In short

Every electromagnetic wave in vacuum has a frequency f and a wavelength \lambda related by

\boxed{\;c \;=\; f\lambda,\qquad c \;=\; 2.998\times 10^8\text{ m/s}\;}

Change f by twenty orders of magnitude — from a 50 Hz power-line hum to a 10^{22} Hz gamma ray — and nothing about the underlying physics changes. Only the human names for the bands, the technology that produces them, and how they interact with matter.

The bands, from lowest frequency to highest.

Band f range \lambda range What makes it What detects it
Radio 3 kHz – 300 MHz 1 km – 1 m Oscillating current in an antenna LC-tuned receiver
Microwave 300 MHz – 300 GHz 1 m – 1 mm Klystron, magnetron Semiconductor diode
Infrared 300 GHz – 400 THz 1 mm – 750 nm Hot body (Wien) Thermopile, bolometer, IR CCD
Visible 400 THz – 790 THz 750 nm – 380 nm Electronic transitions Retina, silicon photodiode
Ultraviolet 790 THz – 30 PHz 380 nm – 10 nm Hot stars, arc discharges Photomultiplier, CCD
X-ray 30 PHz – 30 EHz 10 nm – 10 pm Inner-shell transitions, bremsstrahlung Gas proportional, scintillator
Gamma ray > 30 EHz < 10 pm Nuclear transitions, annihilation Germanium detector, scintillator

Photon energy. Each oscillation delivers a quantum of energy

\boxed{\;E \;=\; hf \;=\; \frac{hc}{\lambda}\;}

with h = 6.626\times 10^{-34} J·s. Radio photons carry 10^{-26} J each (millionths of an eV) — too small to break chemical bonds. X-ray photons carry 10^{-16} J (around 10 keV) — enough to knock electrons out of atoms, which is why X-rays are ionising and radio waves are not.

All EM waves travel at c in vacuum. They slow down in matter by the refractive index n, and the frequency stays fixed while \lambda \to \lambda/n. But in vacuum — the empty air between your phone and the nearest tower, the vacuum of space — every band travels at exactly the same speed, predicted by Maxwell's equations alone: c = 1/\sqrt{\mu_0\varepsilon_0}.

Tune a car radio in Bengaluru across the FM band and a spectrum rolls past: 91.1 FM Radio City, 93.5 Red FM, 98.3 Radio Mirchi, 104.8 Fever. Each station occupies a narrow slice of frequency around 100 MHz — radio waves, \lambda \approx 3 metres. Now point a TV remote at a friend: infrared pulses at \lambda \approx 940 nm (just beyond what your eye can see) command the set-top box. Stand in a patch of afternoon sun at Cubbon Park: visible light, \lambda \approx 500 nm, the wavelength that evolution tuned your eyes to. Stand in the shade of a neem tree and you still feel infrared from the sun's warmth, heating your skin. Walk into a hospital and an AIIMS radiologist images your chest: X-rays, \lambda \approx 0.1 nm. Go into the basement of the BARC Pelletron accelerator: gamma rays from fusion reactions, \lambda \approx 10^{-13} m.

These wildly different things — radio, IR, visible, X-ray — are the same kind of wave. They differ only in how fast the electric and magnetic fields oscillate. Stretch or compress the same wave along the frequency axis and you get the whole spectrum, from 50 Hz mains hum to 10²² Hz gamma rays, in one continuous family. This chapter maps that family, shows what produces each band, what detects it, and why the Indian grid at 50 Hz and the AIIMS radiology department at 10¹⁸ Hz are running on identical physics scaled across twenty decades.

Why every band travels at c

The displacement current article shows that Maxwell's equations combined give a wave equation for \vec{E} and \vec{B} with speed

c \;=\; \frac{1}{\sqrt{\mu_0\varepsilon_0}}

Step 1. Plug in the constants. \mu_0 = 4\pi\times 10^{-7} T·m/A; \varepsilon_0 = 8.854\times 10^{-12} C²/(N·m²).

\mu_0\varepsilon_0 \;=\; (4\pi\times 10^{-7})(8.854\times 10^{-12}) \;=\; 1.113\times 10^{-17}\text{ s}^2/\text{m}^2

Step 2. Take the reciprocal square root.

c \;=\; \frac{1}{\sqrt{1.113\times 10^{-17}}} \;=\; \frac{1}{3.336\times 10^{-9}} \;=\; 2.998\times 10^{8}\text{ m/s}

Why: this is the predicted wave speed from electromagnetism alone. Neither of these constants was measured by timing a light beam — \mu_0 came from measurements of magnetic force between currents, \varepsilon_0 from measurements of electric force between charges. Yet their combination matches the observed speed of light to six decimal places. Light is an electromagnetic wave.

Step 3. Notice what is not in the formula. The wave speed does not depend on frequency, amplitude, polarisation, or direction. A 50 Hz hum from a transformer and a 10^{20} Hz gamma ray from a thorium decay travel through vacuum at exactly the same speed. Any band of the spectrum obeys

\boxed{\;c \;=\; f\lambda\;}

Fix c (a universal constant). Then high frequency means short wavelength, and low frequency means long wavelength, across the whole spectrum.

Explore: wavelength, frequency, and photon energy across the spectrum

Drag the wavelength slider from a kilometre (long-wave radio) down to a picometre (gamma) and watch how frequency and photon energy track together.

Interactive wavelength slider across the electromagnetic spectrum A horizontal axis showing log10 of wavelength in metres from minus 12 (gamma rays) to plus 3 (long-wave radio). Coloured bands mark each region: gamma ray, X-ray, ultraviolet, visible, infrared, microwave, radio. Drag the red point along the axis to read off wavelength in metres, frequency in hertz, and photon energy in electron-volts. log₁₀(λ / metre) — drag the red point γ X-ray UV V IR microwave radio −12 −9 −6 −3 0 +3 1 pm BARC γ 0.1 nm AIIMS X-ray 500 nm sunlight 10 μm body heat 12 cm DTH 3 m FM radio 300 m AIR AM drag to scan the spectrum
Slide through the spectrum. At $\lambda = 500$ nm (visible green), $f \approx 6\times 10^{14}$ Hz, $E \approx 2.5$ eV — exactly enough to excite chlorophyll for photosynthesis. At $\lambda = 0.1$ nm (medical X-ray), $f \approx 3\times 10^{18}$ Hz, $E \approx 12$ keV — enough to ionise atoms, which is why X-ray technicians stand behind lead screens.

The relation c = f\lambda ties the two axes; the photon-energy formula E = hc/\lambda maps wavelength to the energy each quantum of light carries. Every useful property of each band — whether it can penetrate your skin, what detector to use, what source produces it — falls out of these two simple formulas applied at the right \lambda.

A band-by-band tour

Radio waves (3 kHz – 300 MHz; \lambda = 1 km – 1 m)

Source. An alternating current in a wire — literally, an LC oscillator driving an antenna. The All India Radio medium-wave transmitter at Rampur, Uttar Pradesh, running at 630 kHz (\lambda \approx 476 m), feeds a 1-megawatt oscillator into a vertical mast a quarter-wavelength tall (about 119 m). The oscillating current launches radio waves that curve around the earth and cover most of northern India.

Detector. A coil-and-capacitor (LC) tuned circuit resonant at the station's frequency. When a passing radio wave oscillates the electric field across the antenna, electrons in the antenna oscillate at the same frequency. The LC circuit in the receiver selects the wanted station and rejects the rest (see LCR resonance for exactly how that selection works).

Uses.

Sub-band Frequency Use
Long-wave, MW 30 kHz – 3 MHz AIR AM services, marine
Short-wave 3 – 30 MHz AIR External Services, amateur radio
VHF 30 – 300 MHz FM radio (88–108 MHz), VHF TV, AIR FM

Physics note. Radio waves have photon energy E = hf around 10^{-25} J for MHz frequencies — billions of times smaller than thermal kT at room temperature (4 × 10⁻²¹ J). Radio-frequency radiation cannot break chemical bonds or ionise atoms; this is why a 1-megawatt radio tower and a 5-watt mobile phone are considered non-ionising.

Microwaves (300 MHz – 300 GHz; \lambda = 1 m – 1 mm)

Source. A magnetron (the device inside every microwave oven) is the cheap workhorse: cavity resonators in a cylindrical anode, electrons spiralling in an axial magnetic field, resonating at 2.45 GHz. For satellite and radar applications, klystrons and solid-state GaAs-based oscillators take over.

Detector. A semiconductor diode (Schottky barrier) rectifies the microwave, converting the high-frequency signal to a low-frequency or DC signal you can measure.

Uses.

Physics note. At microwave frequencies (10^{10} Hz), E = hf \approx 10^{-5} eV — still vastly below ionisation energies (typically eV). Microwave heating is thermal, not chemical.

Infrared (300 GHz – 400 THz; \lambda = 1 mm – 750 nm)

Source. Any warm body. A blackbody at temperature T emits a continuous spectrum peaking at wavelength \lambda_\text{max} \approx 2.9\times 10^{-3}/T metres (Wien's displacement law). Your body, at 310 K, peaks at \lambda \approx 9 μm — mid-infrared. A 1500 K incandescent bulb filament peaks at 2 μm — near-infrared. The Sun's 5800 K surface peaks at 500 nm — visible, with strong tails into IR and UV.

Detector. Thermal detectors (thermopiles, bolometers, microbolometer arrays in cheap thermal-imaging cameras). Photon detectors (InGaAs photodiodes for near-IR, HgCdTe for mid-IR, doped silicon for far-IR).

Uses.

Physics note. Infrared energy "makes molecules vibrate rather than rotate" — the quanta at IR frequencies (10^{-1} eV) match molecular vibrational-mode spacings, which is why IR spectroscopy is the chemist's fingerprint technique.

Visible light (400 THz – 790 THz; \lambda = 750 nm – 380 nm)

Source. Electronic transitions in atoms. A tungsten filament, a sodium-vapour street lamp (monoenergetic yellow at 589 nm — the Fraunhofer D line), a fluorescent tube (mercury UV emission converted to visible by the phosphor coating), an LED (band-gap recombination in a III-V semiconductor).

Detector. The retina of your eye — rods and cones containing rhodopsin, tuned precisely to this narrow octave. Silicon photodiodes, CCD and CMOS sensors in cameras, photomultiplier tubes in laboratories.

The visible octave. From 380 to 780 nm spans about one octave — a factor of 2 in wavelength, matching exactly the range where water is transparent and sunlight peaks. Evolution designed the eye where the signal was strongest and the atmosphere cleanest.

The visible spectrum expanded A horizontal bar showing the visible spectrum from 380 nm violet to 750 nm red. Labelled regions: violet, blue, cyan, green, yellow, orange, red. Wavelengths in nanometres. Visible spectrum — the one octave the human eye evolved to see 380 450 495 570 590 620 750 wavelength (nm) in vacuum violet blue green Y O red f ≈ 790 THz f ≈ 400 THz
The visible octave, roughly 380–750 nm. Indian sunlight at noon in Bengaluru peaks near 500 nm — the green the eye is most sensitive to. The narrow ranges of yellow and orange between green and red are where sodium (589 nm) and mercury lines land.

Indian connection. C. V. Raman's 1928 discovery of the Raman effect was the observation that when monochromatic light (he used a narrow-band mercury line, ~546 nm) scatters off molecules, a small fraction emerges at shifted wavelengths corresponding to molecular vibrational modes. Raman won the 1930 Nobel Prize in Physics for it — India's first Nobel in the sciences, and still the only unaided-instrument Nobel awarded for work done entirely in India. Raman scattering is today the basis of non-destructive molecular identification, used in pharmaceutical quality control, art conservation at the National Museum in Delhi, and lunar mineralogy on Chandrayaan-3's Vikram lander.

Ultraviolet (790 THz – 30 PHz; \lambda = 380 nm – 10 nm)

Source. Hotter bodies (stars hotter than the Sun), electric arcs (mercury vapour in germicidal lamps), plasmas, some LEDs.

Detector. Photomultiplier tubes with UV-transparent windows (quartz), solar-blind silicon carbide or GaN photodiodes, CCDs with UV-enhanced coatings.

Sub-bands.

Physics note. UV photon energies (3–10 eV) overlap the energies of outer-shell electrons in atoms, which is why UV excites fluorescence (banknote security features glowing under a UV lamp) and ionises some molecules (why UV-C sterilises water — it cracks DNA).

X-rays (30 PHz – 30 EHz; \lambda = 10 nm – 10 pm)

Source. Inner-shell electron transitions (characteristic K-alpha lines of copper at 0.154 nm, the standard X-ray diffraction wavelength) and bremsstrahlung — the continuous X-ray spectrum emitted when high-energy electrons are abruptly decelerated (typically by slamming into a tungsten target inside a sealed glass tube). An AIIMS chest X-ray machine accelerates electrons through 80 kV onto a tungsten anode; the electrons decelerate in the metal and emit a broad continuum peaked near 30 keV, corresponding to 0.04 nm wavelength.

Detector. Historically, photographic film (still common in some rural Indian PHCs). Now, scintillator-coupled photomultipliers, gas proportional counters, and silicon or germanium solid-state detectors.

Uses.

Physics note. X-rays at 10 keV carry enough energy to eject inner-shell electrons from atoms (hence "ionising radiation"). Cumulative exposure damages DNA, which is why a CT scan is prescribed only when needed and radiologists stand behind a lead-lined wall.

Gamma rays (> 30 EHz; \lambda < 10 pm)

Source. Nuclear transitions (cobalt-60's 1.17 and 1.33 MeV lines, used in cancer radiotherapy at AIIMS and Tata Memorial Hospital), electron-positron annihilation (511 keV line), and astrophysical sources (pulsars, gamma-ray bursts from distant galaxies).

Detector. Scintillator crystals (sodium iodide doped with thallium) coupled to photomultipliers, high-purity germanium solid-state detectors for precision spectroscopy.

Uses.

Physics note. A gamma-ray photon carries MeV of energy — enough to break apart a nucleus (photodisintegration) or create an electron-positron pair in the field of a nucleus (pair production, above 1.022 MeV threshold). This is the extreme end of the spectrum where the wave picture starts to fail and the particle picture (photon) becomes essential.

The unified picture

The full electromagnetic spectrum in one picture Horizontal spectrum bar with seven labelled regions from gamma (left) to radio (right), with frequency and wavelength scales below. Sample sources and detectors annotated for each band. The electromagnetic spectrum — logarithmic, with a few Indian examples γ-ray X-ray UV visible IR microwave radio 10²⁰ Hz 10¹⁶ Hz 10¹³ Hz 10¹⁰ Hz 10⁷ Hz frequency → 10⁻¹² m 10⁻⁸ m 10⁻⁵ m 10⁻² m 10 m ← wavelength AIIMS γ-ther AIIMS X-ray UV water filter sunlight TV remote Tata Play DTH AIR, FM MeV keV 10 eV 2 eV 0.1 eV μeV neV photon energy E = hf ionising ← → non-ionising
The full spectrum on logarithmic axes. Every band is the same kind of wave — oscillating $\vec{E}$ and $\vec{B}$ at speed $c$. The dashed line near UV marks the rough boundary where photon energy exceeds atomic ionisation thresholds (a few eV). Radio, microwave, and IR are non-ionising; UV, X-ray, and gamma are ionising.

The single most important idea in this chapter: the physics is the same across 20+ decades of frequency. The equations (c = f\lambda, E = hf, Maxwell's four equations) are identical whether you are designing an FM radio or a gamma-ray telescope. What differs is the detector, the source, and the wavelength-dependent interactions with matter (which determine whether the wave reflects, refracts, absorbs, or ionises).

Worked examples

Example 1: Photon energy — FM radio vs medical X-ray

An FM radio station (Radio Mirchi 98.3) broadcasts at f_1 = 98.3 MHz. A chest X-ray machine at AIIMS emits photons at roughly \lambda_2 = 0.10 nm (corresponding to ~12 keV). Compute the photon energy in eV for each, and the ratio.

Step 1. Radio photon energy.

E_1 \;=\; hf_1 \;=\; (6.626\times 10^{-34})(98.3\times 10^{6})
E_1 \;=\; 6.51\times 10^{-26}\text{ J}

Convert to eV by dividing by 1.602\times 10^{-19} J/eV:

E_1 \;=\; \dfrac{6.51\times 10^{-26}}{1.602\times 10^{-19}} \;=\; 4.07\times 10^{-7}\text{ eV} \;=\; 0.41\,\mu\text{eV}

Why: the FM photon carries less than a microvolt of energy. Thermal energy at room temperature is kT \approx 25 meV — about 6\times 10^{10} times larger. An FM photon lands on a molecule like a snowflake on a warm highway.

Step 2. X-ray photon energy.

E_2 \;=\; \dfrac{hc}{\lambda_2} \;=\; \dfrac{(6.626\times 10^{-34})(2.998\times 10^{8})}{0.10\times 10^{-9}}
E_2 \;=\; 1.986\times 10^{-15}\text{ J} \;=\; \dfrac{1.986\times 10^{-15}}{1.602\times 10^{-19}} \;\approx\; 12{,}400\text{ eV} \;=\; 12.4\,\text{keV}

Step 3. Ratio.

\dfrac{E_2}{E_1} \;=\; \dfrac{1.986\times 10^{-15}}{6.51\times 10^{-26}} \;\approx\; 3.05\times 10^{10}

Result. The X-ray photon is about 3\times 10^{10} times more energetic than the FM photon. Same kind of wave, thirty billion times the per-photon energy.

What this shows. Energy per photon, not total power, is what determines whether radiation is dangerous. A 50 kW FM transmitter pumps huge power into the air but each photon is harmless. A 10 mA X-ray tube at 100 kV emits far less total power but each photon can smash a DNA strand. This is why medical dosimetry is done in energy-per-mass (grays, sieverts), not in watts.

Example 2: Sizing an AIR medium-wave antenna

All India Radio Rohtak broadcasts on 594 kHz from a single vertical tower. For efficient radiation, a vertical monopole antenna should be a quarter of a wavelength tall. Find the wavelength, the required tower height, and the time it takes the radio wave to travel from Rohtak to a receiver in Delhi, 70 km away.

Step 1. Wavelength from c = f\lambda.

\lambda \;=\; \dfrac{c}{f} \;=\; \dfrac{2.998\times 10^{8}\text{ m/s}}{594\times 10^{3}\text{ Hz}} \;\approx\; 504.7\text{ m}

Why: low-frequency radio means long wavelength — about half a kilometre in this case. That is why AM broadcast antennas are towers, not the handheld rods of FM radios.

Step 2. Quarter-wave monopole height.

h \;=\; \dfrac{\lambda}{4} \;=\; \dfrac{504.7}{4} \;\approx\; 126\text{ m}

Why: a vertical monopole fed against a ground plane acts as a half-wave dipole when you mirror it in the ground. A quarter-wave height produces a standing-wave current pattern with a maximum at the base and a node at the top — the efficient radiation configuration.

Step 3. Propagation time to Delhi.

t \;=\; \dfrac{d}{c} \;=\; \dfrac{70\times 10^{3}\text{ m}}{2.998\times 10^{8}\text{ m/s}} \;=\; 2.34\times 10^{-4}\text{ s} \;\approx\; 234\,\mu\text{s}

Step 4. For comparison, sound in air would take

t_\text{sound} \;=\; \dfrac{70{,}000}{343} \;\approx\; 204\text{ s} \;\approx\; 3.4\text{ min}

Result. \lambda \approx 505 m; h \approx 126 m; light-travel time 234 μs, about 870,000 times faster than sound.

What this shows. The physical size of every radio tower encodes the frequency it works at. The tall AIR medium-wave masts (100+ m) are needed because MW wavelengths are long; FM towers are much shorter because FM wavelengths are only a few metres; mobile-phone base-station antennas are a few centimetres because 2 GHz wavelengths are 15 cm. The antenna has to be roughly the size of the wave it radiates.

Example 3: Wien's law — the colour of a Diwali candle vs the Sun

A Diwali diya burns at about T_1 = 1500 K. The Sun's surface is at T_2 = 5778 K. Find the wavelength of peak emission for each, and classify which band each peaks in.

Step 1. Wien's displacement law.

\lambda_\text{max} \;=\; \dfrac{2.898\times 10^{-3}\text{ m·K}}{T}

Why: Wien's law comes from maximising the Planck blackbody emission formula with respect to wavelength; the constant 2.898 mm·K falls out of that calculus.

Step 2. Diya peak.

\lambda_{1,\text{max}} \;=\; \dfrac{2.898\times 10^{-3}}{1500} \;=\; 1.93\times 10^{-6}\text{ m} \;=\; 1.93\,\mu\text{m}

This is near-infrared — invisible to your eye. What makes the diya look yellow-orange is the tail of the blackbody curve that extends into the visible, not the peak. Most of the candle's power is going into heat you feel on your hand, not light you see with your eye.

Step 3. Sun peak.

\lambda_{2,\text{max}} \;=\; \dfrac{2.898\times 10^{-3}}{5778} \;=\; 5.02\times 10^{-7}\text{ m} \;=\; 502\text{ nm}

This is green-cyan — the middle of the visible band. The Sun's peak coincides almost exactly with the peak sensitivity of the human eye at 555 nm.

Step 4. Compare.

Ratio of peak wavelengths: \lambda_1/\lambda_2 = 1930/502 \approx 3.85. Ratio of temperatures: T_2/T_1 = 5778/1500 \approx 3.85. Wien's law predicts \lambda_\text{max}\propto 1/T, and the ratios match perfectly.

Result. Diya peaks at 1.93 μm (near-IR); Sun peaks at 502 nm (visible green); the ratio is exactly the inverse temperature ratio.

What this shows. Your eye's sensitivity is not a coincidence — evolution tuned the visible octave to where the Sun has most of its power. A candle is warm but dim because most of its energy leaves as infrared that you don't see. A tungsten bulb at 2700 K splits the difference: peak at 1.07 μm (IR), but enough visible tail to act as a (very inefficient) light source — which is why LEDs at 25% electrical-to-visible efficiency replaced tungsten bulbs at 3%.

Common confusions

If you can convert \lambda \leftrightarrow f \leftrightarrow E across the spectrum and name each band's sources and detectors, you have the JEE-level content. The deeper treatment addresses the Poynting vector, polarisation, wave momentum, and how the speed of EM waves in a medium relates to the vacuum speed.

Energy flow — the Poynting vector

The energy density of an electromagnetic wave is the sum of electric and magnetic contributions:

u \;=\; \tfrac{1}{2}\varepsilon_0 E^2 + \dfrac{B^2}{2\mu_0}

For a plane wave, E and B are related by B = E/c, and the two terms are equal. The total density is

u \;=\; \varepsilon_0 E^2

Power per unit area (the intensity of the wave) is given by the Poynting vector:

\vec{S} \;=\; \dfrac{1}{\mu_0}\,\vec{E}\times\vec{B}

For a plane wave, |\vec{S}| = E^2/(\mu_0 c) = c\varepsilon_0 E^2, pointing in the direction of propagation. The time-averaged intensity for a sinusoidal wave of peak field E_0 is

\langle S\rangle \;=\; \tfrac{1}{2}c\varepsilon_0 E_0^2

For sunlight at Earth's surface (S_\text{sun} \approx 1000 W/m²), this gives E_0 \approx 870 V/m. The corresponding B_0 = E_0/c \approx 2.9\,\muT — about 5% of Earth's static magnetic field.

Momentum — radiation pressure

Electromagnetic waves carry not only energy but also momentum. The momentum per unit volume in a plane wave is p/V = u/c, and the radiation pressure on a perfectly absorbing surface is

P_\text{rad} \;=\; \dfrac{S}{c}

For sunlight, P_\text{rad} = 1000/(3\times 10^{8}) \approx 3\,\muPa — vanishingly small in everyday terms, but enough to drive a solar sail, to shape comet tails, and to cool atoms in atomic-physics experiments. ISRO's Aditya-L1 spacecraft, launched 2023 to monitor the Sun, continuously accounts for solar-radiation pressure in its trajectory corrections.

Light has momentum because E = pc for massless particles — a relation from special relativity that Maxwell's equations anticipated by several decades.

Polarisation

An EM wave has a direction of propagation (say \hat{k}) and two orthogonal transverse axes along which \vec{E} can oscillate. A wave with \vec{E} oscillating only along one of those axes is linearly polarised. A wave with \vec{E} rotating in the transverse plane is circularly or elliptically polarised. Unpolarised light (from a hot source like the Sun) is an incoherent mixture of all polarisations.

Polarisation matters because it determines how waves interact with oriented structures — polaroid filters (selective absorption), birefringent crystals (different n for different polarisation), antennas (must be aligned with the incoming polarisation). FM radio in India uses horizontal polarisation; DTH satellite uses circular; polaroid sunglasses preferentially block horizontally polarised sunlight reflected off roads.

Speed in matter — the refractive index

In a non-magnetic dielectric with permittivity \varepsilon = \varepsilon_r\varepsilon_0 and magnetic permeability \mu_0, Maxwell's equations still predict waves, now at speed

v \;=\; \dfrac{1}{\sqrt{\mu_0\varepsilon}} \;=\; \dfrac{c}{\sqrt{\varepsilon_r}} \;=\; \dfrac{c}{n}

where n = \sqrt{\varepsilon_r} is the refractive index. Crucially, the frequency f is set by the source and does not change as the wave crosses a boundary — it is the wavelength that shrinks by a factor of n:

\lambda_\text{medium} \;=\; \dfrac{\lambda_\text{vacuum}}{n}

For water (n \approx 1.33 at visible wavelengths), 500 nm green light in vacuum has wavelength 376 nm inside the water. For glass (n \approx 1.5), the wavelength is 333 nm. This wavelength-shortening is what makes lenses work — the wave bends at a boundary because it must match frequency on both sides.

Different frequencies see different refractive indices (dispersion), which is why a glass prism separates white light into colours — blue slows more than red, so it refracts more. The rainbow after a Pune monsoon shower is dispersion in suspended water droplets combined with one or two internal reflections.

The cosmic microwave background

At the longest wavelengths routinely measured, at a thermal temperature T_\text{CMB} = 2.725 K, there is a cosmic background of microwave radiation left over from the early universe. Its peak wavelength (from Wien's law) is 1.06 mm — at the boundary of microwave and far-infrared. It fills the sky uniformly to one part in 10^{5}, and its anisotropies (mapped by the WMAP and Planck satellites) encode the initial conditions of the Big Bang. The existence of this thermal microwave bath, and its precise spectrum, is itself a test of Maxwell's equations applied to the entire visible universe.

Where this leads next