AcademyElectromagnetic Waves
Academy
Plane Waves and Light Speed
Level 1 - Physics topic page in Electromagnetic Waves.
Principle
A plane electromagnetic wave has uniform wavefronts, travels at \(c\) in vacuum, and carries electric and magnetic fields transverse to the direction of travel.
Notation
\(\lambda\)
wavelength
\(\mathrm{m}\)
\(f\)
frequency
\(\mathrm{Hz}\)
\(\omega\)
angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(k\)
wave number
\(\mathrm{rad\,m^{-1}}\)
\(n\)
refractive index
1
Method
Derivation 1: Plane-wave geometry
A plane wave moving in the \(+x\) direction has fields that depend on \(x\) and \(t\), not on \(y\) or \(z\).
Wave phase
\[\phi=kx-\omega t\]
Wave speed
\[v=\frac{\omega}{k}\]
Derivation 2: Vacuum light speed
For electromagnetic waves in vacuum, Maxwell's equations set the wave speed.
Frequency-wavelength form
\[c=f\lambda\]
Angular form
\[c=\frac{\omega}{k}\]
Medium speed
\[v=\frac{c}{n}\]
Derivation 3: Direction
The propagation direction is the direction of \(\\vec E\\times\\vec B\).
Transverse fields
\[\vec E\perp\vec B\perp\vec v\]
Field magnitudes
\[E=cB\]
Rules
Vacuum speed
\[c=f\lambda=\frac{\omega}{k}\]
Refractive index
\[n=\frac{c}{v}\]
Wave number
\[k=\frac{2\pi}{\lambda}\]
Angular frequency
\[\omega=2\pi f\]
Examples
Question
A radio wave has frequency
\[100\,\mathrm{MHz}\]
Find its wavelength in vacuum.Answer
\[\lambda=\frac{c}{f}=\frac{3.0\times10^8}{1.00\times10^8}=3.0\,\mathrm{m}\]
Checks
- Frequency is set by the source and usually does not change when light enters a new medium.
- Speed and wavelength change in a medium according to \(v=f\\lambda\).
- For a wave moving in \(+x\), a possible orientation is \(\\vec E\) along \(+y\) and \(\\vec B\) along \(+z\).
- The field ratio \(E/B=c\) is for vacuum plane waves.