A changing electric field creates a magnetic field. A changing magnetic field creates an electric field. This mutual, self-sustaining relationship — not four separate rules — is what actually lets a wave propagate through empty space.
Maxwell's equations are four relationships that together describe every classical electric and magnetic phenomenon — from a compass needle to a Wi-Fi signal. The two shown here, Faraday's Law and the Ampère–Maxwell Law, are the pair responsible for wave propagation: each field's rate of change in time drives the other field's spatial circulation, so a disturbance keeps regenerating itself and travels outward without needing a physical medium.
Gauss's Law for E describes how electric field lines originate on positive charge and terminate on negative charge. Gauss's Law for B states that magnetic field lines never begin or end — there are no magnetic monopoles, only loops. Faraday's Law says a time-varying magnetic field induces a circulating electric field. The Ampère–Maxwell Law says a time-varying electric field (or a real current) induces a circulating magnetic field. Individually these describe static and quasi-static behavior; together, the last two form a feedback loop that supports self-propagating waves.
When Maxwell combined these relationships mathematically, the wave speed that fell out of the equations matched the already-measured speed of light almost exactly — which is how it was first recognized that visible light itself is an electromagnetic wave, governed by the same equations as radio waves, microwaves, and X-rays, differing only in frequency.
Antenna design, transmission line behavior, electromagnetic compatibility (EMC) analysis, motor and transformer design, and RF/microwave engineering all trace back to these four relationships. Most day-to-day engineering work uses simplified derived forms (circuit theory, transmission line equations, radiation patterns) rather than solving Maxwell's equations directly, but those simplified tools are only valid within the approximations Maxwell's equations justify.
Rarely in full differential form — most applied work uses derived tools (circuit theory, transmission line equations, antenna gain formulas) that already bake in the relevant physics. Full-wave solutions are reserved for specialized RF, antenna, and EMC simulation work, typically done in dedicated electromagnetic simulation software.
The other three relationships were largely known before Maxwell. His key theoretical contribution was recognizing that Ampère's Law was incomplete — he added a term (the 'displacement current') representing a changing electric field's own ability to generate a magnetic field, which was necessary to make the equations mathematically consistent and which is what makes wave propagation possible.
In a propagating plane wave in free space, the electric and magnetic field components are in phase with each other in time at a given point, but their peak magnitudes occur perpendicular to each other in space (E and B are orthogonal). The visualization above shows the underlying driving relationship — each field's time-derivative sourcing the other's spatial curl — rather than a literal snapshot of a single propagating wave.
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