|MadSci Network: Physics|
These equations are the results of Maxwell's successful 1865 effort to systematize and work through the logical implications of the experimental facts then known about electricity and magnetism: conservation of electric charge, Coloumb's Law, Ampere's Law, and Faraday's Law. The vacuum equations tell us how the electric field E and the magnetic field B propagate through empty space, in the absence of any charges or currents. Maxwell noticed a logical inconsistency in Ampere's Law, which led him to recognize the need for the final term above, on the RH side of the last of the four equations.
But a beginning student does not really even have to understand clearly what "div" and "curl" mean to notice that these equations are symmetric between the E and B fields, in the sense that if you swap E and B, you get essentially the same set of equations. (Except for the minus sign, which is very important. It seems to break the symmetry but actually does not, because it can be placed on either of the two curl equations, depending on the arbitrary choice of a right-hand or left-hand rule of definition.) This symmetry means that E and B propagate and behave in a vacuum in exactly the same way; they are indistinguishable (yet distinct) in this regard, like some cosmic Tweedledee and Tweedledum. And since a changing magnetic flux gives rise to an electric field (before Maxwell, this was known only as Faraday's Law of Induction), and vice versa (due to Maxwell's modification of Ampere's Law), they are intimately coupled and in some sense inseparable. That is, you really can't have one without the other.
The 90° angle mentioned in the question is not truly fundamental but just comes out of one particular, but very important, solution of Maxwell's Equations, the free sinusoidal plane waves. In this situation the constantly changing flux of one field (E or B) is the source for the other, and energy is continuously passed back and forth between them. Such waves are what we usually mean when we say "electromagnetic waves", the basis for radio, microwave, infra-red, visible light (and thus all optics), ultraviolet light, X rays, and gamma rays. The motions of electrons in their atomic orbits, mentioned in the question, do necessarily generate electric currents, and are thus sources for the full (non-vacuum) set of equations, but are not themselves related to the symmetry. The intrinsic spins of charged particles, notably the electron, have intrinsic magnetic moments associated with them; but that is a more difficult matter, coming out of relativistic quantum mechanics and the Dirac equation.
Finally, despite their deep symmetry and interrelatedness, magnetic fields and electric fields are obviously not the same, and we must just mention the reason for the difference. The version of Maxwell's Equations given above is for empty space. If there are currents or charges, then there are also additional terms on the right-hand sides. These extra terms are sometimes called the source terms for the fields. (For example, electric lines of force typically begin and end on electric charges, and so we say the charges are the source for the E field.) It is easy to add the source terms to the equations, preserving the symmetry, but the full set of equations so obtained has never been subjected to experimental test.
The reason is that, despite many experimental efforts to find them, there seem to be no magnetic charges (magnetic monopoles) in nature; nor the magnetic currents they would create when moving. Thus the terms in the equations due to magnetic charges and currents are always zero, so far as has ever been observed to date; Maxwell's Equations lose their beautiful symmetry; and electricity and magnetism are in practice not the same. Many, maybe even most, theories of elementary particle physics (beyond that part, called "The Standard Model", which is presently fairly well established and fairly well understood) predict that magnetic monopoles should actually exist, possibly in presently inaccessible parts of the Universe, such as immediately after the Big Bang. The resolution of this question is one of the outstanding problems that remains to be solved in fundamental physics.
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