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The Wiki article is about 2 graduate years of physics beyond my understanding. What is a good high-school rendition of his thought process: regarding his use of the "distributed capacitance and inductance of the vacuum" to reach his conclusion?"

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    $\begingroup$ The value of the speed of light had been figured out well before Maxwell. Do you want to know how Maxwell explained that value in terms of other things? $\endgroup$ – G. Smith May 12 at 22:00
  • $\begingroup$ Maxwell himself was a very impressive mathematician, and built the theory up without the aid of lots of modern notation. I'm certain his thought process was much harder to follow than the modern exposition which has been honed over 150 years. $\endgroup$ – jacob1729 May 12 at 22:44
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The speed of light has been accurately measured by Foucault in 1862. See http://www.speed-light.info/measurement.htm for a historical overview of deyermi actions of the speed of light.

Maxwell unified electromagnetism and one of the consequences of his equations was that electromagnetic fields could propagate at the speed of $1/\sqrt{\mu_0 \epsilon_0} $ and theorized that light is an electromagnetic wave. Hertz confirmed the existence of such waves.

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Maxwell derived that the speed of propagation of electromagnetic waves was $$c = \sqrt \frac{1}{\epsilon_0 \mu_0},$$ where epsilon and mu were known from experiment.

His result for $c$ was close to the experimental value of the speed of light in his days, so he could conclude that also light was an electromagnetic wave. (For better history, there is https://hsm.stackexchange.com/ .)

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