Electromagnetic waves are self-sustaining oscillations of coupled electric (E) and magnetic (B) fields that propagate through space without requiring a medium. Maxwell's four equations predict that c = 1/√(ε₀μ₀) ≈ 3 × 10⁴ m/s in vacuum, unifying optics and electromagnetism. Radio waves, microwaves, visible light, X-rays, and gamma rays are all electromagnetic waves differing only in frequency, spanning over 24 orders of magnitude.
This simulation lets you adjust polarisation direction, frequency, and medium permittivity to visualise how the transverse E and B fields oscillate in phase, perpendicular to each other and to the propagation direction. You can compare phase velocity with group velocity in dispersive media.
Why do E and B fields oscillate perpendicular to each other?
Maxwell's curl equations require it: a changing E field produces a B field perpendicular to it (Ampere-Maxwell law), and a changing B field produces an E field perpendicular to it (Faraday's law). In a plane wave both fields must also be perpendicular to the propagation direction, making electromagnetic waves transverse.
What is the speed of light and how is it derived?
The speed of light in vacuum is exactly c = 299,792,458 m/s (by definition since 1983). Maxwell derived it theoretically as c = 1/√(ε₀μ₀), where ε₀ = 8.854 × 10⁻¹² F/m and μ₀ = 4π × 10⁻⁷ H/m. The close match with the measured speed of light led Maxwell to identify light as an electromagnetic wave.
What is polarisation?
Polarisation describes the orientation of the electric field vector. In linear polarisation the E field oscillates in a fixed plane; in circular polarisation it rotates at the wave frequency, tracing a helix. Polarisation is exploited in LCD screens, polarising sunglasses, radio antennas, and 3D cinema glasses.
Phase velocity (vp = ω/k) is the speed at which a single-frequency wave crest moves. Group velocity (vg = dω/dk) is the speed at which the envelope of a wave packet—and thus information or energy—travels. In vacuum both equal c. In a dispersive medium they differ; in a waveguide, phase velocity can exceed c without violating relativity because information travels at group velocity.
In a dielectric with relative permittivity εr, the wave speed becomes v = c/√(εr) = c/n, where n is the refractive index. Glass has n ≈ 1.5, so light travels at about 2 × 10⁴ m/s inside it. This slowing, combined with constant frequency, causes the wavelength to shorten, which is the origin of refraction and total internal reflection.
Energy flux is described by the Poynting vector S = (1/μ₀)(E × B) in W/m². The time-averaged irradiance of a plane wave is I = ½cε₀E₀², where E₀ is the peak electric field amplitude. Near Earth, sunlight delivers about 1361 W/m² (the solar constant), corresponding to E₀ ≈ 1011 V/m.
All EM waves travel at c in vacuum, but their interaction with matter depends on frequency. Radio waves (kHz–GHz, wavelengths from km to cm) diffract around obstacles and reflect off the ionosphere, enabling long-range communication. Higher-frequency light is scattered by atmospheric particles and absorbed or reflected by solid objects, limiting practical range.
The spectrum spans from radio waves (below 300 MHz, wavelengths above 1 m) through microwaves, infrared, visible light (400–700 nm), ultraviolet, X-rays, to gamma rays (above 10¹⁹ Hz). Visible light occupies less than one octave of this vast range. Medical X-rays typically use photon energies of 20–150 keV; gamma rays from nuclear reactions can exceed 1 MeV.
Accelerating electric charges emit electromagnetic radiation. In a radio antenna, alternating current accelerates electrons back and forth, radiating at the drive frequency. Synchrotron sources accelerate electrons to near-light speeds in circular paths, producing intense tunable X-ray beams. Thermal radiation from warm objects arises from the random acceleration of charged particles.
For EM waves: c = fλ, so wavelength λ = c/f. A photon's energy is E = hf = hc/λ, where h = 6.626 × 10⁻³⁴ J·s (Planck's constant). A green photon at 550 nm has energy ≈ 2.25 eV, while an X-ray photon at 0.1 nm has energy ≈ 12.4 keV—about 5500 times more energetic.