How this instrument works
KE_max is the fastest a photoelectron can leave a surface once light of a given frequency knocks it loose. Each photon carries a fixed packet of energy, hf, set only by frequency, never by brightness. Some of that energy pays the toll the material charges to let an electron escape, called the work function, φ. Whatever energy remains becomes kinetic energy, so KE_max = hf − φ is a one-line statement of conservation of energy applied to a single photon meeting a single electron.
Before 1905, light was treated as a continuous wave, and a wave's energy should build up gradually as intensity rises, eventually loosening any electron given enough time. Real photocells refused to behave that way: below a sharp threshold frequency, no electrons came out no matter how long or how strongly the surface was lit. Einstein's fix was to treat light as arriving in discrete quanta of energy hf, extending Planck's 1900 idea about radiation to light itself. A photon below the threshold simply doesn't carry enough energy to pay φ, however many of them arrive.
The formula has a hard edge: when hf is less than φ, the honest answer is that no photoelectrons exist at all, not a negative kinetic energy — the equation only describes the regime above threshold. It also reports a maximum, not a typical value, because electrons sit at different depths and lose energy escaping from below the surface, so a real photocurrent shows a spread of kinetic energies topping out at KE_max. And φ is not a fixed property of an element alone — a contaminated or differently cut crystal face of the same metal can shift it by several tenths of an electronvolt, which is why photocathode manufacturers control surface preparation so carefully.
- Enter the incident light's frequency into Light frequency, in hertz — visible light runs roughly 4×10¹⁴ to 8×10¹⁴ Hz.
- Enter the target surface's Material work function in electronvolts; look up the value for the specific metal or coating, since it varies by material and surface condition.
- Read the result in Maximum photoelectron kinetic energy — it reports in eV, the natural unit for single-electron energies this small.
- A negative reading means the frequency sits below that material's threshold; no electrons are ejected at all, no matter how intense the light.
Worked example — cesium-like photocathode at 500 THz
Take a surface with a work function of 1.9 eV — close to cesium's, among the lowest of any metal, which is exactly why cesium coatings show up in real photocathodes and photomultiplier tubes. Shine light on it at a frequency of 5×10¹⁴ Hz, in the green part of the visible spectrum.
First find the energy per photon: hf = 4.135667696×10⁻¹⁵ eV·s × 5×10¹⁴ Hz = 2.067833848 eV. Subtract the work function: KE_max = 2.067833848 − 1.9 = 0.167833848 eV. That is Einstein's photoelectric equation run end to end, the exact result his 1905 paper predicted and that Robert Millikan's careful photocell measurements confirmed years later, before quantum theory had matured enough to explain why it worked.
Questions
Why doesn't brighter light raise the maximum kinetic energy?
Because intensity means more photons arriving per second, not more energy per photon. Each photon still carries hf, fixed by frequency alone, so KE_max stays the same. A brighter beam knocks loose more electrons per second, raising the photocurrent, but it cannot push any single electron out faster than a dimmer beam of the same frequency does.
What happens below the threshold frequency?
Nothing is ejected, no matter how long or how intensely the surface is lit. Below threshold, hf is less than φ, so no single photon carries enough energy to free an electron, and photons don't pool their energy for this interaction. A negative reading here signals the physical answer is zero electrons, not a negative energy.
How do I find the threshold frequency for a material?
Divide the work function by Planck's constant: f0 = φ / h. For a work function of 1.9 eV, f0 = 1.9 ÷ 4.135667696×10⁻¹⁵ eV·s ≈ 4.59×10¹⁴ Hz, near the red end of the visible spectrum. Any frequency above that value frees electrons from this material; anything below never will, regardless of intensity.
Does this prove light behaves as a particle?
It shows light energy arrives in discrete packets during this interaction, something a continuous wave cannot explain. A wave picture predicts photoelectrons at any frequency once enough energy accumulates, with energy scaling by intensity — neither happens here. Einstein's 1905 explanation of that exact mismatch, not his later relativity work, is what earned him the 1921 Nobel Prize in Physics.
Why does the work function vary for the same metal?
Because it depends on the exact surface, not just the bulk element. Different crystal faces, adsorbed gas layers, and oxide films each shift the energy needed to free an electron, sometimes by several tenths of an electronvolt. Published tables give typical clean-surface values — cesium near 1.9 to 2.1 eV, sodium near 2.36 eV, platinum near 5.6 eV — but a real photocathode is measured, not assumed.
Why is the result reported in electronvolts instead of joules?
Because single-electron energies in joules are awkwardly small — 0.167833848 eV works out to about 2.688×10⁻²⁰ joules. One electronvolt is the energy an electron gains crossing a one-volt potential, a natural scale for atomic and photoelectric processes, so physicists quote results directly in eV rather than carrying factors of 10⁻¹⁹ through every step.