AcademyElectromagnetic Waves
Academy
Energy and Momentum in Electromagnetic Waves
Level 1 - Physics topic page in Electromagnetic Waves.
Principle
Electromagnetic waves carry energy and momentum. Their energy flow is described by the Poynting vector, and their momentum transfer produces radiation pressure.
Notation
\(u\)
electromagnetic energy density
\(\mathrm{J\,m^{-3}}\)
\(\vec S\)
Poynting vector
\(\mathrm{W\,m^{-2}}\)
\(I\)
average intensity
\(\mathrm{W\,m^{-2}}\)
\(p_{\mathrm{rad}}\)
radiation pressure
\(\mathrm{Pa}\)
\(A\)
area receiving the wave
\(\mathrm{m^{2}}\)
Method
Derivation 1: Energy density
Electric and magnetic fields both store energy.
Electric contribution
\[u_E=\frac12\epsilon_0E^2\]
Magnetic contribution
\[u_B=\frac{B^2}{2\mu_0}\]
Plane wave
\[u=u_E+u_B=\epsilon_0E^2=\frac{B^2}{\mu_0}\]
Derivation 2: Energy flow
The Poynting vector points in the direction of energy transport.
Poynting vector
\[\vec S=\frac{1}{\mu_0}\vec E\times\vec B\]
Instantaneous magnitude
\[S=\frac{EB}{\mu_0}=c\epsilon_0E^2\]
Derivation 3: Average intensity and pressure
For sinusoidal waves, average over a full cycle.
Average intensity
\[I=\langle S\rangle=\frac12c\epsilon_0E_0^2=\frac{E_0B_0}{2\mu_0}\]
Absorbing surface
\[p_{\mathrm{rad}}=\frac{I}{c}\]
Reflecting surface
\[p_{\mathrm{rad}}=\frac{2I}{c}\]
Rules
Poynting vector
\[\vec S=\frac{1}{\mu_0}\vec E\times\vec B\]
Average intensity
\[I=\frac12c\epsilon_0E_0^2\]
Power on area
\[P=IA\]
Radiation pressure
\[p_{\mathrm{rad}}=\frac{I}{c}\ \text{or}\ \frac{2I}{c}\]
Examples
Question
A beam has intensity
\[800\,\mathrm{W\,m^{-2}}\]
and hits \[0.20\,\mathrm{m^2}\]
Find the power delivered.Answer
\[P=IA=(800)(0.20)=160\,\mathrm{W}\]
Checks
- Intensity is average power per unit area.
- A perfect reflector receives twice the pressure of a perfect absorber at the same intensity.
- In a plane wave, electric and magnetic energy densities are equal at every instant.
- Momentum transport matters even when the pressure is very small.