Predicting the Existence of Electromagnetic Waves

Probably the greatest achievement of Maxwell's career was his prediction of the existence of EM waves. Maxwell asked the question, what do the equations say about a situation where there are no charges or conduction currents? In that case the last two equations have the solution:

E(x,t) = Emax cos(kx - wt)

B(x,t) = Bmax cos(kx - wt)

predicting that oscillating electric and magnetic fields travel as waves. He even predicted their speed, which he called c:
c =
1
(mo eo)1/2
= 2.998 x 108 m/s

Maxwell realized that this was in the ballpark of what scientists were measuring for the speed of light, and that his oscillating electric and magnetic fields, traveling as waves, must in fact be what light is.

So, in one fell swoop he unified electricity, magnetism, and optics.