The story of how physics broke open at the turn of the twentieth century — where classical ideas failed to explain a glowing hot iron, how Planck rescued the situation by selling energy in fixed "coins", how light knocks electrons off a metal (the photoelectric effect), Einstein's equation for it, the stopping-potential measurement, Einstein's special relativity with its slow clocks and shrinking lengths, de Broglie's daring claim that matter is also a wave, and finally where all of this lives in everyday technology. As you scroll, the panel on the right plays each idea out with a real object you already know — a glowing iron, a coin slot, a torch beam, a voltage hill, a fast spaceship.
1 — Where classical physics failed
Heat a piece of iron in a forge and it tells you its temperature by its colour: first a dull red, then bright orange, then a dazzling yellow-white. Every hot body radiates a smooth spread of wavelengths — its black-body radiation — and as it gets hotter the peak shifts towards the blue end and the body glows brighter.
- Black body — an ideal object that absorbs all radiation falling on it and, when hot, is also the best possible emitter.
- Spectral curve — intensity plotted against wavelength; it rises to a peak then falls away on both sides.
- The peak shifts — hotter bodies peak at shorter wavelengths (red → white), which is why colour reveals temperature.
The crisis: classical (Rayleigh–Jeans) physics predicted the intensity should keep rising without limit as wavelength got shorter — infinite energy in the ultraviolet. This absurd prediction was named the ultraviolet catastrophe. The real curve clearly turns over and falls, so something in classical physics was badly wrong.
2 — Planck's quantum hypothesis
In 1900 Max Planck found the only way to fit the real curve: assume that the oscillators in the walls of a hot body can give out energy only in whole packets, never in arbitrary amounts. Energy is quantised. Think of a vending machine that refuses loose change and accepts only fixed coins — you can pay 1, 2 or 3 coins, but never two-and-a-half.
Planck's energy of a quantumE = h f (= h c / λ)
h = 6.63 × 10⁻³⁴ J·s (Planck's constant) · f = frequency
- Energy comes in discrete lumps called quanta; each has energy E = hf.
- Higher frequency (bluer light) → bigger energy per quantum.
- Planck's constant h is tiny, so in everyday life the steps look smooth — the graininess only shows at the atomic scale.
Why it worked: high-frequency (ultraviolet) quanta are so expensive that a warm body can rarely afford even one, so the predicted intensity falls instead of exploding — the catastrophe disappears. This was the birth of quantum physics.
3 — The photoelectric effect
Shine light on a clean metal surface and, if its frequency is high enough, electrons fly out. This is the photoelectric effect. Einstein pictured light as a stream of energy packets — photons, each carrying E = hf — and one photon knocks out at most one electron, like a single ball-bearing flicking one marble off a ledge.
- Threshold frequency (f₀) — the minimum frequency below which no electrons escape, no matter how bright the light.
- Instant emission — electrons appear the moment the light arrives; there is no time lag.
- Brightness vs frequency — brighter light (more photons) ejects more electrons; higher frequency makes each ejected electron faster.
The classical puzzle: a wave should pour in energy steadily, so even dim light should eventually free electrons — but it never does below f₀. The photon picture explains it: one photon must carry enough energy in a single hit, so frequency, not brightness, decides whether emission happens.
4 — Einstein's photoelectric equation
Einstein (1905) wrote the energy bookkeeping exactly. A photon delivers hf. Part of it, the work function Φ, is the minimum energy needed to drag an electron out of the metal; the rest leaves as the electron's kinetic energy. The fastest electrons (from the surface) carry the maximum.
Einstein's photoelectric equationh f = Φ + ½ m v²max
⇒ KEmax = h f − Φ · Φ = h f₀ (escape fee at the threshold)
| If you increase… | …then KEmax |
|---|
| Frequency f (bluer light) | increases — more energy per photon |
| Brightness (same colour) | no change — more electrons, same speed |
| Work function Φ (different metal) | decreases — a bigger escape fee |
Exam point: a graph of KEmax against frequency f is a straight line of slope h, cutting the f-axis at the threshold f₀ and the energy-axis at −Φ. The same Planck's constant h appears for every metal.
5 — Stopping potential
How do we measure that maximum kinetic energy? Put the ejected electrons in a circuit and make the collecting plate negative, so it repels them. As you raise this reverse voltage, slower electrons are turned back first. At the stopping potential Vs, even the fastest electron is just stopped — the current falls to zero. It is a voltage hill exactly high enough to halt the quickest runner.
Stopping potential ↔ maximum kinetic energye Vs = KEmax = ½ m v²max = h f − Φ
e = 1.6 × 10⁻¹⁹ C (electron charge)
- The work done against the field, eVs, equals the electron's maximum kinetic energy.
- Higher frequency light → a taller hill needed → larger Vs.
- Brighter light → more current, but the same Vs (speed unchanged).
- Plot Vs against f: slope = h/e, a clean way to find Planck's constant.
Picture it: the electrons are balls rolling up a hill made of voltage. Raise the hill until the fastest ball just fails to reach the top and rolls back — that height measures its launch energy precisely.
6 — Special relativity
Einstein's special relativity (1905) starts from two postulates: the laws of physics are the same for all observers in uniform motion, and the speed of light c is the same for everyone. The consequences are strange but real. To a watcher on Earth, a clock on a spaceship travelling near c runs slow (time dilation), and the ship itself appears shortened along its direction of motion (length contraction).
Time dilation, length contraction, mass–energyt = t₀ / √(1 − v²/c²) · L = L₀ √(1 − v²/c²)
E = m c² (mass is a frozen store of energy)
- The factor γ = 1/√(1 − v²/c²) is ≈ 1 at ordinary speeds, so we never notice; it blows up only as v → c.
- Moving clocks tick slow; moving lengths contract — but only along the motion.
- Nothing with mass can reach c; it would need infinite energy.
- E = mc²: a tiny mass holds a huge energy — the principle behind the Sun and nuclear power.
Real evidence: fast muons made high in the atmosphere should decay before reaching the ground, yet they arrive — their internal clocks run slow exactly as relativity predicts. GPS satellites must correct for time dilation or they would soon be wrong by kilometres.
7 — Wave–particle duality & de Broglie wavelength
Light behaves as a wave (interference, diffraction) yet also as particles (the photoelectric effect) — it has a dual nature. In 1924 Louis de Broglie turned the idea around: if waves can be particles, then particles can be waves. Every moving body has a wavelength set by its momentum.
de Broglie wavelengthλ = h / p = h / (m v)
p = momentum · h = Planck's constant
- The faster (more massive) the particle, the shorter its wavelength.
- For a cricket ball λ is fantastically tiny, so we never see the ball "wave" — but for an electron it is comparable to atomic spacings.
- Proof: electrons fired at a crystal produce a diffraction pattern, just like a wave (Davisson–Germer experiment).
Big idea: the electron is neither a tiny ball nor a pure wave — it is something that shows whichever face the experiment asks for. This wave nature is what lets us build electron microscopes that see far smaller detail than light ever could.
8 — Applications & recap
These early-quantum ideas are not museum pieces — they power everyday devices.
- Solar cells — photons lift electrons across a junction, turning sunlight directly into electric current (the photoelectric effect at work).
- Photocells — light switches a current on or off; used in automatic doors, light meters, burglar alarms and street-lamp sensors.
- Electron microscope — uses the tiny de Broglie wavelength of fast electrons to resolve viruses and atoms, far beyond an optical microscope.
- Nuclear & the Sun — E = mc² converts a sliver of mass into enormous energy.
photon energy
Find the energy of a photon of green light, f = 5.0 × 10¹⁴ Hz.
E = hf = 6.63 × 10⁻³⁴ × 5.0 × 10¹⁴ = 3.3 × 10⁻¹⁹ J (≈ 2.1 eV)
photoelectric KEmax
A metal has Φ = 2.0 eV. Light of energy hf = 3.5 eV strikes it. Find KEmax and Vs.
KEmax = hf − Φ = 3.5 − 2.0 = 1.5 eV
eVs = KEmax ⇒ Vs = 1.5 V
de Broglie wavelength
An electron (m = 9.1 × 10⁻³¹ kg) moves at v = 1.0 × 10⁶ m/s. Find λ.
λ = h/(mv) = 6.63 × 10⁻³⁴ / (9.1 × 10⁻³¹ × 1.0 × 10⁶) = 7.3 × 10⁻¹⁰ m
- Classical physics failed for black-body radiation — the ultraviolet catastrophe.
- Planck: energy is quantised, E = hf.
- Photoelectric effect: light is photons; a threshold frequency f₀ exists.
- Einstein: hf = Φ + ½mv²max, so KEmax = hf − Φ.
- Stopping potential: eVs = KEmax.
- Relativity: moving clocks slow, lengths contract; E = mc².
- de Broglie: matter is also a wave, λ = h/p.