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James Clerk[sic] Maxwell was able to show that an electric field vibrating at a specific frequency would result in a magnetic field vibrating at exactly that same frequency, and that the latter magnetic field would result in an electric field vibrating at the same frequency -- ad infinitem. He was also able to show that this continual vibration would travel away from its starting point at the speed of light -- thus showing that light, at its basic nature, is a vibrating EM field. Experiments showed that pretty much all predictions based on Maxwell's Equations did, indeed, occur -- such as this EM field travelling without any media needed.

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