Photoelectric Effect Calculator — Einstein’s Equation, Solved Both Ways
Shine light on a clean metal surface in vacuum and electrons come off — but only if the light is blue enough. That single qualification is what broke classical physics. This photoelectric effect calculator works through E = h·f = φ + KE_max in every direction: give it the light and the material and it returns the maximum kinetic energy and stopping potential; give it a measured stopping potential and it returns the work function; give it a table of measurements and it returns Planck’s constant.
What the equation actually says
A photon carries h·f of energy and hands all of it to one electron, or none of it. Freeing that electron from the surface costs the work function φ, a property of the surface that runs from about 2.1 eV for caesium to 5.7 eV for platinum. Whatever is left over becomes kinetic energy: KE_max = h·f − φ. Two consequences follow immediately, and both were experimentally confirmed before anyone believed the model.
First, there is a threshold frequency f₀ = φ/h, and a corresponding cutoff wavelength λ₀ = h·c/φ. Below the threshold, nothing happens at any brightness. Sodium’s cutoff is 543.8 nm, so ordinary green light ejects electrons from it; copper’s is 263.8 nm, deep in the ultraviolet, which is why a copper photocathode needs a laser rather than a lamp. Second, brightness changes how many electrons come off, never how fast. That is the observation classical wave theory could not survive.
Working an example by hand
Take 400 nm violet light on sodium, φ = 2.28 eV. The photon energy is E = h·c/λ = 3.100 eV, so KE_max = 3.100 − 2.28 = 0.820 eV, which is 1.313 × 10⁻¹⁹ J. Because a stopping potential of V volts is exactly V electronvolts of energy, V₀ = 0.820 V without any further arithmetic. The fastest electron leaves at √(2·KE/mₑ) = 5.370 × 10⁵ m/s, about 0.18 % of light speed, carrying a de Broglie wavelength of 1.355 nm. Only 26.4 % of the photon’s energy survives as motion; the rest went into escaping the metal.
Measuring Planck’s constant in an afternoon
Rearranged for a photocell experiment, Einstein’s equation becomes V₀ = (h/e)·f − φ/e — a straight line. Plot stopping potential against frequency for several filters, take the slope, multiply by the elementary charge and you have Planck’s constant; the intercept gives the work function for free. Millikan did exactly this in 1916, trying to disprove Einstein, and measured h to within half a percent instead. The least-squares fit in this tool reports the slope, the fitted h, its signed deviation from the defined SI value, the fitted φ, and R² to six decimal places — because a clean data set lands at 0.99999, which ordinary rounding would flatten into a meaningless bare 1.
Photocurrent, quantum efficiency and real detectors
For photodiodes and photomultipliers the interesting quantity is current, not energy. Beam power divided by photon energy gives the photon flux, and multiplying by the quantum efficiency and the elementary charge gives the saturation photocurrent. A 5 W/m² beam at 400 nm on one square centimetre delivers about 1.007 × 10¹⁵ photons/s; at 10 % efficiency that is 16.13 µA. This is the calculation behind every photocathode datasheet, and it is why bialkali and Cs₃Sb surfaces — with work functions near 2 eV — dominate blue-sensitive detectors.
When the classical speed formula stops working
v = √(2·KE/mₑ) assumes the electron is slow. Once the kinetic energy passes about one percent of the 511 keV electron rest energy — roughly 5.1 keV, which needs soft X-rays rather than visible light — the calculator switches to v = c·√(1 − 1/γ²) and says so. For every ordinary photoelectric problem the two agree far beyond display precision, but at X-ray energies the classical formula happily returns speeds above c, which is a good reminder that it was only ever an approximation.