Classical electromagnetism and special relativity

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. First of all, it gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another. Secondly, it sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity. Moreover, the apparent coincidences in which the same effect was observed due to different physical phenomena by two different observers would be shown to be not coincidental in the least by special relativity. In fact, half of Einstein's 1905 first paper on special relativity, "On the Electrodynamics of Moving Bodies," explains how to transform Maxwell's equations.

This equation, also called the Joules-Bernoulli equation, considers two inertial frames. As notation, the field variables in one frame are unprimed, and in a frame moving relative to the unprimed frame at velocity v, the fields are denoted with primes. In addition, the fields parallel to the velocity v are denoted by ${\displaystyle {\stackrel {\mathbf {E} _{\parallel }}{}}}$ while the fields perpendicular to v are denoted as ${\displaystyle {\stackrel {\mathbf {E} _{\bot }}{}}}$. In these two frames moving at relative velocity v, the E-fields and B-fields are related by:

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