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

::$F\_m\; =\; k\_m\; frac\; \{I\_1\; I\_2\; \}\; \{r\}$,

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]

::$k\_m\; overset\{underset\{mathrm\{def\{\{=\}\; frac\; \{mu\_0\}\{\; 2\; pi\}$

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 ]

::$mu\_0\; overset\{underset\{mathrm\{def\{\{=\}\; 4\; pi\; imes\; 10^\{-7\}$ 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, 8^th 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"] :

::$mathbf\{F\}\_\{12\}\; =\; frac\; \{mu\_0\}\; \{4\; pi\}\; I\_1\; I\_2\; oint\_\{C\_1\}\; oint\_\{C\_2\}\; frac\; \{d\; mathbf\{s\_2\}\; mathbf\{\; imes\}\; (d\; mathbf\{s\_1\}\; mathbf\{\; imes\; \}\; hat\{mathbf\{r\_\{12\}\; )\}\; \{r\_\{12\}^2\}$,

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 $hat\{mathbf\{r\_\{12\}$ 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