Photoelectric Effect & Stopping Potential Calculator: Free Photoelectric Effect & Stopping Potential Calculator tool: calculate maximum photoelectron kinetic energy, stopping potential V0, cutoff threshold.
1. The Photoelectric Effect & Albert Einstein's Light Quantum Theory
First discovered experimentally by Heinrich Hertz in 1887 and systematically investigated by Philipp Lenard in 1902, the Photoelectric Effect is the instantaneous emission of electrons from a conductive metal surface when illuminated by electromagnetic radiation.
Throughout the late 19th century, classical wave optics (Maxwell's electrodynamics) utterly failed to account for four consistent experimental anomalies:
- Independence from Light Intensity: Classical theory predicted that brighter light (greater electric field amplitude) would deposit more energy into orbital electrons, increasing their kinetic energy. Experiments revealed that increasing intensity increases only the number of emitted photoelectrons (photocurrent), while the maximum kinetic energy remains completely unaffected!
- The Cutoff Threshold Frequency ($nu_0$): For any metal, there exists a sharp minimum frequency $nu_0$ below which absolutely zero photoelectrons are emitted, no matter how intensely or continuously the surface is irradiated.
- Immediate Emission: Emission occurs within less than $10^{-9}text{ seconds}$ of illumination, contradicting wave theory which calculated that dim light would require hours of continuous wave energy accumulation.
- Linear Dependence on Frequency: The maximum kinetic energy $E_{k,max}$ scales strictly linearly with optical frequency $nu$.
In 1905, in his landmark Annus Mirabilis paper, Albert Einstein resolved this crisis by proposing that light itself propagates not as a continuous wave, but as localized packets of energy called Photons, each carrying discrete energy:
When a single photon strikes a cathode electron in a one-to-one collision, a portion of the photon's energy is consumed to liberate the electron from the metallic lattice (the Work Function $Phi$), and the remaining energy appears as the liberated photoelectron's maximum kinetic energy:
2. Stopping Potential ($V_0$) & Robert Millikan's Experimental Proof
To measure the maximum kinetic energy of photoelectrons without destructive velocity analysis, experimentalists apply an adjustable retarding (reverse) electric potential difference $V$ between the illuminated photocathode and the collector anode.
As the anode is made increasingly negative relative to the cathode, photoelectrons are repelled. The exact potential that halts even the fastest photoelectron, extinguishing measured photocurrent to zero, is the Stopping Potential ($V_0$):
In 1916, American experimental physicist Robert A. Millikan set out to disprove Einstein's photon hypothesis by constructing an elaborate ultra-high-vacuum "machine shop in vacuo" to shave pure metal surfaces clean of oxides. To his astonishment, plotting stopping potential $V_0$ against frequency $nu$ yielded an unmistakable straight line across every tested metal:
- The slope was universal across all metals, measuring exactly $text{Slope} = frac{h}{q} approx 4.136 times 10^{-15}text{ V}cdottext{s}$, providing the world's first direct experimental measurement of Planck's constant $h$!
- The horizontal intercept yielded the threshold frequency $nu_0 = Phi / h$.
- The vertical intercept yielded $-Phi / q$.
3. Cathode Work Functions Across the Periodic Table
The work function $Phi$ depends on the crystal surface structure and metallic valence electron configuration:
| Metal Cathode Material | Work Function ($Phi$, eV) | Threshold Wavelength ($lambda_0$, nm) | Threshold Optical Regime | Common Technological Role |
|---|---|---|---|---|
| Cesium (Cs) | 2.14 eV | 579.4 nm | Visible (Yellow) | Night-vision image intensifiers & phototubes |
| Potassium (K) | 2.30 eV | 539.1 nm | Visible (Green) | Alkali photocathodes |
| Sodium (Na) | 2.36 eV | 525.4 nm | Visible (Green) | Historical Millikan validation tests |
| Calcium (Ca) | 2.87 eV | 432.0 nm | Visible (Violet) | Visible light detectors |
| Zinc (Zn) | 4.30 eV | 288.3 nm | Ultraviolet (UV-C) | Electroscope discharge demonstrations |
| Copper (Cu) | 4.65 eV | 266.6 nm | Ultraviolet (UV-C) | Particle accelerator RF electron photoinjectors |
| Gold (Au) | 5.10 eV | 243.1 nm | Far Ultraviolet | Spacecraft electrostatic discharge grounding |
| Platinum (Pt) | 5.65 eV | 219.4 nm | Deep Ultraviolet | High chemical inertness reference cathodes |
4. Worked Physics Problem: Ultraviolet Illumination of Zinc
A freshly prepared zinc metal plate ($Phi = 4.30text{ eV}$) is placed inside an evacuated photocell and illuminated by monochromatic ultraviolet radiation of wavelength $lambda = 250.0text{ nm}$.
- Planck Constant: $h = 6.62607 times 10^{-34}text{ J}cdottext{s} = 4.13567 times 10^{-15}text{ eV}cdottext{s}$
- Speed of Light: $c = 2.99792 times 10^8text{ m/s}$
- Electron Charge: $q = 1.60218 times 10^{-19}text{ C}$
- Electron Mass: $m_e = 9.10938 times 10^{-31}text{ kg}$
Step A: Incident Photon Energy
E_{text{photon}} = 4.95937 times 1.60218 times 10^{-19}text{ J} = 7.9458 times 10^{-19}text{ Joules}
Step B: Maximum Kinetic Energy of Photoelectrons
E_{k,max} = 0.6594 times 1.60218 times 10^{-19}text{ J} = 1.0565 times 10^{-19}text{ Joules}
Because $E_{text{photon}} (4.96text{ eV}) > Phi (4.30text{ eV})$, photoemission proceeds readily with a positive surplus kinetic energy of $0.66text{ eV}$.
Step C: Stopping Potential ($V_0$)
Applying an opposing reverse bias of $-0.66text{ V}$ will completely extinguish the measured circuit current.
Step D: Maximum Photoelectron Ejection Velocity ($v_{max}$)
= 481,622text{ m/s} approx 481.6text{ km/s}
As a fraction of light speed: $frac{v_{max}}{c} = frac{481,622}{2.9979 times 10^8} approx 0.0016$ ($0.16% c$). Classical Newtonian kinematics is exceptionally accurate here; relativistic corrections are negligible.
Step E: Threshold Cutoff Parameters
nu_0 = frac{c}{lambda_0} = frac{2.99792 times 10^8}{2.8834 times 10^{-7}} = 1.0397 times 10^{15}text{ Hz} = 1,039.7text{ THz}
If illuminated by visible blue light ($450text{ nm}$), $E_{text{photon}} = 2.76text{ eV} < 4.30text{ eV}$. Zero electrons will escape the zinc surface, no matter how bright the beam!
5. Photocathode Quantum Efficiency & Photomultiplier Tubes (PMTs)
Not every incident photon produces an emitted electron; many are reflected or dissipate as lattice heat. The Quantum Efficiency (QE) is:
In modern scientific Photomultiplier Tubes (PMTs), photoelectrons ejected from an alkali photocathode are accelerated across a cascade of electrostatic dynodes (each held at $+100text{ V}$ higher potential). Secondary electron multiplication yields a gain of $10^6$ to $10^8$, allowing single-photon detection in neutrino observatories (Super-Kamiokande) and astronomical spectrophotometry.