Plane Waves, Reflection and Skin Effect

Maxwell's equations predict waves. This lesson covers how plane waves travel, lose energy in conductors, and reflect at boundaries. The static and time-varying field laws are in Electromagnetic Fields.

Uniform plane waves

In a source-free medium, E and H satisfy the wave equation and travel together, perpendicular to each other and to the direction of travel (TEM).

• Phase velocity v = 1/√(με) = c/√(μ_r ε_r).
• Intrinsic impedance η = √(μ/ε); free space η₀ = 120π ≈ 377 Ω. In a dielectric with μ_r = 1: η = 377/√ε_r.
• Propagation constant γ = α + jβ, with β = ω√(με) = 2π/λ in a lossless medium.

Polarisation: linear (E stays on a line), circular (equal, 90° out-of-phase components), elliptical (general).

Lossy media and skin depth

• Loss tangent σ/(ωε): ≪ 1 for a good dielectric, ≫ 1 for a good conductor.
• In a good conductor: α = β = √(πfμσ); skin depth δ = 1/√(πfμσ). Fields fall to 1/e (37%) in one skin depth; at 1 MHz in copper, δ ≈ 66 µm.
• Surface resistance R_s = 1/(σδ).

Power flow

Poynting vector P = E × H. For a plane wave, the time-average power density = E₀²/(2η).

Reflection at normal incidence

From medium 1 (η₁) to medium 2 (η₂):

• Reflection coefficient Γ = (η₂ − η₁)/(η₂ + η₁); transmission coefficient τ = 1 + Γ = 2η₂/(η₂ + η₁).
• A perfect conductor (η₂ = 0) gives Γ = −1: a standing wave.
• Standing wave ratio SWR = (1 + |Γ|)/(1 − |Γ|).
• Reflected power fraction = |Γ|².

Oblique incidence

• Snell's law: n₁ sin θ₁ = n₂ sin θ₂ (n = √ε_r).
• Critical angle (from denser to rarer): sin θ_c = n₂/n₁; beyond it, total internal reflection (optical fibres).
• Brewster angle (parallel polarisation is fully transmitted): tan θ_B = n₂/n₁ = √(ε₂/ε₁).

Worked example

A plane wave goes from air into a lossless dielectric with ε_r = 4 at normal incidence.

• η₂ = 377/2 = 188.5 Ω; Γ = −1/3; reflected power = 1/9 = 11.1%; SWR = 2.

Practice