See [my answer here](https://physics.stackexchange.com/a/74999/26076): Maxwell's big contribution was the notion of *displacement current*, which then changed the equations of electromagnetism in a way that foretold electromagnetic radiation whereby the Cartesian components of the fields all fulfilled [D'Alembert's Wave equation](http://en.wikipedia.org/wiki/D%27Alembert%27s_formula) and moreover that the wavespeed $c$ would be $c = 1/\sqrt{\mu_0\,\epsilon_0}$. The latter's ($c$, that is) surprising nearness to the experimentally known value as found by the [Fizeau experiment](http://en.wikipedia.org/wiki/Fizeau%E2%80%93Foucault_apparatus) led Maxwell to assert that light is one such electromagnetic wave.

Historians of physics widely consider that Maxwell's foretelling was first vindicated by the [Hertz Spark Gap experiment](http://en.wikipedia.org/wiki/Spark-gap_transmitter).

So, without being too glib, the great J C Maxwell's main gig was the second term on the right hand side of your equation.