Shine light on a metal and electrons fly off — but only if the light's colour is right, never mind how bright it is. That one stubborn fact forced light to arrive in packets, each carrying energy hc/λ. Einstein's little equation catches every detail; a decade later the favour was returned, and matter itself was found to ripple. Wire up the photocell and chase the stopping potential live.
Radiation is emitted and absorbed in indivisible quanta — photons. Each one of wavelength λ carries energy E = hc/λ and momentum p = h/λ, travels at c, and has zero rest mass. Turning up the intensity at fixed colour sends more photons per second, not more energetic ones. Slide the wavelength and watch a single photon's energy climb as the wave tightens.
A photocell turns the effect into a graph. Pick a metal (sets φ), a wavelength, and an intensity, then read the V–I curve. Forward bias saturates once every emitted electron is collected — that height scales with intensity. Reverse bias fights the electrons; at −V₀ even the fastest is turned back, so the stopping potential is where the current dies. Change only the intensity and V₀ never budges — the ghost curve proves it.
Plot K_max against frequency and every metal gives a straight line. They all share the same slope — Planck's constant h — and differ only in where they cross the axis: the threshold frequency ν₀ = φ/h. Below ν₀ nothing comes off, at any brightness. That single graph, with its universal slope, is how Millikan pinned down h. Slide the frequency and pick a metal to walk along its line.
If waves can act like particles, de Broglie asked, why not the reverse? Every moving particle carries a wavelength λ = h/p. Accelerate a charge q through a voltage V and its de Broglie wavelength is λ = h/√(2mqV) — for an electron, a tidy 1.227/√V nm. Pick a particle and dial the voltage: at the same V the heaviest particle has the shortest wave, which is exactly why electron microscopes see so much finer than light.
A photon of energy E carries momentum E/c, so a beam that lands on a surface presses on it. A black plate absorbs that momentum once (F = P/c); a mirror reverses each photon and feels twice as much (F = 2P/c). Curve the surface into a sphere and the geometry averages the factor of 2 away — a reflecting sphere feels exactly what an absorbing one does, F = πr²I/c. Swap the target and watch the force live.
Six solved problems that run the whole module — from the bench readouts to the traps waiting in the exam bank below. Keep hc = 1240 eV·nm in reach.
280 nm on lithium (φ = 2.5 eV): E = 1240/280 = 4.43 eV, so K_max = 4.43 − 2.5 = 1.93 eV and eV₀ = 1.93 eV gives V₀ ≈ 1.9 V — untouched by how bright the beam is.
On the same lithium: V₀ = 1.63 V at 300 nm, 0.60 V at 400 nm. Slope = ΔV₀ / Δ(1/λ) = hc/e = 1.03 / (8.3×10⁻⁴ nm⁻¹) ≈ 1240 V·nm → h ≈ 6.6×10⁻³⁴ J·s. The intercept alone recovers φ.
Isolated Cu sphere, r = 2 cm, φ = 4.5 eV, lit by 150 nm: K_max = 8.27 − 4.5 = 3.77 eV, so emission dies at V = 3.77 V — set by K_max, not the radius. Charge held: Q = 4πε₀rV ≈ 8.4×10⁻¹² C, only ~5×10⁷ electrons lost.
K_max = 2.0 eV electrons into B = 1.0×10⁻⁴ T (⊥ v): p = √(2mK) = 7.6×10⁻²⁵ kg·m/s, so r = p/(eB) = 4.8 cm. Watch the eV→J step — one lost factor of ten sends you to 4.8 mm.
Sunlight I = 1.4 kW/m² on a perfectly reflecting sphere, r = 1 m: integrating over the lit hemisphere gives F = πr²I/c = 1.5×10⁻⁵ N — the same as an absorbing sphere. A flat mirror would double it; the curvature washes that out.
Electron vs α-particle through the same V: λ = h/√(2mqV), so λ_e/λ_α = √((m_α/m_e)(q_α/q_e)) = √(7344 × 2) ≈ 121. Drop the charge factor and you get ≈ 86 — a favourite trap.