As a side note there was a historical debate, and the shape of these equations is the end result of the chosen system.
In geometric algebra which did not prevail, these are one equation.
Start here for the ideas https://en.wikipedia.org/wiki/Electromagnetic_tensor
Geometric algebra produces two equations too, by the way, not one.
*d*
To get an intuition of the physics, I think the traditional 4 equation form is actually more useful, as you can construct toy examples and study the equations one at a time in isolation.
Where the more advanced formulations are useful, and actually are used, is for stuff like relativistic physics where 4-vectors, curved spacetime etc. are needed and not just a gimmick.
But for more down-to-earth applications of electrodynamics like antennas, field propagation in various forms of matter etc., the classical version is fine.
Also there is a new community for geometric algbera people https://bivector.net/.
Check out the demo https://observablehq.com/@enkimute/animated-orbits
Watch the SIGGRAPH talk https://youtu.be/tX4H_ctggYo
Join the discord https://discord.gg/vGY6pPk