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Applying the curl operator to Maxwell's third and fourth equations, combined with the first and second equations and partial derivative rules, reveals that both electric and magnetic field components satisfy the same wave equation. This mathematical derivation shows that electromagnetic fields propagate as three-dimensional waves in a vacuum with a speed determined by the permittivity and permeability of vacuum.

The propagation speed of electromagnetic waves is determined by combining the natural constants: vacuum permittivity and vacuum permeability. When these values are substituted into the wave equation, the calculated speed matches the experimentally measured speed of light, leading Maxwell to hypothesize that light itself is an electromagnetic wave.

Applying Maxwell's first and second equations to the general wave solutions reveals that the z-components of the electric and magnetic fields are zero when the wave travels in the z-direction. This means both fields oscillate perpendicular to the propagation direction, defining them as transverse waves rather than longitudinal waves.

Maxwell's third equation applied to wave solutions shows that the electric and magnetic field components are related and can be expressed in vector form. The traveling electric and magnetic fields are mutually perpendicular to each other and both perpendicular to the wave's propagation direction, while remaining in phase with one another.

Maxwell discovered that the speed calculated from electromagnetic theory using vacuum permittivity and permeability matched the experimentally measured speed of light. This remarkable coincidence led him to hypothesize that light is nothing but electromagnetic waves, a prediction later experimentally verified and demonstrating mathematics' power in explaining nature.

Yes. Maxwell's equations in a vacuum reveal that electric and magnetic fields can propagate without any source charges present. The wave equation solutions show that electromagnetic fields are not purely mathematical constructs but have physical meaning, traveling through empty space as self-sustaining waves determined by the wave equation.

Since electromagnetic fields follow the superposition principle, more general solutions to the wave equations can be written as linear superpositions of the known general solutions. Maxwell's equations then apply additional constraints to these superposed solutions, revealing that the resulting fields maintain the properties of being in phase, mutually perpendicular, and transverse.