Ampère's force law
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[1]) the definition of the ampere, the SI unit of electric current, states that the magnetic force per unit length between two straight parallel conductors is
is the total force on either wire per unit length of the shorter (the longer is approximated as infinitely long relative to the shorter),
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).
depends upon the system of units chosen, and the value of
decides how large the unit of current will be.
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 To determine the force between wires in a material medium, the magnetic constant is replaced by the actual permeability of the medium.
For the case of two separate closed wires, the law can be rewritten in the following equivalent way by expanding the vector triple product and applying Stokes' theorem:[7]
The form of Ampere's force law commonly given was derived by James Clerk Maxwell in 1873 and is one of several expressions consistent with the original experiments of André-Marie Ampère and Carl Friedrich Gauss.
The x-component of the force between two linear currents I and I', as depicted in the adjacent diagram, was given by Ampère in 1825 and Gauss in 1833 as follows:[8]
{\displaystyle dF_{x}=kII'ds'\int ds{\frac {\cos(xds)\cos(rds')-\cos(rx)\cos(dsds')}{r^{2}}}.}
Following Ampère, a number of scientists, including Wilhelm Weber, Rudolf Clausius, Maxwell, Bernhard Riemann, Hermann Grassmann,[9] and Walther Ritz, developed this expression to find a fundamental expression of the force.
As Maxwell noted, terms can be added to this expression, which are derivatives of a function Q(r) and, when integrated, cancel each other out.
Thus, Maxwell gave "the most general form consistent with the experimental facts" for the force on ds arising from the action of ds':[10]
Q is a function of r, according to Maxwell, which "cannot be determined, without assumptions of some kind, from experiments in which the active current forms a closed circuit."
Integrating around s' eliminates k and the original expression given by Ampère and Gauss is obtained.
Thus, as far as the original Ampère experiments are concerned, the value of k has no significance.
In the non-ethereal electron theories, Weber took k=−1 and Riemann took k=+1.
Ritz left k undetermined in his theory.
Using the vector identity for the triple cross product, we may express this result as
When integrated around ds' the second term is zero, and thus we find the form of Ampère's force law given by Maxwell:
In other words, the differential element of wire 1 is at
If wire 1 is also infinite, the integral diverges, because the total attractive force between two infinite parallel wires is infinity.
In fact, what we really want to know is the attractive force per unit length of wire 1.
Therefore, assume wire 1 has a large but finite length
As expected, the force that the wire feels is proportional to its length.
The direction of the force is along the y-axis, representing wire 1 getting pulled towards wire 2 if the currents are parallel, as expected.
The magnitude of the force per unit length agrees with the expression for
