Playground
K_max vs frequency with threshold and ejection indicator.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Max kinetic energyoutput Maximum kinetic energy of ejected electrons | J | M·L²·T⁻² | 0 – 1e-17 | |
| Frequency Frequency of incident light | Hz | T⁻¹ | 100000000000000 – 10000000000000000 | |
| Work function Minimum energy to free an electron from the metal | J | M·L²·T⁻² | 1.6e-19 – 1e-18 |
Deep dive
Derivation
Einstein postulated that light of frequency ν consists of quanta of energy E = hν. When a photon strikes a metal, it transfers all its energy to a single electron. The electron must overcome the binding energy φ (work function) to escape; whatever is left becomes kinetic energy: K_max = hν − φ. Below ν₀ = φ/h, no escape is possible.
Experimental verification
Millikan's painstaking 1916 measurements (he was trying to *disprove* Einstein) yielded a perfectly linear K_max vs ν plot with slope h, measuring Planck's constant to ~0.5% accuracy. Lenard's earlier observations of frequency-dependent threshold and intensity-independent K_max were already inexplicable classically.
Common misconceptions
- Brighter light does NOT give electrons more energy — it only ejects more of them. K_max depends on frequency, not intensity.
- There is no time delay even at very low intensity — the photon model predicts instantaneous emission, contradicting classical wave theory.
- The work function depends on the metal and surface conditions, not the light source.
Real-world applications
- Photomultiplier tubes and night-vision devices.
- Solar cells (photovoltaic effect — extension to semiconductors).
- X-ray photoelectron spectroscopy (XPS) for chemical analysis.
- CCD/CMOS camera sensors converting photons to electrons.
Worked examples
400 nm light on a 2.0 eV metal
Given:
- λ:
- 4e-7
- φ_eV:
- 2
Find: K_max in eV
Solution
Photon energy = 1240/400 = 3.10 eV; K_max = 3.10 − 2.0 = 1.10 eV
Threshold wavelength for cesium (φ = 2.1 eV)
Given:
- φ_eV:
- 2.1
Find: λ₀
Solution
λ₀ = hc/φ = 1240/2.1 ≈ 590 nm (yellow-green)
Scenarios
What if…
- scenario:
- What if you double the light intensity?
- answer:
- Twice as many electrons are ejected per second, but each one still has the same K_max. Photoelectric current doubles; stopping voltage is unchanged.
- scenario:
- What if the metal is replaced by one with double the work function?
- answer:
- The threshold frequency doubles. Light that previously worked may no longer eject electrons; remaining ejected electrons have lower K_max.
- scenario:
- What if classical wave theory were right?
- answer:
- K_max would depend on intensity, there would be a measurable buildup time at low intensity (seconds to hours), and there would be no sharp threshold — none of which is observed.
Limiting cases
- condition:
- hν < φ
- result:
- K_max = 0 (no emission)
- explanation:
- Below the threshold frequency, no electrons are ejected regardless of intensity — pure quantum behavior.
- condition:
- hν = φ
- result:
- K_max = 0 (threshold)
- explanation:
- Defines the cutoff frequency ν₀ = φ/h characteristic of the metal.
- condition:
- hν ≫ φ
- result:
- K_max ≈ hν
- explanation:
- At high photon energies the work function is negligible; electron KE tracks photon energy directly.
Context
Albert Einstein · 1905
Einstein proposed light quanta to explain Lenard's puzzling photoelectric data, earning him the 1921 Nobel Prize.
Hook
Why does dim blue light eject electrons but bright red light doesn't?
Light of wavelength 400 nm strikes a metal with work function 2.0 eV. What is the maximum kinetic energy of the ejected electrons?
Dimensions: [K_max] = [h]·[ν] = (M·L²·T⁻¹)·(T⁻¹) = M·L²·T⁻² ✓ (energy)
Validity: Valid for single-photon absorption in clean metallic surfaces. Breaks down for very intense lasers (multi-photon ionization), and must be modified for semiconductors and insulators where band structure matters.
Related formulas
De Broglie Wavelengthplanck_radiation