Electromagnetic spectrum & polarization

Derive the wave equation from Maxwell's equations and out drops a single speed, c = 1/√(ε₀μ₀), together with a fixed geometry in which E, B and the propagation direction are mutually perpendicular. No frequency appears anywhere in that derivation. Any f solves the equations, so the spectrum is continuous and the band names are human conventions laid over it. In vacuum c = f λ converts between them: 1.0 MHz is 300 m, 2.45 GHz is 12.2 cm, 1.0 × 10¹³ Hz is 30 μm, 5.0 × 10¹⁴ Hz is 600 nm, 1.0 × 10¹⁸ Hz is 0.30 nm. Notice the scale. From long-wave radio near 10⁵ Hz to gamma rays above 10²⁰ Hz is fifteen decades, and the visible band, 400 nm to 700 nm, is less than one octave inside it. Note also that λ depends on the medium while f does not: light entering glass of index 1.50 keeps its frequency, because the driven charges oscillate at the drive frequency, and shortens its wavelength to λ/1.50.

A band is characterised by the motion it can drive, and matter responds strongly only when the drive lands near one of its own frequencies. Radio moves free electrons along conductors, which is why an efficient antenna is a fraction of a wavelength long. Microwaves at 2.45 GHz push on the permanent dipole of the water molecule faster than the surrounding liquid lets it turn, and the lag dumps energy as heat. Infrared near 10¹³ Hz matches molecular vibration: the bending mode of CO₂ sits at 667 cm⁻¹, which is 2.0 × 10¹³ Hz and λ = 15 μm, the wavelength it removes from outgoing thermal radiation. Visible and ultraviolet quanta of 1.8 eV to 12 eV move valence electrons between levels, which is colour and photochemistry. X-rays reach the core electrons, and at λ ≈ 0.1 nm they diffract from the atomic planes of a crystal.

Rank a band by what one quantum can do and the tool is E = hf, or E = hc/λ with the shortcut hc = 1240 eV nm. Red light at 700 nm is 1.77 eV, violet at 400 nm is 3.10 eV, a 250 nm germicidal line is 4.96 eV, and 124 nm is exactly 10.0 eV. Set those against the energies matter cares about: a C–C bond is about 3.6 eV and the first ionization energy of water is 12.6 eV. That comparison is where the non-ionizing boundary near 10 eV comes from, and it is a per-quantum statement, so intensity cannot move it. A 900 W oven at 2.45 GHz delivers 5.5 × 10²⁶ photons per second, each carrying 1.0 × 10⁻⁵ eV; a 4 W lamp at 250 nm delivers 5.0 × 10¹⁸ per second, each carrying 4.96 eV. The oven has 225 times the power and half a million times less energy per quantum.

Transversality leaves E free in a two-dimensional plane, so write it in components: E = Eₓ cos(kz − ωt) x̂ + E_y cos(kz − ωt + δ) ŷ. Two amplitudes and one relative phase δ describe every state there is. With δ = 0 the components rise and fall together and the tip of E runs back and forth along a fixed line tilted at arctan(E_y/Eₓ); with δ = π the y component is simply flipped in sign, so the line tilts the other way, at −arctan(E_y/Eₓ). Either value gives linear polarization. With δ = ±π/2 and Eₓ = E_y the tip travels a circle once per period, one way or the other according to the sign: circular polarization. Everything else is an ellipse. Thermal light from a filament or the Sun is none of these for long, because it arrives as an uncorrelated flood from enormous numbers of independent atoms; orientation and δ scramble on a nanosecond timescale, and averaged over any detection time the beam is unpolarized.

An ideal polarizing sheet transmits only the component of E along its transmission axis. Projection acts on amplitude, E = E₀ cosθ, and intensity goes as amplitude squared, which is Malus's law I = I₀ cos²θ. Feed it unpolarized light instead and every orientation occurs equally often; the average of cos²θ over all angles is ½, so the first sheet passes half whichever way it is turned, and what leaves it is polarized along its axis. Chain three sheets and the consequence is sharp. Crossed sheets at 0° and 90° pass nothing. Slide a third sheet at 45° between them and the output is ½ × ½ = ¼ of what left the first sheet, because the middle sheet does not merely subtract: light leaving it is polarized at 45°, and 45° is only 45° away from the analyzer.

Polarization is itself the evidence that the field is transverse. A longitudinal wave has no direction to choose across its line of travel, which is why sound cannot be polarized, and why rotating a sheet about the beam axis could never change what passes if E lay along the propagation direction. Two everyday effects confirm it. Light reflected near Brewster's angle, tan θ_B = n₂/n₁, which is 56.3° for air on glass of index 1.50, is strongly polarized parallel to the surface, and polarized sunglasses live on that fact. Sunlight scattered through 90° in the sky is polarized too, because the driven air molecules radiate nothing along their own oscillation direction. Two limits bound the model: real sheets absorb, so a Polaroid passes about 38% of unpolarized light rather than 50%, and no wave picture explains why absorption arrives in quanta of hf.