Section 02
The Idea, Step by Step
A car radio, a microwave oven, the sunlight on your face, and the X-ray at the dentist feel like four unrelated things. They are not. They are all the same stuff — electromagnetic waves — racing along at the same speed, just "shaken" at different rates. Think of piano keys: same string-and-hammer idea, wildly different pitch.
Naming the pieces
Two numbers describe any wave. The frequency $f$ counts how many times the wave wiggles each second (in hertz, Hz). The wavelength $\lambda$ is the length of one full wiggle (in metres). For light they are locked together by one rule:
The one rule
$$c=f\lambda,\qquad c=3\times10^{8}\;\text{m/s}$$
Because $c$ is fixed, a high frequency must mean a short wavelength, and vice-versa. Worked number: an FM station at $f=100\;\text{MHz}=1\times10^{8}\;\text{Hz}$ has $\lambda=c/f=3$ m — a wave as long as a room. Green light at $\lambda=550$ nm wiggles at $f=c/\lambda\approx5.5\times10^{14}$ Hz — about half a million billion times a second.
Why the spectrum has an order
Each wave also arrives in tiny packets called photons, and every photon carries an energy set by its frequency:
Photon energy
$$E=hf=\frac{hc}{\lambda},\qquad h=6.626\times10^{-34}\;\text{J·s}$$
So shorter wavelength means higher frequency means a more energetic photon. That single chain orders the whole spectrum and explains the stakes: radio photons (micro-eV) pass through you harmlessly, visible photons ($\sim$2 eV) are just energetic enough to trigger vision and photosynthesis, while UV, X-ray and $\gamma$ photons (eV up to MeV) carry enough punch to break chemical bonds and ionise atoms — which is exactly why we shield against them. A hot object glows because it pours out a whole spread of these waves at once; Planck's law sets the shape of that spread and Wien's law, $\lambda_{\max}T=2.898\times10^{-3}$ m·K, says where it peaks. In the sim, the $\log_{10}(f)$ slider walks you along the spectrum and the temperature sliders set the blackbody curves.
Try this in the sim above
First, in Spectrum mode drag $\log_{10}(f)$ from low to high and watch the band readout march Radio → Visible → $\gamma$ while $\lambda$ shrinks and $E$ climbs — that is $c=f\lambda$ and $E=hf$ happening live. Second, switch to Blackbody mode, set $T=5778$ K (the Sun) and read $\lambda_{\max}\approx500$ nm; then raise $T$ and watch the peak slide left into the ultraviolet (Wien's law). Third, compare $T_2=3000$ K against 5778 K and see the hotter curve both tower over the cooler one and shift toward shorter wavelengths.
Section 03
Equations & Derivation
EM Wave Properties
$$c=f\lambda=3\times10^8\;\text{m/s},\quad E_0=cB_0,\quad I=\frac{E_0^2}{2\mu_0 c}$$
Photon Energy
$$E=hf=\frac{hc}{\lambda},\quad h=6.626\times10^{-34}\;\text{J·s}=4.136\times10^{-15}\;\text{eV·s}$$
Planck's Blackbody Law
$$B(\lambda,T)=\frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/\lambda k_BT}-1}$$
Wien's Law & Stefan-Boltzmann
$$\lambda_{\max}T=2.898\times10^{-3}\;\text{m·K},\quad P=\sigma T^4,\quad\sigma=5.67\times10^{-8}\;\text{W\,m}^{-2}\text{K}^{-4}$$
EM Spectrum Bands
| Band | Photon energy | Wavelength |
|---|
| $\gamma$ | > 10 MeV; nuclear transitions | λ < 10 pm |
| $X\text{-ray}$ | 1 keV–10 MeV | 10 pm–1 nm |
| $UV$ | 3–1000 eV | 1–400 nm |
| $\text{Visible}$ | 1.65–3.1 eV | 380–750 nm |
| $IR$ | 0.001–1.65 eV | 750 nm–1 mm |
| $\text{Microwave/Radio}$ | < 1 meV | > 1 mm |
1
EM waves. Oscillating $\vec{E}$ and $\vec{B}$ fields, mutually perpendicular, propagating at $c$. $E_0=cB_0$. The Poynting vector $\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B}$ gives direction and magnitude of energy flux (intensity $I=E_0^2/(2\mu_0 c)$ W/m²).
2
Photon energy. $E=hf=hc/\lambda$. Visible photons: 1.65–3.1 eV. A 550 nm green photon: $E=3.61\times10^{-19}$ J = 2.26 eV. X-ray photon at 0.1 nm: 12.4 keV. One mole of 550 nm photons carries 217 kJ — comparable to chemical bond energies.
3
Blackbody radiation. Every object emits EM radiation thermally. Planck (1900) solved the ultraviolet catastrophe by quantising oscillator energy: $E_n=nhf$. This birthed quantum mechanics. Wien's law gives peak wavelength; Stefan-Boltzmann gives total power.
4
Rayleigh scattering. $I_{\text{scattered}}\propto1/\lambda^4$. Explains blue sky (blue scatters most), red sunset (blue scattered away), and white clouds (all wavelengths scatter equally from large water droplets).
Ref: Halliday, Resnick & Walker 10th Ed., §33-1; Griffiths — Introduction to Electrodynamics (4th Ed.), Ch. 9; HRW §38-2.
Section 05
Common Misconceptions
❌ X-rays travel faster than radio waves in vacuum.
✅ All EM waves travel at exactly $c=2.998\times10^8$ m/s in vacuum, regardless of frequency. Einstein's special relativity requires this constancy. In a medium, phase velocity can vary with frequency (dispersion), but in vacuum all EM waves are absolutely identical in speed.
📖 HRW 10th Ed., §33-3.
❌ Hotter objects are always brighter to the human eye.
✅ Hotter objects radiate more power ($P\propto T^4$) but shift peak to shorter wavelengths (Wien's Law). Above ~6000 K, peak emission is in UV — invisible to humans. The total radiated power is enormous, but the visible fraction decreases. A 50,000 K star emits mostly UV and appears less brilliant optically than its total power suggests.
📖 HRW 10th Ed., §38-2.
❌ Microwave ovens cook food by heating the water molecules with radio waves.
✅ Microwave ovens (typically 2.45 GHz) do heat water, but not by resonance. The frequency 2.45 GHz was chosen for regulatory reasons (ISM band), not because it matches any water resonance. Water absorbs microwaves broadly due to its polar nature and rotational relaxation, not at a specific resonant frequency.
📖 Hecht — Optics (5th Ed.), Appendix 1.
Misconception research: Arons — A Guide to Introductory Physics Teaching.