Corrected Ampère's circuital law: the Ampère-Maxwell equation 2.1 Displacement current 3 Ampère's law in cgs units 4 See also 5 References 6 External links

In its historically original form, Ampère's Circuital law relates the magnetic field to its source, the current density . The equation is not in general correct (see "Maxwell's correction" below), but is correct in the special case where the electric field is constant (unchanging) in time.

The law can be written in two forms, the "integral form" and the "differential form". The forms are equivalent, and related by the Kelvin-Stokes theorem.

In SI units, (the version in cgs units is in a later section), the "integral form" of the original Ampère's Circuital law is:

or equivalently,

where:

• is the closed line integral around contour (closed curve) C.
• is the magnetic field in teslas.
• is the vector dot product.
• is an infinitesimal element (differential) of the contour C (i.e. a vector with magnitude equal to the length of the infinitesimal surface element, and direction equal to the direction of integration, see below),
• denotes an integral over the surface S enclosed by contour C (see below). The double integral sign is meant simply to denote that the integral is two-dimensional in nature.
• is the magnetic constant.
• is the current density (in amperes per square meter), both bound and free, through the surface S enclosed by contour C
• is the vector area of an infinitesimal element of surface S (i.e. a vector with magnitude equal to the area of the infinitesimal surface element, and direction normal to surface S),
• is the net current that penetrates through the surface S, both bound and free.

There are a number of ambiguities in the above definitions that warrent elaboration.

First, three of these terms are associated with sign ambiguities: the line integral could go around the loop in either direction (clockwise or counterclockwise); the vector area could point in either of the two directions normal 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 the right-hand rule: When the index-finger of the right-hand points along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area , and current passing in that same direction must be counted as positive. The right hand grip rule can also be used to determine the signs.

Second, there are infinitely many possible surfaces S that have the contour C as their border. (Imagine a soap film on a wire loop, which can be deformed by blowing gently at it.) 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.

By the Kelvin-Stokes theorem, this equation can also be written in a "differential form". Again, this equation only applies in the case where the electric field is constant in time; see below for the more general form. In SI units, the equation states:

where

is the curl operator.

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 On Physical Lines of Force.

The generalized law (in SI), as corrected by Maxwell, takes the following integral form:

where is the vacuum permittivity and E is the electric field.

This Ampère-Maxwell law can also be stated in differential form (with the Kelvin-Stokes theorem):

Main article: Displacement current

The displacement current is defined so as to make these equations more transparent. It is defined by

and then the equation is:

(integral form), or

(differential form).

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.

In cgs units, the integral form of the equation, including Maxwell's correction, reads

where c is the speed of light.

The differential form of the equation (again, including Maxwell's correction) is

• Maxwell's equations
• Biot-Savart law
• Faraday's law of induction
• Electric current
• Vector calculus
• Stokes' theorem

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 013805326X. 
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0716708108. 

• a section on Ampere's law from an online textbook
• MISN-0-138 Ampere's Law (PDF file) by Kirby Morgan for Project PHYSNET.
• MISN-0-145 The Ampere-Maxwell Equation; Displacement Current (PDF file) by J.S. Kovacs for Project PHYSNET.
• The Ampère's Law Song (PDF file) by Walter Fox Smith; Main page, with recordings of the song.