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Ampère's force $d^2\vec{F_{21}}$ of current element $i_2d\vec{\ell_2}$ on $i_1d\vec{\ell_1}$ is$$d^2\vec{F_{21}}=-\frac{\mu_0}{4\pi}i_1i_2\frac{\hat{r}}{r^2}\left[2(d\vec{\ell_1}\cdot d\vec{\ell_2})-3(\hat{r}\cdot d\vec{\ell_1})(\hat{r}\cdot d\vec{\ell_2})\right]=-d^2\vec{F_{12}}$$(cf. Ampère's 1826 Théorie mathématique des phénomènes électrodynamiques uniquement déduite de l’expérience).

Can one derive Ampère's circuital law

$$\vec{\nabla}\times\vec{B}-\epsilon_0\mu_0\frac{\partial\vec{E}}{\partial t}=\mu_0\vec{J},$$as it appears as one of Maxwell's equations, from Ampère's force law above? What assumptions might one need to make?

Geremia
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  • For something that I think basically amounts to the converse, see http://physics.stackexchange.com/questions/20477/can-the-lorentz-force-expression-be-derived-from-maxwells-equations –  Sep 15 '14 at 20:27
  • There are probably many different ways to slice this kind of thing up into a set of axioms and theorems. One possible approach is given in ch. 10 of my SR book, http://www.lightandmatter.com/sr/ . I explicitly list four assumptions in section 10.3.1. This leads through a plausibility argument to the existence of a magnetic field and the Lorentz force law. Section 10.7 then gives a plausibility argument for Maxwell's equations. I don't claim it's a rigorous proof, but I think it shows that if you're trying to construct a reasonable theory, you end up with Maxwell's equations. –  Sep 15 '14 at 20:35

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