# Ampère force law, action, reaction

by lalbatros
Tags: action, ampere, force, reaction
 P: 1,235 This is apparently a well known topic, but I did not know it before today. Let us consider the Ampère law for the force experience by a current element (1) in the magnetic fields of another current elment (2): $$\mathbf{dF}_{12} = \frac {\mu_0} {4 \pi} I_1 I_2 \frac {d \mathbf{s_2}\ \mathbf{ \times} \ (d \mathbf{s_1} \ \mathbf{ \times } \ \hat{\mathbf{r}}_{12} )} {r_{12}^2}$$ You can easily check that the "action and reaction" are not balanced by the Ampère law since: $$\mathbf{dF}_{12} + \mathbf{dF}_{21} <> 0$$ How should we understand that?
 Sci Advisor P: 1,250 That force law is only valid if ds_1 and ds_2 are integrated over closed circuits. Then a little vector calculus can be used to put it into a different form that does have F_12=F_21. This form is called the Neumann form. It depends on the dot product of ds_1 and ds_2. This is shown in most EM textbooks.
 P: 192 It's a fascinating fact that the electromagnetic field itself carries momentum in classical electrodynamics, and this causes Newton's third law to appear to fail. For this reason it is intractable to use Newton's laws in electrodynamics (it is not impossible, there are pseudo-mechanical formulations of E&M, including Maxwell's own model, that allow us to apply Newtonian mechanics to find the missing reaction forces and momentum, but no one does this). Instead it is more convenient to use Lagrangian methods and/or the covariant formulation of E&M i.e. the faraday tensor, etc.