Plane Electromagnetic Waves from Maxwell's Equations
Combining the two induction equations in empty space gives a wave equation whose speed is fixed by two measured electric and magnetic constants. That speed came out equal to the measured speed of light, which is why light is an electromagnetic wave.
What a learner can do afterwards
- Shows how the two curl equations combine into a wave equation for the fields
- States the speed of the wave in terms of the electric and magnetic constants and evaluates it
- Describes the geometry of a plane wave: fields perpendicular to each other and to the direction of travel
1 · Read
In empty space the Gauss laws make E and B divergence-free. Take the curl of Faraday's law. The identity curl of curl E equals grad of div E minus the Laplacian of E, with div E zero, leaves minus the Laplacian. Substitute Ampere-Maxwell for the curl of B: every component obeys a wave equation, second time derivative equals a constant times second space derivative. The same steps work for B.
The constant sets the speed: c equals 1 over the square root of mu0 times epsilon0. Both constants are measured with charges and currents, and the result is about 300 million metres per second, the measured speed of light. Light never enters the calculation, so the match shows light is an electromagnetic wave. Every band from radio to gamma rays travels at this c in vacuum, and frequency times wavelength equals c.
The standard plane wave has E and B oscillating in step at right angles to each other and both at right angles to the travel direction: a transverse wave with E divided by B equal to c everywhere. A 300 MHz radio wave then spans 1 metre, since 300 million times 1 equals c.
One equation covers the whole spectrum; only the frequency changes. Sources differ: antennas for radio, electron transitions in atoms for visible light, fast electron collisions for X-rays.
Two curls combine into one wave, and its speed is the speed of light.
The rewritten readings were written for this dataset. Video by Khan Academy, linked from YouTube.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.