I have often heard it said that several problems in the theory of electromagnetism as described by Maxwell's equations led Einstein to his theory of Special Relativity. What exactly were these problems that Einstein had in mind, and how does Special Relativity solve them?
Answer
There was no problem with electromagnetism. The problem was that Maxwell's equations are invariant under Lorentz transformations but are not invariant under Galileo transformations whereas the equations of classical mechanics can be easily made invariant under Galileo transformations.
The question was: how to reconcile both in a universe in which Maxwell's equations had been tested much more thoroughly than the equations of classical mechanics when $v$ is in the same order of $c$ and not much smaller.
Einstein basically solved the problem by deciding that electromagnetism is more fundamental in physics, and then showing that classical mechanics could be modified in such a way, that it, too, became Lorentz invariant. As a side effect, he recovered classical mechanics as a natural limit for $v/c\to0$, which perfectly explained almost all observations of macroscopic dynamics available at that time (leaving Mercury's perihelion precession to be explained by general relativity ten years later).
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