From Wikipedia, the free encyclopedia
This article is about Ampère's force law. For
the law relating the integrated magnetic field around a closed loop
to the electric current passing through the loop, see
Ampère's circuital law.
Figure 1: Circuit 1 with current
I_{1} exerts
force F
_{12} on Circuit 2
via its
Bfield
B
_{1}, and conversely
In magnetostatics, the force of attraction
or repulsion between two currentcarrying 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 BiotSavart law),
and the other wire experiences a Lorentz force as a consequence.
The bestknown 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, infinitely long, stationary, parallel wires, the
force per unit length one wire exerts upon the other in the vacuum
of free space is

 ,
where k_{A} 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. This is a good approximation for finite
lengths if the distance between the wires is small compared to
their lengths. The value of k_{A} depends upon the
system of units chosen, and the value of k_{A}
decides how large the unit of current will be. In the SI system,^{[1]}
^{[2]}

with μ_{0} the magnetic
constant, defined in SI units as^{[3]}^{[4]}


newtons
/ (ampere)^{2}.
Thus, for two parallel wires carrying a current of 1 A, and
spaced apart by 1 m in vacuum,^{[5]} 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 ^{[6]} ^{[7]}^{[8]}:

 ,
where
 F_{12} 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,
 ds_{1} and
ds_{2} 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
is a vector of unit length along the line connecting the element
pair [from s_{1} to
s_{2}], 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
 ^ Raymond A Serway & Jewett JW (2006). Serway's principles of
physics: a calculus based text (Fourth Edition ed.).
Belmont, CA: Thompson Brooks/Cole. p. 746. ISBN 053449143X. http://books.google.com/books?id=1DZz341Pp50C&pg=RA1PA746&dq=wire+%22magnetic+force%22&lr=&as_brr=0&sig=4vMV_CH6Nm8ZkgjtDJFlupekYoA#PRA1PA746,M1.
 ^ Paul M. S. Monk (2004). Physical chemistry:
understanding our chemical world. New York: Chichester:
Wiley. p. 16. ISBN 0471491810. http://books.google.com/books?vid=ISBN0471491802&id=LupAi35QjhoC&pg=PA16&lpg=PA16&ots=IMiGyIL67&dq=ampere+definition+si&sig=9Y0k0wgvymmLNYFMcXodwJZwvAM.
 ^
BIPM
definition
 ^ "Magnetic constant".
2006 CODATA recommended values. NIST. http://physics.nist.gov/cgibin/cuu/Value?mu0. Retrieved
20070808.
 ^
By vacuum is meant the unattainable vacuum of free space used as a
reference state in electromagnetic theory.
 ^
The integrand of this expression appears in the official
documentation regarding definition of the ampere BIPM SI Units brochure,
8^{th} Edition, p. 105
 ^ Tai L. Chow (2006). Introduction to
electromagnetic theory: a modern perspective. Boston:
Jones and Bartlett. p. 153. ISBN 0763738271. http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&lpg=PA153&dq=%22ampere's+law+of+force%22&source=web&ots=uZOFz9dWv7&sig=NJp3UQvbCOvcVm7eJN4IUdlC9bs.
 ^
Ampère's Force Law
Scroll to section "Integral Equation" for formula.
Further
reading
See also