The story of Maxwell's equations is different from the development of other equations or laws. It is a purely theatrical result. James Clerk Maxwell didn't do any experiment before he got the equations. He compared previous works and tried to solve the relationship between electricity and magnetism. Now a days, you can derive Maxwell's equations in different ways as evidenced by the huge amount of text books. But theory is just mathematics, what is the physics behind the equations. First, it predicts the electromagnetic wave. Maxwell's student Heinrich Rudolf Hertz' demonstrated the existence of electromagnetic waves by building an apparatus to produce and detect VHF or UHF radio waves. Secondly, in Maxwell's equations, there exists a special speed known as the speed of light c. So this means there exists a special reference frame, traveling at the speed of light c. People began to search for this special reference frame.

James Clerk Maxwell

This picture comes from http://en.wikipedia.org/wiki/Image:James_Clerk_Maxwell.png

Maxwell's work: Electricity and Magnetism

Photo by Xuanye Ma at UAF library

Reference

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