EM Fields & Wave Propagation

An electromagnetic wave needs no medium — it carries itself. A changing electric field creates a magnetic field, whose change recreates the electric field, and the pair leapfrogs through space at the speed of light. The two fields sit at right angles, oscillate in step, and their cross-product points the way the energy flows. Set the frequency and the medium below and watch the geometry, speed, and power respond.

Wave
Frequency f
Field amplitude E₀
Medium
Presets
Permittivity εr
Permeability µr
Conduction
Conductivity σ
Playback
Phase / time
Transverse Wave — E ⊥ B
—
Electric field E Magnetic field B → Propagation / Poynting S = E×H
Fields & Power Flow vs. Position
E / E₀ H / H₀ Poynting S (power)
Readout

The Physics

Maxwell's coupling. Two of Maxwell's equations make a wave possible. Faraday's law says a changing magnetic field curls up an electric field; the Ampère–Maxwell law says a changing electric field (the displacement current) curls up a magnetic field. Each field regenerates the other, so the disturbance propagates on its own:

∇×E = −∂B/∂t  ·  ∇×H = ∂D/∂t  →  ∂²E/∂z² = µε·∂²E/∂t²

The wave equation's speed falls straight out of the material constants. In a vacuum it's c; a medium with relative permittivity εr and permeability µr slows it and shrinks the wavelength:

v = 1/√(µε) = c/√(εrµr)  ·  λ = v/f  ·  n = √(εrµr)

The two fields aren't independent — their ratio is the medium's intrinsic impedance η, the free-space value being 377 Ω. And their cross-product is the Poynting vector, the power crossing a unit area in the propagation direction:

E/H = η = √(µ/ε) = 377Ω·√(µr/εr)  ·  S = E×H  ·  Savg = ½·E₀²/η
📡 Power-system tie-ins. The same physics sets the velocity factor of cables — a signal in XLPE (εr≈2.3) travels at ~66% of c, which is what traveling-wave fault locators time. A transmission line's surge impedance is the wave-impedance idea applied to a guided wave (~400 Ω overhead, ~40 Ω in cable).
🛡 Conductors & skin depth. Switch the medium to Conductive. In a good conductor the wave is gobbled up as it enters: the amplitude decays as e−z/δ over the skin depth δ = √(2/ωµσ), and E and H fall ~45° out of phase. That's why fields don't penetrate metal — the basis of shielding, and why AC current crowds onto a conductor's surface (skin effect).