Maxwell's equations represent one of the greatest achievements in physics. These four fundamental equations unify electricity and magnetism into a single elegant theory. They describe how electric and magnetic fields interact and propagate through space, and remarkably, they predict the existence of electromagnetic waves, including light itself.
Maxwell didn't create these equations from nothing. He built upon four key empirical laws discovered by earlier scientists. Coulomb's law led to Gauss's law for electricity. The absence of magnetic monopoles gave us Gauss's law for magnetism. Faraday's experiments revealed electromagnetic induction. And Ampère discovered the relationship between currents and magnetic fields. Maxwell's genius was unifying these laws and adding the crucial missing piece: the displacement current term.
Ampère's original law worked perfectly for steady currents, but it failed when electric fields were changing. Consider a charging capacitor: current flows in the wires, but no current flows between the plates. Yet the electric field is changing between the plates. This violated charge conservation! Maxwell realized that something was missing from Ampère's law.
Maxwell's brilliant insight was to add a new term to Ampère's law: the displacement current. This term represents the 'current' created by a changing electric field, even where no charges are moving! With this addition, the equations became consistent and symmetric. Most remarkably, they predicted electromagnetic waves traveling at the speed of light, revealing that light itself is an electromagnetic phenomenon.
To summarize what we've learned: Maxwell didn't derive his equations from first principles, but rather unified four empirical laws discovered by earlier scientists. His crucial contribution was adding the displacement current term to resolve inconsistencies in Ampère's law. This unified theory predicts electromagnetic waves traveling at the speed of light, revealing that light itself is electromagnetic radiation. Maxwell's equations remain the foundation of classical electromagnetism and modern physics.