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If Maxwell's electromagnetic synthesis was the high-water mark of 19th-century classical physics (lesson 3), the photoelectric effect was the crack through which the 20th century walked in. The story is short, contained, and pedagogically clean: a single experimental observation — electrons emitted from a metal surface only when light of frequency above a threshold strikes it — refused to fit the classical wave theory of light. Max Planck's introduction of energy quanta in 1900 (to fix a separate problem, the blackbody-radiation spectrum) and Albert Einstein's 1905 application of those quanta to the photoelectric effect together transformed light into a particle (or rather, a quantum entity with both wave and particle aspects). Robert Millikan's experimental verification of Einstein's equation between 1914 and 1916 completed the package. By 1923, the photon was a routine concept in physics. This lesson covers the experimental observations that classical wave theory could not explain, Planck's energy quantisation, Einstein's photoelectric equation, the work-function concept, and the standard A-Level calculations of maximum kinetic energy and stopping potential.
Spec mapping: This lesson covers part of AQA 7408 section 3.12.2 — the photoelectric effect, Planck's quantum hypothesis, Einstein's photoelectric equation, work function and threshold frequency, and Millikan's experimental verification of the equation. (Refer to the official AQA specification document for exact wording.)
Synoptic links:
- Section 3.2.2 (particles and quantum phenomena): the photoelectric effect is taught in Year 12 as the canonical motivation for photons. This lesson revisits it through a historical lens — placing the same physics in its 1900-1916 context.
- Section 3.12.2 (Maxwell): the photoelectric effect is the failure mode of the wave theory of light that had triumphed in the 19th century. Without Maxwell's success, the photoelectric "crisis" would not have been so philosophically jarring.
- Section 3.12.2 (electron diffraction, lesson 5): the wave-particle duality is asymmetric — photons get particle properties (this lesson), and electrons get wave properties (next lesson). Both halves are needed for the modern picture.
Hertz had noticed in 1887 that ultraviolet light shining on the spark gap of his transmitter made the sparks form more easily. He did not pursue the observation; his student Philipp Lenard, working in Heidelberg in 1900-1902, ran a systematic experimental programme on what we now call the photoelectric effect. Light shone on a clean metal surface (a freshly polished cathode in a vacuum tube) can liberate electrons, which can then be collected by an anode at a positive potential and measured as a photoelectric current.
Lenard reported four observations that — taken together — destroyed any classical wave-theory account of the effect:
Each of these is at right-angles to what a classical wave theory predicts.
| Observation | Classical wave-theory prediction | Experimental reality |
|---|---|---|
| Threshold frequency | None — any frequency could eventually liberate electrons given enough intensity or time | Sharp f₀ for each metal |
| KE_max dependence | Should depend on intensity | Depends only on frequency |
| Photocurrent | Should grow with both intensity and time of exposure | Proportional to intensity, instantaneous |
| Time delay | Seconds-to-minutes at low intensities | Essentially zero (< 10⁻⁹ s) |
Lenard, Hertz's student, was a sceptic of the emerging quantum interpretations and remained so for decades. But his experimental results are unimpeachable and they framed the problem to which Einstein supplied the answer.
Planck had been working on an entirely different problem — the blackbody radiation spectrum. A hot object emits a continuous spectrum of EM radiation whose intensity-versus-frequency curve peaks at a wavelength determined by the temperature (Wien's displacement law). Classical EM theory predicted that the curve should rise without limit at short wavelengths — the so-called ultraviolet catastrophe. The observed curve, in contrast, falls smoothly to zero at high frequencies.
Planck found a mathematical fix in 1900 that exactly reproduced the observed curve at all temperatures and frequencies. The cost was a radical assumption: the oscillating charges in the cavity wall could emit or absorb EM radiation only in discrete packets of energy
E = hf
where h is a new constant — now Planck's constant, h ≈ 6.63 × 10⁻³⁴ J s — and f is the frequency of the oscillator. Planck himself viewed this as a mathematical trick, not a statement about the nature of light. He spent years trying to derive the formula classically and failed.
Einstein, in one of the four papers of his 1905 annus mirabilis, took Planck's quantisation seriously and applied it to the photoelectric effect with a single bold step: light itself comes in packets of energy E = hf, not just the energy emitted by hot oscillators. A "photon" of frequency f carries energy hf. Each photon hitting an electron in the metal either has enough energy to liberate it or it does not.
Liberating an electron costs a fixed minimum energy — the work function φ — which is a property of the metal surface (typically 2-6 eV for clean metals). Any excess energy goes into the kinetic energy of the emitted electron. Hence:
hf = φ + KE_max
— Einstein's photoelectric equation. Rearranged:
KE_max = hf − φ
This single equation explains all four of Lenard's observations:
graph LR
A["Photon<br/>(E = hf)"] --> B["Metal surface"]
B --> C{"hf vs φ?"}
C -->|"hf < φ"| D["No emission"]
C -->|"hf ≥ φ"| E["Electron emitted<br/>KE_max = hf - φ"]
style D fill:#e74c3c,color:#fff
style E fill:#27ae60,color:#fff
It is striking that the photoelectric equation was not universally accepted on Einstein's authority. Many physicists — including Planck himself — were uncomfortable with the idea of light as a particle. Robert Millikan (whom we met in lesson 1 for the oil-drop experiment) set out in 1908 to disprove Einstein's equation by a careful experimental measurement of the KE_max-versus-f relationship.
Over six years of meticulous work — Millikan had to develop a method to keep the photoelectric cathode atomically clean inside a vacuum, since oxide layers severely affected the work function — he produced a series of straight-line KE_max-versus-f graphs for several alkali metals (sodium, potassium, lithium). The slope of each graph was h to within 0.5%, in perfect agreement with the value Planck had derived from blackbody radiation. The x-intercept gave the threshold frequency f₀ for each metal, and the y-intercept gave the work function φ.
Millikan, despite having set out to refute Einstein, concluded in his 1916 paper that the equation had been verified to high precision. Einstein received the 1921 Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect" — explicitly not for relativity, which remained controversial in 1921. Millikan received the 1923 Nobel Prize, partly for the oil-drop experiment and partly for the photoelectric verification.
A practical way to measure KE_max is to apply a reverse potential V_s across the photocell — anode negative, cathode positive — that just stops the most energetic photoelectrons from reaching the anode. At the stopping point, all the electron's initial KE is converted to electrical PE:
KE_max = eV_s
So a graph of V_s versus f has slope h/e and intercept −φ/e. This is the standard A-Level form of the photoelectric experiment and produces a measurement of h/e of high accuracy.
A clean caesium surface has work function φ = 2.10 eV. Light of wavelength λ = 400 nm (violet) is incident on it.
(a) Calculate the photon energy in joules and in eV. (b) Calculate the threshold frequency of caesium. (c) Calculate the maximum kinetic energy of emitted photoelectrons. (d) Calculate the stopping potential.
Use h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, e = 1.60 × 10⁻¹⁹ C.
Solution.
(a) f = c/λ = (3.00 × 10⁸)/(400 × 10⁻⁹) = 7.50 × 10¹⁴ Hz.
E = hf = (6.63 × 10⁻³⁴)(7.50 × 10¹⁴) = 4.97 × 10⁻¹⁹ J In eV: E = (4.97 × 10⁻¹⁹)/(1.60 × 10⁻¹⁹) = 3.11 eV.
(b) φ = 2.10 eV = (2.10)(1.60 × 10⁻¹⁹) = 3.36 × 10⁻¹⁹ J.
f₀ = φ/h = (3.36 × 10⁻¹⁹)/(6.63 × 10⁻³⁴) = 5.07 × 10¹⁴ Hz (which corresponds to λ₀ = c/f₀ ≈ 592 nm — yellow-orange light, on the threshold of visible).
(c) KE_max = hf − φ = (4.97 × 10⁻¹⁹) − (3.36 × 10⁻¹⁹) = 1.61 × 10⁻¹⁹ J = 1.01 eV.
(d) V_s = KE_max / e = (1.61 × 10⁻¹⁹)/(1.60 × 10⁻¹⁹) = 1.01 V.
(Equivalently: V_s = KE_max in eV ÷ 1 in coulombs of one electron = 1.01 V; the eV-and-V identification is direct.)
In a photoelectric experiment, light of wavelength 250 nm produces photoelectrons whose maximum kinetic energy is measured as 2.5 eV. Calculate the work function of the metal and identify the most likely candidate from the table below.
| Metal | Work function (eV) |
|---|---|
| Caesium | 2.1 |
| Potassium | 2.3 |
| Sodium | 2.4 |
| Zinc | 4.3 |
| Silver | 4.7 |
| Tungsten | 4.5 |
Solution.
f = c/λ = (3.00 × 10⁸)/(250 × 10⁻⁹) = 1.20 × 10¹⁵ Hz. E_photon = hf = (6.63 × 10⁻³⁴)(1.20 × 10¹⁵) = 7.96 × 10⁻¹⁹ J = 4.97 eV. φ = hf − KE_max = 4.97 − 2.5 = 2.47 eV.
This is closest to sodium (2.4 eV). The slight overshoot is within experimental error of a typical measurement. Caesium and potassium are also close but the best match is sodium.
The contrast with the heavier metals (Zn, Ag, W) is informative: 250 nm light on tungsten would give KE_max = 4.97 − 4.5 = 0.47 eV (small but positive), while on silver it would give 4.97 − 4.7 = 0.27 eV. The same photons that produce 2.5 eV electrons from sodium produce a much smaller KE from tungsten — because tungsten holds its electrons more tightly.
The classical-wave puzzle: on a wave theory, the energy of light is spread uniformly over the wavefront, and an electron in the metal absorbs energy at a rate proportional to the intensity (energy per unit area per unit time) of the light. To liberate an electron, the absorbed energy needs to accumulate to roughly the work function. For a moderately weak intensity — say a low-power laser pointer at 1 mW spread over 1 mm² → intensity 10³ W m⁻² — and an atomic cross-section of ~10⁻²⁰ m², the absorbed power per atom is 10⁻¹⁷ W. To accumulate the ~3 × 10⁻¹⁹ J of a typical work function therefore takes ~3 × 10⁻² s — 30 milliseconds.
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