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Maxwell's equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose trajectory can thus be determined using suitable boundary conditions. The objective of electromagnetism is thus theoretically complete.
However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations: What happens to the electric and magnetic fields when there are no source charges?
Consider the differential forms of the four Maxwell's equations in a vacuum. A natural symmetry is observed in the first-order partial differential equations except for the natural constants.
Mathematical explorations revealed that each electric and magnetic field component follows the same wave equation. The speed is determined by a combination of the natural constants: the permittivity of the vacuum and the permeability of the vacuum. Substituting their numbers, Maxwell discovered that this speed is very close to the experimental value of the speed of light.
The result is startling given the independence of the observations: one from the theory of electrodynamics, which emerged from studying charges and currents in the laboratory, and the other an experimental determination of light's speed.
Connecting the dots, Maxwell hypothesized that light is nothing but electromagnetic waves. The hypothesis was later experimentally verified. This course of history is an essential lesson on the importance of mathematics in explaining nature. Moreover, electric and magnetic fields are not purely mathematical constructs; they can travel without any material present, which gives them a physical meaning.
The general solutions to the wave equations are known. Since electromagnetic fields follow the principle of superposition, more general solutions of the wave equations can be written as linear superpositions of these solutions.
Maxwell's equations apply additional constraints on the general solutions. They reveal other remarkable facts: the electric and magnetic fields are in phase with each other, are mutually perpendicular, and are both perpendicular to the wave's propagation direction. Thus, electromagnetic waves are transverse waves.
