Ampère's force law

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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 I1 exerts force F12 on Circuit 2 via its B-field B1, and conversely
Figure 1: Circuit 1 with current I1 exerts force F12 on Circuit 2 via its B-field B1, and conversely

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 km is the magnetic force constant, r is the separation of the wires, and I1, I2 are the DC currents carried by the wires. The value of km depends upon the system of units chosen, and the value of km decides how large the unit of current will be. In the SI system,[1] [2]

k_m \ \overset{\underset{\mathrm{def}}{}}{=}\ \frac {\mu_0}{ 2 \pi} \

with μ0 the magnetic constant, defined in SI units as[3][4]

\mu_0 \ \overset{\underset{\mathrm{def}}{}}{=}\ 4 \pi \times 10^{-7} \ 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]:

\mathbf{F}_{12} = \frac {\mu_0} {4 \pi} I_1 I_2 \oint_{C_1} \oint_{C_2} \frac {d \mathbf{s_2}\ \mathbf{ \times} \ (d \mathbf{s_1} \ \mathbf{ \times } \ \hat{\mathbf{r}}_{12} )} {r_{12}^2} \ ,

where

F12 is the total force on circuit 2 exerted by circuit 1 (usually measured in newtons),
I1 and I2 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,
ds1 and ds2 are infinitesimal vector elements of the paths C1 and C2, 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 r12 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.

[edit] References and notes

  1. ^ Raymond A Serway & Jewett JW (2006). Serway's principles of physics: a calculus based text, Fourth Edition, Belmont, CA: Thompson Brooks/Cole, p. 746. ISBN 053449143X. 
  2. ^ Paul M. S. Monk (2004). Physical chemistry: understanding our chemical world. New York: Chichester: Wiley, p. 16. ISBN 0471491810. 
  3. ^ BIPM definition
  4. ^ Magnetic constant. 2006 CODATA recommended values. NIST. Retrieved on 2007-08-08.
  5. ^ By vacuum is meant the unattainable vacuum of free space used as a reference state in electromagnetic theory.
  6. ^ The integrand of this expression appears in the official documentation regarding definition of the ampere BIPM SI Units brochure, 8th Edition, p. 105
  7. ^ Tai L. Chow (2006). Introduction to electromagnetic theory: a modern perspective. Boston: Jones and Bartlett, p. 153. ISBN 0763738271. 
  8. ^ Ampère's Force Law Includes animated graphic of the force vectors. Scroll to bottom for formulas

[edit] See also