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How I know and read in the Internet that Ampere realized that two current-carrying wires can exert force on each other. Ampere's force law is: $$d^2\vec{F_{21}}=-\frac{\mu_0}{4\pi}i_1i_2\frac{\hat{r}}{r^2}\left[2(d\vec{\ell_1}\cdot d\vec{\ell_2})-3(\hat{r}\cdot d\vec{\ell_1})(\hat{r}\cdot d\vec{\ell_2})\right]=-d^2\vec{F_{12}}$$

Where $i_2d\vec{\ell_2}$ and $i_1d\vec{\ell_1}$ are the current elements of the wires. And the circuital law is: $$\oint \vec B\cdot d\vec l=\mu_oI_{net}$$

My question is which law was first written? Does Ampere extract the circuital law from the force equation above? Does the mathematical form of the magnetic field equation is the original form that Ampere wrote or it came later by another scientists and if so, then what is the original formulation.

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    $\begingroup$ This seems more like a question about the history of Ampere's law, so you may get better answers over at HSM.SE: hsm.stackexchange.com $\endgroup$ – d_b Mar 6 at 0:16
  • $\begingroup$ I’m voting to close this question because it belongs on History of Science and Mathematics $\endgroup$ – Dale Mar 6 at 22:36
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As far as I know, Ampére tried to explain the force between conductors by the formula you wrote. The dominant notion of the time was action at a distance, that was successful for gravity and also for electrostatic.

Even the magnetic forces were explained by him as resultant of forces between micro currents in the materials.

The idea of a magnetic field that mediates the force between wires prevails with time, maybe because microcurrents were too speculative, and Faraday's iron fillings demonstrations too persuasive about the existence of a magnetic field.

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