--- title: "Plane Electromagnetic Waves from Maxwell's Equations" description: "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 ligh" canonical: https://lightmysky.com/learn/science/plane-electromagnetic-waves-from-maxwells-equations-mt_27tbaYcxCM source: https://lightmysky.com/learn/science/plane-electromagnetic-waves-from-maxwells-equations-mt_27tbaYcxCM.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Electricity & Magnetism · Ages 21 to 22 Page: https://lightmysky.com/learn/science/plane-electromagnetic-waves-from-maxwells-equations-mt_27tbaYcxCM ## Ready when they can - 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 ## Lesson: Light from two equations In empty space the two curl equations lock together. Take the curl of the Faraday law and of the Ampere Maxwell law, substitute each into the other, and every component of E and B obeys a wave equation: second time derivative equals a constant times second space derivative. No new physics enters beyond those two laws plus vector calculus. The constant sets the speed: c equals 1 over the square root of mu0 times epsilon0, about 300 million metres per second. Evaluating those two measured electric and magnetic constants gives the measured speed of light, which is why light is an electromagnetic wave. Frequency times wavelength always equals c, so higher frequency means shorter wavelength. **Example.** 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. **Tip.** One equation rules the whole spectrum from kilometre radio to tiny gamma rays; only the frequency changes. Read each band by its sources and uses: antennas for radio, atomic jumps for visible, slammed electrons for X-rays. The fields always oscillate transversely at c. **Recap.** Two curls combine into one wave, and its speed is the speed of light. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [The Wave Equation and Its Travelling-Wave Solutions](https://lightmysky.com/learn/science/the-wave-equation-and-its-travelling-wave-solutions-mt_Ue-v1cSxNx) - [Displacement Current and Maxwell's Equations](https://lightmysky.com/learn/science/displacement-current-and-maxwells-equations-mt_YR8m8TNfyy) ## Opens up - [The Poynting Vector and Radiation Pressure](https://lightmysky.com/learn/science/the-poynting-vector-and-radiation-pressure-mt_dwTbNlzBgE)