@helenS Tunneling has nothing to do with Maxwell, but his work certainly embodies "Physics is cool!" I think of Maxwell's equations as sort of the gateway to relativity, because they beg the question "wave velocity relative to what?" But I can't tie them into the ultraviolet catastrophe or the photoelectric effect or the other precursors of quantum physics.
@ElwoodBlues Isn't quantum mechanics a critic of classical electrodynamics? According to the equations above, the H atom should not exist. The electron should spiral into the core while emitting radiation of ever-increasing frequency.
@helenS Sure, and yet classical electrodynamics describes antennas and waveguides and radar and EM emissions of circuit boards so accurately. Each theory has its place. Even though relativity supplanted Newton's laws over a century ago, I bet you haven't abandoned F=ma.