Why a Brighter Light Never Makes Photoelectrons Fly Out Faster

By the end of the nineteenth century, Maxwell's equations and Hertz's own experiments generating and detecting electromagnetic waves had thoroughly established that light is a wave -- yet, in a strange twist, it was Hertz's very own apparatus that first stumbled onto the effect that would eventually challenge that same wave picture. Around the same period, passing an electric discharge through gas at very low pressure inside a glass tube produced a fluorescent glow, traced to rays streaming from the negative electrode (the cathode); William Crookes suggested in 1879 that these cathode rays were fast-moving, negatively charged particles, and J. J. Thomson confirmed this directly in 1897 by applying crossed electric and magnetic fields to measure the particles' speed and their charge-to-mass ratio, e/m, finding this ratio to be exactly the same no matter which metal or gas was used -- a universal particle, which Thomson named the electron. In 1913, the American physicist R. A. Millikan's celebrated oil-drop experiment pinned down the electron's exact charge, finding that every charged oil droplet's charge was always a whole-number multiple of one fixed, elementary amount, 1.602x10⁻¹⁹ C -- direct, precise proof that electric charge itself comes only in discrete, indivisible packets, never in a continuously adjustable amount. Knowing both the electron's charge (from Millikan) and its charge-to-mass ratio (from Thomson) together pinned down its mass as well, completing the identification of the electron as matter's own fundamental, universal building block.

Metals are full of free electrons, responsible for their conductivity, yet these electrons cannot simply wander out of the metal's surface -- attempting to escape leaves the surface positively charged, which pulls the electron straight back in. A certain minimum amount of energy, called the work function of the metal (φ0, usually measured in the convenient unit of electron volts, where 1 eV = 1.602x10⁻¹⁹ J, the energy an electron gains crossing a 1-volt potential difference) must be supplied before an electron can escape at all, and this minimum energy can be delivered by any of three distinct physical processes: thermionic emission, heating the metal until thermal energy alone is enough (the working principle inside older vacuum-tube electronics); field emission, applying an extremely strong electric field (around 10⁸ V/m) to directly pull electrons out, as in a spark plug; and photoelectric emission, illuminating the metal surface with light of a suitable frequency, ejecting electrons now called photoelectrons -- the specific process this entire chapter is built around.

Heinrich Hertz discovered photoelectric emission entirely by accident in 1887, noticing that ultraviolet light striking one plate of his spark-gap apparatus enhanced the sparks jumping across the detector -- light was somehow helping electrons escape the metal. Wilhelm Hallwachs and Philipp Lenard investigated the effect in detail over the following years: Hallwachs found that a negatively charged zinc plate lost its charge when illuminated by ultraviolet light, while an uncharged plate became positively charged under the same light -- direct evidence that negative particles (electrons) were being ejected. A genuinely puzzling detail emerged alongside this: no electrons were ejected AT ALL below a certain minimum frequency of incident light, called the threshold frequency, no matter how bright that light was made -- and this threshold varied by metal, with zinc, cadmium and magnesium responding only to short-wavelength ultraviolet light, while alkali metals like sodium, potassium and caesium responded even to ordinary visible light.

A proper apparatus for studying the effect uses an evacuated tube with a photosensitive emitter plate C and a collector plate A, a battery to set the voltage between them (reversible in polarity), a microammeter measuring the resulting photocurrent, and a way to vary the incident light's intensity and frequency independently. Keeping frequency and the accelerating (positive) voltage fixed, varying only the intensity of the incident light shows the photocurrent increasing exactly linearly with intensity -- more light means proportionally more photoelectrons ejected per second, confirming that intensity governs the NUMBER of electrons emitted. Keeping intensity and frequency fixed instead and varying the collector's own voltage reveals a second pattern: increasing the positive (accelerating) voltage increases the photocurrent only up to a point, called the saturation current, where every single ejected electron is already being collected; reversing the voltage to negative (retarding) instead shows the current dropping rapidly to exactly zero at one particular negative voltage, the stopping potential V0, defined precisely as the retarding voltage strong enough to repel back even the single MOST energetic photoelectron, so that eV0 = Kmax, the maximum photoelectron kinetic energy. Repeating this with brighter light of the SAME frequency raises the saturation current (more electrons overall) but leaves the stopping potential completely unchanged -- a first, genuinely strange clue that a photoelectron's own maximum energy has nothing to do with how bright the light is.

Repeating the same experiment across different frequencies of light, at the same intensity, reveals the crucial second pattern: the stopping potential DOES change with frequency, growing more negative (needing a stronger retarding voltage) as frequency increases, while the saturation current stays exactly the same across all these frequencies. Plotting stopping potential against frequency produces a clean straight line for any one photosensitive material, one that crosses zero stopping potential at exactly the threshold frequency ν0 already noticed by Hallwachs and Lenard. Four experimental facts, taken together, now stood as a complete, precise description of the photoelectric effect: photocurrent is directly proportional to intensity (for frequency above threshold); the stopping potential (and hence maximum photoelectron energy) is completely independent of intensity; no photoelectrons are ejected at all below the threshold frequency, however intense the light; and, most strikingly, photoelectric emission begins essentially instantaneously, with no measurable time lag whatsoever, even for extremely dim light.

The classical wave picture of light, which had explained interference, diffraction and polarisation so successfully, predicts that a metal's free electrons absorb radiant energy CONTINUOUSLY from an incoming light wave, with brighter light (larger wave amplitude) delivering energy faster to each electron -- so the wave picture directly predicts that a photoelectron's maximum kinetic energy should INCREASE with intensity, and that a threshold frequency should not exist at all, since even very dim light, given enough time, ought to eventually deliver enough energy to any electron to let it escape. Both predictions are flatly contradicted by the four observed facts. Worse still, the wave picture predicts continuous absorption spread thinly across every electron near the surface simultaneously, meaning the energy delivered to any SINGLE electron per unit time should be tiny -- explicit calculation shows it should take hours, not an instant, for one electron to accumulate enough energy this way to escape, directly contradicting the observed, genuinely instantaneous emission. The wave theory of light, despite its enormous earlier successes, simply cannot explain photoelectric emission at all.

In 1905, Einstein proposed a genuinely radical alternative: rather than being absorbed continuously, radiant energy comes in discrete packets, or quanta, each carrying an energy hν (h being Planck's constant, ν the light's frequency), and photoelectric emission happens when a SINGLE electron absorbs a SINGLE quantum in one shot, rather than gradually accumulating energy from many. If this absorbed quantum's energy exceeds the work function, the electron escapes carrying the leftover energy as kinetic energy, giving Einstein's photoelectric equation: Kmax = hν - φ0. This single, compact equation accounts for every observed feature at once: since Kmax depends only on ν and φ0 (a property of the metal), it is genuinely independent of intensity, exactly as observed; since Kmax cannot be negative, emission requires hν > φ0, meaning a threshold frequency ν0 = φ0/h genuinely must exist below which no emission is possible at all, however intense the light; increasing intensity increases only the NUMBER of quanta arriving per second, and hence the number of electrons absorbing one each, explaining the intensity-current proportionality without touching Kmax at all; and, since absorbing one quantum is a single, instantaneous event for one electron rather than a slow accumulation, emission is instantaneous, exactly as observed. Rewritten using eV0=Kmax, the equation becomes V0 = (h/e)ν - φ0/e, predicting the exact straight-line V0-versus-ν graph already observed, with slope h/e independent of the metal -- and Millikan, originally setting out over a decade of careful experiments specifically to DISPROVE Einstein's equation, instead measured this slope precisely enough to confirm it completely, extracting a value of Planck's constant matching its value from entirely unrelated experiments.

This quantum of radiant energy earned its own name, the photon, once Einstein showed it could also be assigned a definite momentum, p = hν/c -- a particle needs both energy and momentum, and light quanta genuinely have both. The photon's particle-like nature was confirmed further in 1924 by Arthur Compton's experiment scattering X-rays off electrons, behaving exactly as a particle-particle collision would, conserving both total energy and total momentum (though not necessarily the total NUMBER of photons, since one photon can be absorbed while a different one is created). Photons are electrically neutral, travel always at exactly the speed of light c, and every photon of one particular frequency ν carries exactly the same energy and momentum, however intense or dim the overall beam is -- intensity simply counts how many identical photons arrive per second, never changing any single photon's own individual properties. This picture is directly useful, too: a laser's total power output, divided by the energy of each individual photon (hν), gives the number of photons emitted every second, and, working the other direction, a metal's measured work function together with a measured stopping potential pins down the exact wavelength of light that produced it.

Louis de Broglie, in 1924, asked a genuinely bold, symmetrical question: if radiation (long assumed to be purely a wave) turns out to behave like a particle in some situations, might matter (long assumed to be purely particles) behave like a wave in some situations too? He proposed that any moving particle of momentum p has an associated wavelength, λ = h/p = h/mv, the de Broglie wavelength -- and, remarkably, this same formula applies equally well to a photon, where p=hν/c gives back exactly λ=c/ν, the ordinary wavelength of the light itself, showing the relation is genuinely consistent rather than an arbitrary new guess. The formula's real content shows up only at extremely small mass and momentum: an ordinary thrown ball, with its large mass and everyday speed, works out to a de Broglie wavelength far, far smaller than any atomic nucleus, utterly unmeasurable and irrelevant to its motion -- exactly why footballs and cricket balls never show any wave-like behaviour in daily life. An electron, by contrast, with its extraordinarily tiny mass, has a de Broglie wavelength comparable to X-ray wavelengths and to the spacing between atomic planes in a crystal, genuinely measurable and genuinely significant -- confirmed directly by experiments where beams of electrons produce real diffraction patterns passing through crystals, exactly as a wave would, completing de Broglie's originally bold, purely theoretical guess into a fully confirmed physical fact.