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I am learning the Maxwell's equations on my own, and I ran into some questions about Faraday's law $$\nabla \times \mathbf E = -\frac{\partial}{\partial t} \mathbf B.$$

As far as I know, Faraday discovered experimentally that a change in magnetic field induces an electric field. However, the units of $\mathbf E$ and $\mathbf B$ are both defined by the Lorentz force law $\mathbf F = q(\mathbf E + \mathbf v \times \mathbf B)$, so one can only deduce that $\nabla \times \mathbf E = - \alpha \frac{\partial}{\partial t} \mathbf B$ for some constant $\alpha$. How did people figure out that this constant is exactly one?

One possible answer is based on the consideration of special relativity, but that would be an anachronism.

I suspect there were experiments confirming that Faraday's law is compatible with Lorentz force law (under some appropriate unit for velocity in terms of the speed of light). Is this the case? If so, when were these experiments carried out, and were they used as evidence supporting special relativity?

Any comment on the relevant history is also greatly appreciated.

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It's a choice. The SI system of units was defined this way. In Gaussian units, the proportionality constant is $1/c$, a different choice.

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  • $\begingroup$ So you are saying that the unit of $B$ is defined using Faraday's law, either $\nabla \times E = -\frac{d}{dt}B$ or $\nabla \times E = -\frac{1}{c}\frac{d}{dt}B$. But one still needs to verify that Lorentz force law is $F = q(E + v \times B)$, and not $F = q(E + \beta v \times B)$ for some constant $\beta$. $\endgroup$
    – Tian Xia
    Sep 19, 2023 at 19:46
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    $\begingroup$ @TianXia In Gaussian units, your $\beta$ is $1/c$. $\endgroup$
    – John Doty
    Sep 19, 2023 at 19:50
  • $\begingroup$ But I don't think that is true by convention. If no units are specified for $E$ and $B$, Faraday's law and Lorentz force law should be $\nabla \times E = -\alpha \frac{d}{dt}B$ and $F = q(\gamma E + \beta v \times B)$. The unit of $E$ is chosen such that $\gamma = 1$, but the unit of $B$ cannot be chosen to simultaneously specify both $\alpha$ and $\beta$. One of these numbers must be determined theoretically or experimentally. $\endgroup$
    – Tian Xia
    Sep 19, 2023 at 19:56
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    $\begingroup$ The experiment is taking a magnet and a coil. Move the coil relative to the magnet, measure the voltage due to Lorentz force. Now move the magnet relative to the coil, measure the EMF due to Faraday's law. And then read On the Electrodynamics of Moving Bodies (users.physics.ox.ac.uk/~rtaylor/teaching/specrel.pdf) for the theoretical significance. $\endgroup$
    – John Doty
    Sep 19, 2023 at 20:46
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    $\begingroup$ I already told you. That's the experiment. Move the coil or move the magnet, it makes no difference. Compatible with Galilean relativity. But forget the idea that physics is derived. Experiments are fundamental. $\endgroup$
    – John Doty
    Sep 20, 2023 at 0:20

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