Maxwell's equation governing light and all other electromagnetic phenomena would not have been possible without the modification of Ampere's law to include the displacement current. However, I'm a bit curious about the historical origin of the displacement current term. While I know the term is necessary for the continuity equation, in class it was presented as though Maxwell simply added the term for mathematical beauty: it made the form of the equations governing the electric and magnetic fields symmetric. And from there, he was able to derive the electromagnetic wave and all the other phenomena associated with classical electrodynamics.
But physicists are not historians, and I cannot believe that there was no more compelling reason for the addition of the displacement current term. Historians, what was the real reason Maxwell added the displacement current term to Ampere's law? Or am I off-base, and the term really was added simply for symmetry reasons?
I am going to make the assumption, that my mathematical beauty your professor means Simple and Symmetric. Simple in this context stemming from the fewest axioms or "Laws" and Symmetric, meaning unchanging under rotation or accelerating frames of reference. I am also probably going to over simplify some of the math.
From his paper is was fairly clear*** that Maxwell was attempting to integrate the theory of Magnetism and the Theory of Electricity into a single mathematical framework i.e. the Theory of Electromagnetism.
I propose now to examine magnetic phenomena from a mechanical point of view, and to determine what tensions in, or motions of, a medium are capable of producing the mechanical phenomena observed.
At the time the two phenomena where obviously connected, but there was not mathematical way to describe both simultaneously. The displacement current he postulated as a way of connecting Electricity and Magnetism was based around the concept of a "sea" of magnetic vortices that moved energy from one side of a capacitor to another. Maxwell was correct in how he described the effects, but incorrect in identifying the underlying mechanism.
What is critical for context is that Maxwell was doing his work before the invention of vector calculous. This made his work neither simple or beautiful, but rather a pages upon pages of rigorous calculation performed in a fixed set of coordinates. Without vectors, rotating the coordinate axis becomes almost more complex than the underlying mathematics, and the concept of symmetry is almost meaningless. Nor did Maxwell reduce the number of existing Law's, he in fact extended them.
It wasn't until a couple of decades later, when the same equations were reformulated in vector notation that they beauty of what Maxwell did became apparent. When written in vector form Maxwell's equations were both highly symmetric (unchanging under rotation and frame of reference) and simplified. The Symmetry techniques that Maxwell discovered as part of his unification when on to underlie the totality of modern physics, but the technique was the bi-product of a different goal.
In short, Maxwell invented a phenomena, which he described with the displacement current, so we could combine electricity and magnetism. His phenomena was wrong, but his math wasn't. It wasn't until they the E&M equations were reformulated in vector calculous, that they became beautiful. It's concept what very similar to Einstein's Cosmological Constant, in that's its a correct concept implemented for the wrong reasons.
*** It occurs to me that this is not actually clear. Buts its mentioned in both Griffeths text and on the Wiki page.