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
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,
with μ0 the magnetic constant, defined in SI units as
- 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
- N / m.
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.
- 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,
- 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 In is relative to the orientation (for example, if 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:
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.