Explain the photoelectric effect using photons, work function, threshold frequency and Einstein's equation

What this dot point is asking

WACE wants you to describe the photoelectric effect, explain why it forced a particle (photon) model of light, and apply Einstein's photoelectric equation including the work function, threshold frequency and stopping voltage. The central argument is that the experimental results contradict the wave model and are explained naturally if light is quantised.

Photons

Light is emitted and absorbed in discrete packets called photons, each carrying energy

where $h=6.63\times10^{-34}\ \text{J s}$ is Planck's constant. A brighter beam contains more photons but each photon still has the same energy fixed by the frequency. This is the key break from classical physics, where energy was thought to be spread continuously across a wave.

The photoelectric effect

When light of high enough frequency strikes a metal surface, electrons are ejected. The experimental facts are:

• Emission happens only above a threshold frequency $f_0$, no matter how intense the light.
• Above threshold, increasing intensity increases the number of electrons but not their maximum kinetic energy.
• The maximum kinetic energy rises linearly with frequency.
• Emission is essentially instantaneous.

A wave model cannot explain these: it predicts that any frequency should work given enough time and intensity, and that brighter light should give faster electrons. None of that is observed.

The work function and threshold

The work function $W$ is the minimum energy needed to free an electron from the surface. A single photon must supply at least this much in one interaction. The threshold frequency is the frequency at which a photon just provides $W$,

Below $f_0$ no electrons escape however bright the light, because no single photon has enough energy.

Einstein's photoelectric equation

For photons above threshold, the photon energy splits into the work function plus the kinetic energy of the freed electron. The most energetic electrons (those at the surface) have

Plotting $E_{k,max}$ against $f$ gives a straight line of gradient $h$ and frequency-intercept $f_0$, a classic graph question. The vertical intercept is $-W$.

Stopping voltage

The maximum kinetic energy can be measured with a stopping voltage $V_0$, the reverse potential difference that just halts the fastest electrons,

Because $V_0$ depends on frequency but not intensity, it provides direct evidence for the photon model.

Working in joules and electronvolts

Work functions are usually quoted in electronvolts, where $1\ \text{eV}=1.60\times10^{-19}\ \text{J}$. Decide on one unit and convert everything to it before substituting into $E_{k,max}=hf-W$. Keep frequencies in hertz and wavelengths in metres.