How did Maxwell Derive his equations? Can we derive the same by conducting any experiment? $$
\begin{aligned}
\nabla \cdot \mathbf{E} &=\frac{\rho}{\varepsilon_{0}} \\
\nabla \cdot \mathbf{B} &=0 \\
\nabla \times \mathbf{E} &=-\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \times \mathbf{B} &=\mu_{0} \mathbf{j}+\frac{1}{c^{2}} \frac{\partial \mathbf{E}}{\partial t}
\end{aligned}
$$
Maxwell Equations are as shown in the picture
 A: Maxwell was guided in the formulation of his equations by an analogy, which he worked out in mathematical detail, between the behaviour of electric and magnetic fields as then known and the behaviour of a hypothetical space-filling medium consisting of tiny spinning vortices separated by ball-bearing like idlers. The axis of spin of the vortices was analogous to the local direction of the magnetic field, so a magnetic field line was analogous to a 'string' of spinning vortices. Stress on the idlers was analogous to an electric field.
From the analogy, Maxwell was able to predict the existence of a displacement current (roughly speaking the $\frac{\partial\vec E}{\partial t}$ term in the modern presentation of Maxwell's equations). This was not a generalisation of previous experimental results. The especial triumph was that the vortex model predicted transverse waves, which, according to the analogy, were of electric and magnetic fields. Recent experimental measurements (equivalent to the determination of $\mu_0$ and $\epsilon_0$) enabled Maxwell to calculate the speed of the waves. It came out close to the (already known) speed of light! Maxwell famously pointed out that the conclusion seemed 'scarcely to be avoided': light was an electromagnetic wave.
Maxwell published his work on the vortex  model (or 'ether') in 1861. The paper was called On Physical Lines of Force. He realised that his vortex ether, hugely fruitful as it had been, was only an analogy, and in his later work he didn't refer to it, but he had garnered the equations that it yielded. In his 1865 paper, A Dynamical Theory of the Electromagnetic Field he presented them as equations for the cartesian components of the electromagnetic vectors, which would apply either in a vacuum or in a material medium. [The notation of vector calculus had not yet been invented.]  $\vec \nabla.\vec B=0$ doesn't, in fact, appear explicitly, but is implied by $\vec \nabla \times \vec A$ being used for what we now call $\vec B$. Yes – the vector potential started with Maxwell!
Since the vortex medium modelled the known laws of electromagnetism, it is not surprising that Maxwell's equations are largely generalisations, cast in differential form, of previous experimental discoveries. Looking at each equation in turn... (1) This comes from Gauss's law which,  for stationary charges, can be derived from Coulomb's law. Maxwell himself conducted experiments that confirmed the law to much higher accuracy than Coulomb's experiments. He correctly surmised (without experimental tests) that Gauss's law wasn't restricted to stationary charges. (2) The $\vec \nabla.\vec B=0$ law was equivalent to the well-attested observation that there are no magnetic monopoles. (3) This comes from Faraday's experimental law (for stationary circuits).
(4) For steady currents, when $\frac{\partial\vec E}{\partial t}=0$, the equation is the differential form of Ampère's line integral law, which had been amply supported by experiments with various current configurations. The $\frac{\partial\vec E}{\partial t}$ term had not been dreamt of before Maxwell. Although it emerged originally from the vortex model, Maxwell realised that without it, equation (4) would violate charge conservation except when $\vec\nabla.\vec J=0$. The propagation of electromagnetic waves in free space is now usually regarded as the best experimental support for the $\frac{\partial\vec E}{\partial t}$ term; 16 decades after it was proposed it is still not easy to provide more direct experimental evidence for it.
A: Maxwell didn't derive the equation in their present form (the four equations given in the OP). But yes, like most phsyics laws, these are a generalization of the experimental facts, separately known as Faraday's law, Bio-Savart law, Ampere's law, Gauss law, etc.
