Use the photon model and Einstein's photoelectric equation to explain the photoelectric effect.

What this dot point is asking

The photoelectric effect was one of the first experiments to reveal the quantum nature of light, and it earned Einstein his Nobel Prize. It is central to Unit 4's story of how classical physics gave way to quantum ideas.

The experiment

When light shines on a clean metal surface, electrons can be ejected. These ejected electrons are called photoelectrons. Careful measurements revealed three features that classical wave theory of light could not explain:

1. There is a threshold frequency below which no electrons are emitted, no matter how bright (intense) the light.
2. The maximum kinetic energy of the emitted electrons depends on the frequency of the light, not its intensity.
3. Emission is essentially instantaneous, even for very dim light above the threshold frequency.

Classical theory predicts that any frequency should work given enough intensity or time, so these results demanded a new model.

The photon model

Einstein proposed that light energy is carried in discrete quanta, now called photons. Each photon has energy proportional to the frequency $f$ of the light:

where $h = 6.63 \times 10^{-34}\ \text{J s}$ is Planck's constant and $\lambda$ is the wavelength. One photon interacts with one electron in a single, all-or-nothing event.

Work function and threshold frequency

To escape the metal, an electron must be given at least a minimum energy called the work function $\phi$, a property of the metal. The threshold frequency $f_0$ is the lowest frequency whose photon energy just equals the work function:

If the photon energy $hf$ is less than $\phi$, no electron escapes, however intense the beam. This neatly explains feature 1.

Einstein's photoelectric equation

Energy conservation for one photon ejecting one electron gives Einstein's photoelectric equation:

The photon supplies energy $hf$. Part of it ($\phi$) is used to free the electron, and the rest becomes the maximum kinetic energy of the photoelectron. Increasing the frequency raises $E_{k,\text{max}}$ linearly, while increasing the intensity only increases the number of photoelectrons, not their energy. A graph of $E_{k,\text{max}}$ against $f$ is a straight line of gradient $h$ and horizontal intercept $f_0$.

In the exam, watch your units: energies are often quoted in electronvolts, where $1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$. Convert to joules before substituting, and use the linear graph of $E_{k,\text{max}}$ against $f$ to extract Planck's constant or the work function when asked.