Playground
K_max vs frequency plot with threshold. Slider controls photon frequency; work function adjustable. Shows electron ejection.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Max kinetic energyoutput Maximum KE of ejected photoelectrons | J | M*L^2*T^-2 | 0 – 1e-17 | |
| Planck's constant Fundamental quantum of action | J*s | M*L^2*T^-1 | 6.62607015e-34 – 6.62607015e-34 | |
| Photon frequency Frequency of incident light | Hz | T^-1 | 10000000000000 – 100000000000000000 | |
| Work function Minimum energy to eject an electron from the metal | J | M*L^2*T^-2 | 1e-19 – 1e-18 |
Deep dive
Derivation
Model the photon as a quantum of energy hf. Energy conservation on absorption by a single electron: hf = phi + K_max, where phi is the minimum energy to release the electron. Solving gives K_max = hf - phi. K_max is measured as e*V_0 where V_0 is the stopping potential.
Experimental verification
Millikan's 1916 photoelectric experiments measured h to within 0.5%. Modern angle-resolved photoemission spectroscopy (ARPES) uses this relation to map band structures of solids.
Common misconceptions
- Intensity does NOT affect K_max, only photon count — a classical wave theory would predict otherwise
- The emission is effectively instantaneous (< 10 fs), not delayed as a wave energy-buildup model would require
- The work function is a property of the metal surface, not the bulk
Real-world applications
- Photomultiplier tubes
- CCD and CMOS image sensors
- Night vision devices
- Solar cells and photodiodes
- ARPES for condensed matter research
Worked examples
Sodium under 400 nm light
Given:
- lambda:
- 4e-7
- phi_eV:
- 2.28
Find: K_max
Solution
K_max ≈ 0.82 eV ≈ 1.31e-19 J
Threshold frequency for cesium (phi = 2.1 eV)
Given:
- phi_eV:
- 2.1
Find: f_threshold
Solution
f ≈ 5.08e14 Hz (green light)
Scenarios
What if…
- scenario:
- What if you double the intensity?
- answer:
- Twice as many electrons come out per second, but K_max is unchanged.
- scenario:
- What if you use a laser below threshold?
- answer:
- No emission from single-photon process. With extreme intensity two-photon absorption can kick in (non-linear regime).
- scenario:
- What if the metal were perfectly clean?
- answer:
- Work function drops slightly (surface contamination raises phi); small but measurable shift in threshold.
Limiting cases
- condition:
- hf < phi
- result:
- K_max = 0 (no emission)
- explanation:
- Below threshold no electrons escape, no matter how intense the light.
- condition:
- hf = phi
- result:
- f_threshold = phi/h
- explanation:
- Exactly at threshold electrons barely escape with zero KE.
- condition:
- hf >> phi
- result:
- K_max ≈ hf
- explanation:
- At very high frequency the work function is a small correction.
Context
Albert Einstein · 1905
Einstein's explanation of Hertz/Lenard's observations won him the 1921 Nobel Prize and proved light quantization.
Hook
Why does dim blue light eject electrons but bright red light can't?
Electrons are only ejected from metal if the photon frequency exceeds a threshold. What is the max KE of photoelectrons from sodium (work function 2.28 eV) under 400 nm light?
Dimensions:
- lhs:
- K_max → [M*L^2*T^-2]
- rhs:
- [M*L^2*T^-1][T^-1] - [M*L^2*T^-2] → [M*L^2*T^-2]
- check:
- Both sides are energy. ✓
Validity: Valid for single-photon absorption in metals and semiconductors. Multi-photon regime requires intense lasers (non-linear photoemission).