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🔋 Maxwell Waves — Electromagnetic Wave Propagation

An electromagnetic wave is a coupled oscillation of electric field E (red) and magnetic field B (blue), each perpendicular to the other and to the direction of propagation +z. This is the direct consequence of Maxwell's four equations — every photon of light is one of these waves.

Wavelength λ—
Frequency f—
Speed c3.00×10⁴ m/s
E₀/B₀= c

Polarisation

Wave type

Parameters

Stats

Wavelength λ—
Frequency f—
Wave speed c3.00×10⁴ m/s
E₀/B₀= c

Keyboard

P Pause / Play
R Reset

Maxwell's Equations → Wave Equation

Combining Faraday's law ∇×E = −∂B/∂t with Ampère-Maxwell's law ∇×B = μ₀ε₀ ∂E/∂t gives the wave equation ∂²E/∂z² = μ₀ε₀ ∂²E/∂t². The wave propagates at speed c = 1/√(ε₀μ₀) ≈ 3×10⁴ m/s — the speed of light in vacuum.

The ratio of electric to magnetic field amplitudes is always E₀/B₀ = c. Because c is so large, E-field amplitudes (V/m) are numerically much larger than B-field amplitudes (T) for the same wave.

Polarisation

In linear polarisation the E-field always points along a fixed axis (here vertical). In circular polarisation the E-field vector rotates as the wave travels — the tip traces a helix in space. Circular polarisation is the combination of two linear waves 90° out of phase. Wi-Fi antennas, satellite TV and many optical instruments exploit polarisation.

Electromagnetic Spectrum

All EM waves share the same c but differ in frequency: radio (kHz–GHz), microwave (GHz), infrared, visible light (400–700 THz), ultraviolet, X-rays, gamma rays. The relationship is always λ = c/f.

About this simulation

This simulation visualises an electromagnetic wave exactly as Maxwell's equations predict it: a coupled oscillation of the electric field E (red) and magnetic field B (blue), each perpendicular to the other and to the direction of propagation +z. Combining Faraday's law and the Ampère–Maxwell law yields a wave equation whose solutions travel at c = 1/√(ε₀μ₀) ≈ 3×10⁸ m/s — the speed of light in vacuum — with the field-amplitude ratio E₀/B₀ always equal to c.

🔬 What it shows

Red arrows trace the E field and blue arrows trace the B field along the propagation axis. Toggling Plane wave shows a wave of constant amplitude, while Spherical decay applies a 1/r-style envelope so the arrows shrink further from the source, mimicking how a real point-source wave spreads and weakens with distance.

🎮 How to use

Use the Linear / Circular buttons to switch polarisation, and Plane wave / Spherical decay to switch wave type. The Frequency f slider (1–10 units) changes how tightly packed the oscillations are and updates the live wavelength λ readout, while Amplitude A (0.2–2.0) scales the field strength. Pause (P) freezes the animation and Reset (R) restarts the phase from zero.

💡 Did you know?

Every colour of light, every radio signal and every X-ray is the same physical phenomenon — only the frequency f differs, with wavelength always given by λ = c/f. Circular polarisation, shown here as the E-field vector tracing a helix, is simply two linear waves 90° out of phase, and is exactly how many satellite-TV and Wi-Fi antennas are designed to reduce signal loss from orientation mismatch.

Frequently asked questions

Why are the E and B fields always perpendicular to each other?

This follows directly from Maxwell's equations. Faraday's law (∇×E = −∂B/∂t) and the Ampère–Maxwell law (∇×B = μ₀ε₀ ∂E/∂t) describe how a changing E field generates a curling B field and vice versa. Solving these together for a wave travelling along +z forces E and B to lie in the plane perpendicular to z, and to be perpendicular to each other — this is why electromagnetic waves are called transverse waves.

Why does the wave always travel at the same speed c?

Combining the two curl equations produces the wave equation ∂²E/∂z² = μ₀ε₀ ∂²E/∂t², whose propagation speed is fixed by the constants μ₀ (permeability of free space) and ε₀ (permittivity of free space): c = 1/√(ε₀μ₀) ≈ 3×10⁸ m/s. Because these are universal constants, every electromagnetic wave in vacuum — radio, light, X-rays — travels at exactly this speed, regardless of frequency.

What is the difference between linear and circular polarisation?

In linear polarisation the E-field oscillates back and forth along a single fixed axis as the wave travels. In circular polarisation the E-field vector instead rotates steadily as the wave advances, so its tip traces a helix in space — mathematically this is the sum of two linear waves of equal amplitude, 90° out of phase. The simulation's Linear and Circular buttons switch directly between these two cases.

What does the E₀/B₀ = c ratio mean physically?

It means the electric and magnetic field amplitudes are locked together: for any electromagnetic wave in vacuum, dividing the peak electric field (in V/m) by the peak magnetic field (in T) always gives the speed of light. Because c is such a large number, the numeric value of E is always much bigger than B for the same wave — they carry equal physical importance, just expressed in different units.

What does "spherical decay" mode represent?

A truly localised source, like an antenna or an atom emitting light, radiates energy outward in expanding wavefronts whose amplitude falls off with distance so that total energy is conserved over a growing sphere. The simulation approximates this with an envelope that shrinks as 1/(1 + z/scale) moving away from the source, in contrast to "Plane wave" mode which keeps amplitude constant everywhere, as an idealised approximation for a wave very far from its source or between parallel plates.