The Photoelectric Effect

When electromagnetic radiation of high enough frequency hits a metal surface, electrons are emitted from it — this is the $\text{photoelectric effect}$, and the emitted electrons are called $\text{photoelectrons}$. Three experimental facts define the effect: emission happens (effectively) instantly, with no measurable time delay; for a given metal, no electrons are emitted at all below a certain $\text{threshold frequency}$ $f_0$, no matter how intense the light; and above the threshold, increasing the $\text{intensity}$ increases the number of photoelectrons emitted per second, but not their maximum kinetic energy — only increasing the frequency does that.

These facts are impossible to explain with light as a continuous wave. A wave model says energy should arrive continuously and spread evenly across the whole illuminated surface, so a dim light of any frequency should eventually deliver enough energy to eject an electron — you would just have to wait longer. But experimentally, below $f_0$, waiting longer or shining more intense light changes nothing at all: electrons are never emitted.

Einstein resolved the puzzle by treating light as a stream of photons, each carrying a fixed energy $E=hf$ that depends only on frequency. A photoelectron is emitted when a single photon transfers all its energy to a single electron in one go — there’s no accumulation of energy from many photons over time, which is exactly why emission is instantaneous once it happens at all.

Freeing an electron from the metal surface costs a minimum amount of energy called the $\text{work function}$, $\varphi$, which depends on the metal. If a single photon’s energy $hf$ is less than $\varphi$, that photon simply cannot free an electron — no matter how many such photons arrive (more intensity just means more low-energy photons, none of which are individually enough). Only once $hf\ge \varphi$ can a photon free an electron at all, which defines the $\text{threshold frequency}$.

If a photon’s energy exceeds the work function, the surplus energy becomes kinetic energy of the emitted electron. Electrons bound less tightly than the surface minimum get away with more kinetic energy, so it is the $\text{maximum}$ kinetic energy, $E_{k(\max )}$ — from the least tightly bound electrons — that Einstein’s equation predicts.

This equation is a straight-line relationship if you plot $E_{k(\max )}$ against $f$: the gradient is always $h$ (whichever metal you use), and the intercept on the energy axis is $-\varphi$ — a graphical method that was used to give one of the most precise early confirmations of the value of the Planck constant.

Photoelectrons are emitted with a range of kinetic energies up to $E_{k(\max )}$. To measure this maximum experimentally, the emitted electrons are made to travel towards a collecting electrode held at a increasingly negative (retarding) potential. As this $\text{stopping potential}$ $V_s$ is increased, fewer and fewer electrons have enough kinetic energy to reach the collector against the opposing electric field, until the photocurrent falls to exactly zero.

At that point, even the fastest photoelectrons are just barely stopped, so all of their kinetic energy has been converted into electrical potential energy: $e{V}_s=E_{k(\max )}$. This gives a direct, purely electrical way to measure a quantity that started out as a mechanical kinetic energy.