Electromagnetism · Upper Division

Electromagnetic Wave Propagation

Maxwell's equations predict self-sustaining oscillations of E and B that travel at c. This chapter examines those waves in detail: polarization, energy transport, radiation pressure, and the behavior at boundaries between media.

Prerequisites

Maxwell's equations (Ch. M) \cdot Wave properties (Ch. 8) \cdot Vectors and calculus

Learning Goals

Describe the structure of a plane electromagnetic wave and state the relationships between E, B, and k̂.
Distinguish linear, circular, and elliptical polarization and apply Malus's law to polarizer problems.
Calculate the time-averaged intensity and radiation pressure for an incident and reflected plane wave.
Derive the index of refraction from permittivity and permeability of a medium.
Apply the Fresnel equations at normal incidence to find reflectance at a dielectric interface.

W.1 Plane Wave Solutions

In free space (ρ = 0, J = 0), Maxwell's equations reduce to wave equations for both fields. The general monochromatic plane wave traveling in direction k̂ is:

E(r,t) = E_{0} \varepsilon̂ cos(k\cdotr - \omegat) \qquad B = (k̂ \times E)/c

(W.1)

Here ε̂ is the polarization unit vector(perpendicular to k̂). The constraints from Maxwell's equations require:

\omega = ck \qquad E ⊥ B ⊥ k̂ \qquad |B| = |E|/c

(W.2)

These are purely transverse waves — no longitudinal component. The E and B fields are in phase with each other (both peak at the same x and t) and mutually perpendicular, with B always smaller by a factor of c.

Figure W.1. Animated electromagnetic plane wave. Blue: E field (y-direction). Green: B field (x-direction, projected). The inset shows the polarization state end-on. Toggle polarization states and Poynting vector.

W.2 Polarization

Definition W.1 — States of Polarization

Linear polarization:

E oscillates along a fixed direction. \varepsilon̂ = constant

Circular polarization:

E rotates in the transverse plane at frequency \omega. Formed by two equal-amplitude linear com

ponents with 90° phase difference. \qquad E = E_{0}(x̂ cos(kz-\omegat) \pm ŷ sin(kz-\omegat

Elliptical polarization: The general case — two components with unequal amplitudes and arbitrary phase difference. Linear and circular are special cases.

Malus's law governs a polarizing filter: if light of intensity I₀ is linearly polarized at angle θ to a polarizer axis, the transmitted intensity is:

I = I_{0} cos^{2}\theta \qquad (Malus's law)

(W.3)

This follows directly from the E-field projection: |E_transmitted|² = |E₀ cos θ|², and intensity scales as E². Two crossed polarizers transmit nothing; a third at 45° in between restores partial transmission — a consequence of the projection law applied twice.

W.3 Energy, Momentum, and Radiation Pressure

Electromagnetic waves carry both energy and momentum. The energy flux density is the Poynting vector (from Maxwell's equations chapter):

S = (1/\mu_{0}) E \times B \qquad [W/m^{2}]

(W.4)

For a plane wave, |S| = E²/(μ₀c) = cε₀E². The time-averaged intensity is:

⟨I⟩ = \frac{1}{2}c\varepsilon_{0}E_{0}^{2} = E_{0}^{2}/(2\mu_{0}c)

(W.5)

Electromagnetic waves also carry momentum. The momentum density is:

g = S/c^{2} = \varepsilon_{0} E \times B \qquad [kg/(m^{2}\cdots)]

(W.6)

When a wave is absorbed by a surface, it deposits momentum, exerting a radiation pressure:

P_{rad} = I/c \qquad (absorbed) \qquad P_{rad} = 2I/c \qquad (reflected)

(W.7)

Example W.1 — Solar Radiation Pressure on a Sail

$The solar intensity at Earth's distance is I = 1361 W/m^{2}. Find the radiation pressure on a$ $perfectly reflecting solar sail of area A = 100 m^{2} and mass m = 5 kg$

Radiation pressure:

P_{rad} = 2I/c = 2 \times 1361 / (3\times10^{8}) = 9.07\times10^{-6} Pa

Force:

F = P_{rad} \times A = 9.07\times10^{-6} \times 100 = 9.07\times10^{-4} N

Acceleration:

a = F/m = 9.07\times10^{-4} / 5 = 1.81\times10^{-4} m/s^{2}

Perspective:This tiny acceleration, applied continuously without fuel for months, can accumulate to significant velocity changes. Solar sails are viable for deep-space propulsion — IKAROS (JAXA, 2010) demonstrated the principle.

W.4 Waves in Media — Index of Refraction

In a linear, isotropic medium (permittivity ε, permeability μ), the wave equation gives wave speed v and index of refraction n:

v = 1/\sqrt(\mu\varepsilon) \qquad n = c/v = \sqrt(\mu_{r}\varepsilon_{r}) \qquad (\mu_{r} = \mu/\mu_{0}, \varepsilon_{r} = \varepsilon/\varepsilon_{0})

(W.8)

At optical frequencies, μᵣ ≈ 1 for non-magnetic materials, so n ≈ √εᵣ. This connects the electromagnetic and optical properties: the refractive index is the square root of the relative permittivity.

Theorem W.1 — Fresnel Equations (Normal Incidence)

At a boundary between media n_{1} and n_{2}, the reflection and transmission amplitudes for a wa

ve at normal incidence are:

r = (n_{1} - n_{2})/(n_{1} + n_{2}) \qquad t = 2n_{1}/(n_{1} + n_{2}

Reflectance R = r^{2} = [(n_{1}-n_{2})/(n_{1}+n_{2})]^{2}. For glass (n=1.5) in air: R = (0.5/2.5)^{2} = 4%. An

ti-reflection coatings (quarter-wave layers) use destructive interference to eliminate this loss.

Example W.2 — Skin Depth in a Conductor

$In a good conductor (conductivity \sigma), EM waves decay exponentially. Find the skin depth \delta$ .

Wave equation in conductor:

\nabla^{2}E = \mu\sigma \partialE/\partialt + \mu\varepsilon \partial^{2}E/\partialt^{2} \approx \mu\sigma \partialE/\partialt (\sigma/\varepsilon ≫ \omega

Plane wave ansatz:

E \propto e^{ikz-i\omegat} with complex k^{2} = i\omega\mu\sigma

Solve:

k = (1+i)\sqrt(\omega\mu\sigma/2). Imaginary part gives exponential decay

Skin depth:

\delta = \sqrt(2/\omega\mu\sigma) — e-folding length for field amplitude

Example:

Copper at 60 Hz: \sigma = 6\times10^{7} S/m \to \delta = \sqrt(2/(2\pi\times60 \times 4\pi\times10^{-7} \times 6\times10^{7})) \approx 8.5 mm. At 1 GHz: \delta

\approx 2 \mum — only the surface layer carries current

Definition W.2 — Common Traps

B is smaller than E by c in SI units: compare physical field strengths with the correct unit conversion.
Unpolarized light loses half at the first ideal polarizer: Malus's law applies after a definite polarization direction exists.
Radiation pressure doubles on reflection: reflected light reverses momentum instead of merely depositing it.
Index depends on frequency in real media: dispersion means n is not a single constant for all wavelengths.

Exercises — W.1–W.4 EM Wave Propagation

The solar constant is I = 1361 W/m^{2}. Find E_{0} and B_{0} for sunlight. Verify using the Poyntin

g vector formula.

V/m

Straightforward

Unpolarized light passes through three polarizers at 0°, 45°, 90°. What fraction survives? Why does the middle polarizer matter?

Intermediate

3.Estimate the radiation pressure force on Earth from sunlight. Compare to the Sun's gravitational force on Earth. When does radiation pressure dominate?

Intermediate

Derive the radiation pressure formula P = I/c for an absorbed wave from the Maxwell stress

tensor. Explain the factor of 2 for reflection.

Challenging

Key Takeaways

$Plane wave: E ⊥ B ⊥ k̂, both transverse; |B| = |E|/c; \omega = ck$ .
Polarization: linear (fixed direction), circular (E rotates), elliptical (general case).
$Malus's law: I = I_{0} cos^{2}\theta for a polarizer at angle \theta$ .
$Poynting vector S = E\timesB/\mu_{0}; intensity ⟨I⟩ = E_{0}^{2}/(2\mu_{0}c) = \frac{1}{2}c\varepsilon_{0}E_{0}^{2}$ .
$Radiation pressure: P = I/c (absorbed), 2I/c (reflected). EM waves carry momentum g = S/c^{2}$ .
$In medium: v = c/n, n = \sqrt(\mu_{r}\varepsilon_{r}). Fresnel: R = [(n_{1}-n_{2})/(n_{1}+n_{2})]^{2} at normal incidence$ .