Ampère’s force law equation and its application
– The magnetic force per unit length between two straight parallel conductors is given by Ampère’s force law.
– The equation for Ampère’s force law is Fm/L = 2kma(I1I2/r).
– kmA 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 wire.
– r is the distance between the two wires.
– Ampère’s force law is a good approximation for two parallel wires if one wire is much longer than the other.
– The distance between the wires should be small compared to their lengths and large compared to their diameters.
– The value of kmA determines the unit of current in the SI system.
– In the SI system, kmA = μ0/4π, where μ0 is the magnetic constant.
– μ0 is equal to 1.25663706212(19)×10^-6 H/m in SI units.
– The general formulation of Ampère’s force law for arbitrary geometries combines the Biot-Savart law and Lorentz force in one equation.
– The magnetic constant is replaced by the actual permeability of the medium when determining the force between wires in a material medium.
– The force on wire 1 due to wire 2 is equal and opposite to the force on wire 2 due to wire 1, in accordance with Newton’s third law of motion.
– The equation for the force between two separate closed wires can be rewritten using the vector triple product and applying Stokes theorem: F12 = -(μ0/4π) ∫∫ (I1dℓ1 · I2dℓ2) r21/|r|².
Historical background
– Ampère’s force law was derived by James Clerk Maxwell in 1873 based on the experiments of André-Marie Ampère and Carl Friedrich Gauss.
– The force between two linear currents was given by Ampère and Gauss in the early 19th century.
– Several scientists, including Wilhelm Weber, Rudolf Clausius, Maxwell, Bernhard Riemann, Hermann Grassmann, and Walther Ritz, developed the expression for Ampère’s force law.
– Maxwell gave the most general form consistent with experimental facts, taking into account additional terms that cancel each other out when integrated.
– The force on wire 1 due to the action of wire 2 can be expressed using various mathematical identities and functions.
Calculation example
– To calculate the force between two parallel wires, assume wire 1 is at y=D, z=0 and wire 2 is along the x-axis.
– The force per unit length of wire 1 is given by F12/L1 = (μ0I1I2/2πD)(0,-1,0).
– If both wires are infinite, the total force between them is infinite.
– To obtain the force per unit length, assume wire 1 has a large but finite length L1.
– The force felt by wire 1 is proportional to its length and directed along the y-axis.
Notable derivations
– Ampère’s original 1823 derivation.
– Maxwell’s 1873 derivation.
– Pierre Duhem’s 1892 derivation.
– Alfred O’Rahilly’s 1938 derivation.
Related concepts and references
– Ampere
– Magnetic constant
– Lorentz force
– Ampère’s circuital law
– Free space
– References and notes from various sources. Source: https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_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.
