# Ampère's force law

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Ampère's force law

The force of attraction or repulsion between two current-carrying wires (see Figure 1) is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field (according to the Biot-Savart law), and the other wire experiences a Lorentz force as a consequence.

The best-known and simplest example of Ampère's force law, which underlies the definition of the ampere, the SI unit of current, is as follows: For two thin, straight, stationary, parallel wires, the force per unit length one wire exerts upon the other in the vacuum of free space is

::$F_m = k_m frac \left\{I_1 I_2 \right\} \left\{r\right\}$,

where "k"m is the magnetic force constant, "r" is the separation of the wires, and "I"1, "I"2 are the DC currents carried by the wires. The value of "k"m depends upon the system of units chosen, and the value of "k"m decides how large the unit of current will be. In the SI system,cite book
author=Raymond A Serway & Jewett JW
title=Serway's principles of physics: a calculus based text
url=http://books.google.com/books?id=1DZz341Pp50C&pg=RA1-PA746&dq=wire+%22magnetic+force%22&lr=&as_brr=0&sig=4vMV_CH6Nm8ZkgjtDJFlupekYoA#PRA1-PA746,M1 |publisher=Thompson Brooks/Cole
edition=Fourth Edition
location=Belmont, CA
year=2006
page=p. 746
isbn=053449143X
] cite book
author=Paul M. S. Monk
title=Physical chemistry: understanding our chemical world
url=http://books.google.com/books?vid=ISBN0471491802&id=LupAi35QjhoC&pg=PA16&lpg=PA16&ots=IMiGyIL-67&dq=ampere+definition+si&sig=9Y0k0wgvymmLNYFMcXodwJZwvAM |publisher=Chichester: Wiley
location=New York
year=2004
page=p. 16
isbn=0471491810
]

::$k_m overset\left\{underset\left\{mathrm\left\{def\left\{\left\{=\right\} frac \left\{mu_0\right\}\left\{ 2 pi\right\}$

with μ0 the magnetic constant, "defined" in SI units as [ [http://www.bipm.org/en/si/si_brochure/chapter2/2-1/ampere.html "BIPM definition"] ] cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?mu0 |title=Magnetic constant |accessdate=2007-08-08 |work=2006 CODATA recommended values |publisher=NIST ]

::$mu_0 overset\left\{underset\left\{mathrm\left\{def\left\{\left\{=\right\} 4 pi imes 10^\left\{-7\right\}$ newtons / (ampere)2.

Thus, for two parallel wires carrying a current of 1 A, and spaced apart by 1 m in vacuum, [By "vacuum" is meant the unattainable vacuum of free space used as a reference state in electromagnetic theory.] the force on each wire per unit length is exactly 2 × 10-7 N/m.

A more general formulation of Ampère's force law for arbitrary geometries is based upon line integrals, and is as follows [The integrand of this expression appears in the official documentation regarding definition of the ampere [http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf BIPM SI Units brochure, 8th Edition, p. 105] ] cite book
author=Tai L. Chow
title=Introduction to electromagnetic theory: a modern perspective
url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&lpg=PA153&dq=%22ampere's+law+of+force%22&source=web&ots=uZOFz9dWv7&sig=NJp3UQvbCOvcVm7eJN4IUdlC9bs |publisher=Jones and Bartlett
location=Boston
year=2006
page=p. 153
isbn=0763738271
] [ [http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/ampereForce.html Ampère's Force Law] "Includes animated graphic of the force vectors. Scroll to bottom for formulas"] :

::$mathbf\left\{F\right\}_\left\{12\right\} = frac \left\{mu_0\right\} \left\{4 pi\right\} I_1 I_2 oint_\left\{C_1\right\} oint_\left\{C_2\right\} frac \left\{d mathbf\left\{s_2\right\} mathbf\left\{ imes\right\} \left(d mathbf\left\{s_1\right\} mathbf\left\{ imes \right\} hat\left\{mathbf\left\{r_\left\{12\right\} \right)\right\} \left\{r_\left\{12\right\}^2\right\}$,

where:F12 is the total force on circuit 2 exerted by circuit 1 (usually measured in newtons),:"I"1 and "I"2 are the currents running through circuits 1 and 2, respectively (usually measured in amperes),:The double line integration sums the force upon each element of circuit 2 due to each element of circuit 1,:"d"s1 and "d"s2 are infinitesimal vector elements of the paths "C"1 and "C"2, respectively, with the same direction as the conventional current (usually measured in metres),:The vector $hat\left\{mathbf\left\{r_\left\{12\right\}$ is a vector of unit length along the line connecting the element pair [from s1 to s2] , and "r"12 is the distance separating these elements,:The multiplication × is a vector cross product.

To determine the force between wires in a material medium, the magnetic constant is replaced by the actual permeability of the medium.

References and notes

ee also

* Ampere
* Magnetic constant
* Lorentz force
* Ampère's circuital law
* Free space

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