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The Biot-Savart law describes the magnetic field produced by a current carrying wire, $$\textbf{B}(\textbf{r})=\frac{\mu_{0}}{4\pi}\oint_{C}\frac{Id\textbf{l}\times(\textbf{r}-\textbf{l})}{|\textbf{r}-\textbf{l}|^{3}}$$ where $C$ represents the path around the wire. The law was experimentally derived in 1820. Conversely, the Lorentz force law describing the magnetic force experienced by a particle of charge $q$ move at velocity $\textbf{v}$ in a magnetic field, $$\textbf{F}=q\textbf{v}\times\textbf{B}$$ was derived far later, from Maxwells equations. To me it seems that any measurement of field strength would require a measurement of force. So my question is, how could the Biot-Savart law be derived by experimental measurements of magnetic fields if there was no established relation between magnetic force and magnetic field?

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  • $\begingroup$ This is really a question from history of physics, better ask it on hsm.stackexchange.com . $\endgroup$ Commented Jun 9, 2023 at 23:43
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    $\begingroup$ Magnetic field was originally defined through its action on magnetic bodies and current-carrying wires, not charged particles flying in vacuum. Biot, Savart and Laplace did not need the concept of Lorentz force. $\endgroup$ Commented Jun 9, 2023 at 23:50

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The original law deduced by Biot and Savart was less general than what we now call the Biot-Savart law. It amounted to the following statements:

  • the magnetic field a distance $r$ from a current-carrying wire, measured perpendicular to the wire, was inversely proportional to $r$.
  • the field is mutually perpendicular to the wire and to the line from the wire to the observation point

Biot and Savart found this law by suspending pieces of magnetized steel near a current-carrying wire and measuring their periods of oscillation and equilibrium deflections.

Sources:

  1. E. T. Whitaker, A history of the theories of aether and electricity (1910)
  2. Annales de chimie et de physique, t. XV (1820), p. 223
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