UY1: Energy & Momentum In Electromagnetic Waves

This page gives the UY1 working model/result for Energy & Momentum In Electromagnetic Waves. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

Module path: Electromagnetism (UY1)
Practice: UY1 Electromagnetism Quiz
Full routing: UY1 Assessment Map
Math toolkit: Mathematics for Undergraduate Physics

1) At a glance

EM waves transport both energy and momentum.
Vacuum energy density:

u = frac12ε₀E² + B²/2μ₀.

Energy flux (Poynting vector):

vec S = 1/μ₀vec E × vec B.

For a sinusoidal plane wave, the cycle-average intensity is

I = langle Srangle = frac12 ε₀ c E₀² = E₀B₀/2μ₀.

Radiation pressure for normal incidence:

pᵣₐd = I/c(absorption), pᵣₐd = 2I/c(perfect reflection).

Direction check: vec S points along the propagation direction (right-handed vec E,vec B,hatk).

Prerequisites: Electromagnetic Spectrum & Sinusoidal EM Plane Waves
Next uses: Standing Electromagnetic Waves

2) Setup

Consider a sinusoidal plane wave in vacuum.

vec Eperpvec B,
E = cB,
propagation direction along vec E × vec B.

Use cycle averages for practical power/intensity values.

Be clear whether you are using instantaneous fields or amplitudes (E₀,B₀) or RMS values. Intensity uses a cycle average.
Don’t drop the factor of 1/2 in I = tfrac12ε₀cE₀² for a sinusoid.
Pressure uses intensity: p = I/c for absorption and p = 2I/c for perfect reflection (because momentum change doubles).

3) Core derivation/explanation

In vacuum, electric and magnetic energy densities are equal on average:

langle uErangle = langlefrac12ε₀E²rangle, langle uBrangle = langleB²/2μ₀rangle, langle uErangle = langle uBrangle.

Instantaneous Poynting vector:

vec S = 1/μ₀vec E × vec B.

Its cycle average magnitude is intensity:

I = langle Srangle = E₀B₀/2μ₀ = frac12ε₀cE₀².

Momentum density:

vec g = vec S/c²,

so momentum flow causes measurable radiation pressure.

Units checks:

S in W m⁻²,
I/c in N m⁻² (Pa), matching pressure units.

Checks (sanity)

For a vacuum plane wave, langle uErangle = langle uBrangle and I = langle urangle c.
If you reverse the direction of propagation, vec S reverses because vec E × vec B reverses.

4) Worked example(s)

A plane wave has electric amplitude E₀ = 120 V m⁻¹.

Intensity:

I = frac12ε₀cE₀² = frac12(8.85 × 10⁻¹²)(3.00 × 10⁸)(120)² ≈ 19.1 W m⁻².

Radiation pressure:

perfect absorber:

pᵣₐd = I/c ≈ 6.4 × 10⁻⁸ Pa;

perfect reflector:

pᵣₐd = 2I/c ≈ 1.27 × 10⁻⁷ Pa.

5) Practice set (with hints + answers)

If E₀ doubles, what happens to intensity? Hint: Ipropto E₀². Answer: it quadruples.
A beam has I = 500 W m⁻². Find pressure for perfect absorption. Hint: divide by c. Answer: 1.67 × 10⁻⁶ Pa.
Why is radiation pressure from reflection twice that from absorption at same intensity? Hint: compare momentum change signs. Answer: reflected photons reverse momentum, doubling transfer.

6) Summary + next steps

Poynting vector gives direction and rate of EM energy transport.
EM waves carry momentum with density vec S/c².
Radiation pressure is small but crucial in optics, astrophysics, and solar sails.

Next: Standing Electromagnetic Waves Previous: Electromagnetic Spectrum & Sinusoidal EM Plane Waves Back To Electromagnetism (UY1)