Ampère's circuital law

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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.

Original Ampère's circuital law

In its historically original form, Ampère's Circuital law relates the magnetic field Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}

to its source, the current density Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{J}

. 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.

Integral form

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S}

or equivalently,

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_{\mathrm{enc}}

where:

• Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_C

is the closed line integral around contour (closed curve) C.

• Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}

is the magnetic field in teslas.

• Failed to parse (Missing texvc executable; please see math/README to configure.): \cdot

is the vector dot product.

• Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}\mathbf{l}

is an infinitesimal element (differential) of the contour C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction equal to the direction of the contour C, see below),

• Failed to parse (Missing texvc executable; please see math/README to configure.): \iint_S

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.

• Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_0 \!\

is the magnetic constant.

• Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{J}

is the current density, both bound and free, through the surface S enclosed by contour C

• Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}\mathbf{S} \!\

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. The direction of the normal must correspond with the orientation of C by the right hand rule, see below for further discussion of the naked gopher),

• Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{enc}} \!\

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 warrant elaboration.

First, three of these terms are associated with sign ambiguities: the line integral Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_C

could go around the loop in either direction (clockwise or counterclockwise); the vector area Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}\mathbf{S}
could point in either of the two directions normal to the surface; and Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}\mathbf{S}

, 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.

Differential form

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\nabla} \times \!\

is the curl operator.

Corrected Ampère's circuital law: the Ampère-Maxwell equation

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + \epsilon_0 \mu_0 {\mathrm{d} \over \mathrm{d}t} \iint_S \mathbf{E} \cdot \mathrm{d} \mathbf{A}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \ \epsilon_0

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):

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J} + \epsilon_0 \mu_0 \frac{\partial \mathbf{E}}{\partial t}

Displacement current

Main article: Displacement current

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{J}_D=\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}

and then the equation is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S (\mathbf{J}+\mathbf{J}_D) \cdot \mathrm{d} \mathbf{A}

(integral form), or

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\nabla}\times \mathbf{B} = \mu_0(\mathbf{J}+\mathbf{J}_D)

(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.

Ampère's law in cgs units

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \frac{1}{c} \iint_S \left(4\pi\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t}\right) \cdot \mathrm{d}\mathbf{S}

where c is the speed of light.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\nabla} \times \mathbf{B} = \frac{1}{c}\left(4\pi\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t}\right)

See also

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

Further reading

• 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. 

External links

• 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.
• Ampere's Law on Project PHYSNET.bg:Закон на Ампер

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