Ampère's force law

Two current-carrying wires attract each other magnetically: The bottom wire has current I1, which creates magnetic field B1. The top wire carries a current I2 through the magnetic field B1, so (by the Lorentz force) the wire experiences a force F12. (Not shown is the simultaneous process where the top wire makes a magnetic field which results in a force on the bottom wire.)

In magnetostatics, the force of attraction or repulsion between two current-carrying wires (see first figure below) is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field, following the Biot–Savart law, and the other wire experiences a magnetic force as a consequence, following the Lorentz force law.

Equation

Special case: Two straight parallel wires

The best-known and simplest example of Ampère's force law, which underlies the definition of the ampere, the SI unit of current, states that the force per unit length between two straight parallel conductors is

${\displaystyle {\frac {F_{m}}{L}}=2k_{A}{\frac {I_{1}I_{2}}{r}}}$,

where kA is the magnetic force constant from the Biot–Savart law, Fm/L is the total force on either wire per unit length of the shorter (the longer is approximated as infinitely long relative to the shorter), r is the distance between the two wires, and I1, I2 are the direct currents carried by the wires.

This is a good approximation if one wire is sufficiently longer than the other that it can be approximated as infinitely long, and if the distance between the wires is small compared to their lengths (so that the one infinite-wire approximation holds), but large compared to their diameters (so that they may also be approximated as infinitely thin lines). The value of kA depends upon the system of units chosen, and the value of kA decides how large the unit of current will be. In the SI system, [1] [2]

${\displaystyle k_{A}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\frac {\mu _{0}}{4\pi }}}$

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

${\displaystyle \mu _{0}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ 4\pi \times 10^{-7}}$ N / A2.

Thus, in vacuum,

the force per meter of length between two parallel conductors – spaced apart by 1 m and each carrying a current of 1  A – is exactly
${\displaystyle \displaystyle 2\times 10^{-7}}$ N / m.
Demonstration of Ampère's force law. The force acts between two parallel conductors with constant currents passing through them. If they have the same direction, there is force of attraction, but if the directions are opposite, then the force is repellent. Performed by Prof. Oliver Zajkov at the Physics Institute at the Ss. Cyril and Methodius University of Skopje, Macedonia.

General case

The general formulation of the magnetic force for arbitrary geometries is based on iterated line integrals and combines the Biot–Savart law and Lorentz force in one equation as shown below. [5] [6] [7]

${\displaystyle {\vec {F}}_{12}={\frac {\mu _{0}}{4\pi }}\int _{L_{1}}\int _{L_{2}}{\frac {I_{1}d{\vec {\ell }}_{1}\ \mathbf {\times } \ (I_{2}d{\vec {\ell }}_{2}\ \mathbf {\times } \ {\hat {\mathbf {r} }}_{21})}{|r|^{2}}}}$,

where

• ${\displaystyle {\vec {F}}_{12}}$ is the total magnetic force felt by wire 1 due to wire 2 (usually measured in newtons),
• I1 and I2 are the currents running through wires 1 and 2, respectively (usually measured in amperes),
• The double line integration sums the force upon each element of wire 1 due to the magnetic field of each element of wire 2,
• ${\displaystyle d{\vec {\ell }}_{1}}$ and ${\displaystyle d{\vec {\ell }}_{2}}$ are infinitesimal vectors associated with wire 1 and wire 2 respectively (usually measured in metres); see line integral for a detailed definition,
• The vector ${\displaystyle {\hat {\mathbf {r} }}_{21}}$ is the unit vector pointing from the differential element on wire 2 towards the differential element on wire 1, and |r| is the distance separating these elements,
• The multiplication × is a vector cross product,
• The sign of In is relative to the orientation ${\displaystyle d{\vec {\ell }}_{n}}$ (for example, if ${\displaystyle d{\vec {\ell }}_{1}}$ points in the direction of conventional current, then I1>0).

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

By expanding the vector triple product and applying Stokes' theorem, the law can be rewritten in the following equivalent way: [8]

${\displaystyle {\vec {F}}_{12}=-{\frac {\mu _{0}}{4\pi }}\int _{L_{1}}\int _{L_{2}}{\frac {(I_{1}d{\vec {\ell }}_{1}\ \mathbf {\cdot } \ I_{2}d{\vec {\ell }}_{2})\ {\hat {\mathbf {r} }}_{21}}{|r|^{2}}}.}$

In this form, it is immediately obvious that the force on wire 1 due to wire 2 is equal and opposite the force on wire 2 due to wire 1, in accordance with Newton's 3rd law.

Other Languages
azərbaycanca: Amper qanunu
беларуская: Сіла Ампера
français: Force d'Ampère
Հայերեն: Ամպերի օրենք
latviešu: Ampēra spēks
македонски: Амперова сила
oʻzbekcha/ўзбекча: Amper qonuni
русский: Закон Ампера
татарча/tatarça: Ампер кануны
українська: Закон Ампера