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Concept
Ampère's force law
Summary
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.
The best-known and simplest example of Ampère's force law, which underlaid (before 20 May 2019) the definition of the ampere, the SI unit of current, states that the magnetic force per unit length between two straight parallel conductors is
where is the magnetic force constant from the Biot–Savart law, is the total force on either wire per unit length of the shorter (the longer is approximated as infinitely long relative to the shorter), is the distance between the two wires, and , are the direct currents carried by the wires.
This is a good approximation if one wire is sufficiently longer than the other, so 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 depends upon the system of units chosen, and the value of decides how large the unit of current will be.
In the SI system,
with the magnetic constant, in SI units
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.
where
is the total magnetic force felt by wire 1 due to wire 2 (usually measured in newtons),
and 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,
and are infinitesimal vectors associated with wire 1 and wire 2 respectively (usually measured in metres); see line integral for a detailed definition,
The vector 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 is relative to the orientation (for example, if points in the direction of conventional current, then ).
Official source
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