What are the minimal experiments would one need to perform in order to reconstruct Maxwell's equations from scratch, assuming even the concepts of $\vec E$ and $\vec B$ are unknown?
While I'm not necessarily interested in a historical approach, I'd prefer experiments that might reasonably be performed at the undergraduate level. (I'm not planning to create a curriculum; I'm more interested in the experiments than the pedagogy.)
Jefimenko's book introduces currents with experiments, and just reading his book felt like you were seeing the experiments being done right in front of you. I didn't read the whole book, so I don't know if he finishes the job, but he introduces currents as real things (for your system of units, and as a physical observation of fact), and then gets charge from that.
Physics is inductive, so you still have to generalize from situations like: Since a particular mathematical description worked in these situations we postulate that the same mathematical description would work in other situations. And that postulation has to be born out by experience. So Maxwell's equation is still just postulated after any finite selection of different experiments.
If you want people to think of $\vec{E}$ and $\vec{B}$ as real, then you should start by convincing them to believe in energy conservation and momentum conservation, then have Maxwell's Equation, and the Lorentz Force Law, and then get that the fields are needed to save energy and momentum conservation.
Iron filings are suggestive, but there isn't really evidence that a field is a real thing rather than a calculational tool until you get to saving energy and momentum conservation. So you need more than just Maxwell, you need Maxwell-Lorentz.
If the concepts of E and B are unknown, the four Maxwell equations won't be enough. You also need to find out about the Lorentz force equation F = q(E + v x B). I'm assuming the concept of charge is also unknown. I'm also assuming that you're asking this question because you're planing an undergraduate lecture of some sort.
I'd start with a justification for the concepts of charge and fields, that is : action at a distance. Personnally, I use two labs with my students, one for electrostatics and one for magnetostatics. The lab on electrostatics should at least cover : 1) qualitative observations of electrostatic forces. This alone can easily lead to a model where two and only two types of charges are required to explain every observation. 2) quantitative observations : using a scale and small balls covered in conducting paint, one can obtain Coulomb's law. The r dependance is obvious but one can also obtain the q dependance by using the following trick : if a charged ball is put in contact with an identical uncharged ball, the charge is split in half. This can be repeated several times. 3) The concept of a field can be introduced by asking the question : what happens if r is suddently increased. A delay is required before the force is updated, which shows that the action isn't direct. Of course, such a delay is too small to be observed but it's usually convincing to mention that e-m waves wouldn't exist if that delay would be zero.
As for the lab on magnetostatics, I usually start with some analogy between an electric dipole and a small magnet. The best way to justify the direction of the B field is to say that it is chosen in the direction along which magnetic dipoles tend to align. That is analogus to the fact that electric dipoles align on the E field. Therefore, using this definition, the direction of E and B field can be probed experimentally using grass seed and iron filings, respectively.
Observation of some differences between electric and magnetic dipoles is important to show that two types of fields really are required to explain all phenomena. In particular, it needs to be shown experimentally that "poles" cannot be isolated as opposed to electric charges.
Once the need for a second type of field is established, it needs to be shown that no observations require the introduction of a "magnetic charge" concept. This can be done by showing that current carrying solenoids behave exactly as small magnets do : not only do they align surrounding iron fillings, but small solenoids can also rotate and align themselves to a large magnetic field. This leads to the idea that the fundamental source for a B field is "any moving electric charge" and not just a current. This idea can be confirmed experimentally by setting any charged object in motion : a rotating plastic tore produces a B field analogous to that of a current carrying loop.
The proof that magnetic charges don't exist is complete once you also show that B fields exert forces on electric charges. Any picture of a bubble chamber usually does the trick.
The previous should suffice in establishing that there are two types of fields and two types of electric charges and establishing the Lorentz force equation.
Next, you need to transform that knowledge in the form of two Maxwell equations. Gauss law can be obtained analytically from Coulomb's law and "Gauss law for the B field" can be obtained by analogy with Gauss law and from the fact that "poles" cannot be isolated.
Before I keep going, I'd need more precision regarding your question : 1) how detailled do you want your proof to be. For instance, the validity of Gauss law extends past electrostatics but showing that would require more experiments. 2) how much analytics are you willing to introduce as part of your proof.
I hope this helps :-)
1) Kirchoff Voltage law in circuit(generally with an inductor) for faraday's law in scalar form. 2) Cheap electrification experiments for coulomb's law. 3) Electromagnets in dc and Capacitors in ac circuit for ampere-maxwell law.
You will have to measure force/torque patterns to get the concept of E, B fields for full generality of the above. Iron fillings, compass tracing, gauss-meter needed for that.
