AcademyElectromagnetic Waves
Academy
Maxwell's Equations to Waves
Level 1 - Physics topic page in Electromagnetic Waves.
Principle
Maxwell's equations predict self-sustaining electromagnetic waves: a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field.
Notation
\(\vec E\)
electric field
\(\mathrm{N\,C^{-1}}\)
\(\vec B\)
magnetic field
\(\mathrm{T}\)
\(\epsilon_0\)
permittivity of free space
\(\mathrm{F\,m^{-1}}\)
\(\mu_0\)
permeability of free space
\(\mathrm{N\,A^{-2}}\)
\(c\)
speed of electromagnetic waves in vacuum
\(\mathrm{m\,s^{-1}}\)
Method
Derivation 1: Source-free fields
In empty space there is no charge density and no conduction current.
No electric sources
\[\nabla\cdot\vec E=0\]
No magnetic sources
\[\nabla\cdot\vec B=0\]
Derivation 2: Coupled changing fields
The curl equations are the wave-producing pair.
Changing magnetic field creates electric circulation
\[\nabla\times\vec E=-\frac{\partial\vec B}{\partial t}\]
Changing electric field creates magnetic circulation
\[\nabla\times\vec B=\mu_0\epsilon_0\frac{\partial\vec E}{\partial t}\]
Derivation 3: Wave speed
Combining the two curl equations gives a wave equation for each field.
Electric-field wave equation
\[\nabla^2\vec E=\mu_0\epsilon_0\frac{\partial^2\vec E}{\partial t^2}\]
Magnetic-field wave equation
\[\nabla^2\vec B=\mu_0\epsilon_0\frac{\partial^2\vec B}{\partial t^2}\]
Vacuum wave speed
\[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\]
Rules
These relationships connect Maxwell's equations to light.
Vacuum wave speed
\[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\]
Field ratio in vacuum
\[\frac{E}{B}=c\]
Propagation direction
\[\vec S\ \text{points along}\ \vec E\times\vec B\]
Examples
Question
Use
\[\mu_0=4\pi\times10^{-7}\,\mathrm{N\,A^{-2}}\]
and \[\epsilon_0=8.85\times10^{-12}\,\mathrm{F\,m^{-1}}\]
to estimate \(c\).Answer
\[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\approx3.0\times10^8\,\mathrm{m\,s^{-1}}\]
Checks
- Electromagnetic waves do not need a material medium.
- In vacuum, \(\\vec E\), \(\\vec B\), and the propagation direction are mutually perpendicular.
- The fields are in phase for a traveling plane wave in vacuum.
- The speed from Maxwell's equations matches the measured speed of light.