Classical electromagnetism and special relativity

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. 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. It sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electric or magnetic laws. 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 considers two inertial frames. The primed frame is moving relative to the unprimed frame at velocity v. Fields defined in the primed frame are indicated by primes, and fields defined in the unprimed frame lack primes. The field components parallel to the velocity v are denoted by and while the field components perpendicular to v are denoted as and . In these two frames moving at relative velocity v, the E-fields and B-fields are related by: where is called the Lorentz factor and c is the speed of light in free space. The equations above are in SI. In CGS these equations can be derived by replacing with , and with , except . Lorentz factor () is the same in both systems. The inverse transformations are the same except v → −v. An equivalent, alternative expression is: where is the velocity unit vector. With previous notations, one actually has and .