The photoelectric effect

Key ideas

• The photoelectric effect is the emission of electrons from a metal surface when light shines on it. Its details cannot be explained if light is treated as a continuous wave — they need the photon model.
• In the photon model, light is a stream of discrete energy packets called photons. Each photon carries energy:

$E = hf$

• $E$ — the energy of one photon (J)
• $h$ — the Planck constant, $h = 6.63\times10^{-34}$ J s (equivalently $h = 4.14\times10^{-15}$ eV s)
• $f$ — the frequency of the light (Hz)
• A photon is absorbed by one electron at a time. If the photon's energy is enough, the electron is freed from the metal.
• The work function, $W_0$, is the minimum energy needed to remove an electron from a particular metal's surface (J, or often quoted in eV).
• The threshold frequency, $f_0$, is the minimum light frequency that can eject electrons at all: $W_0 = hf_0$. Below $f_0$, no electrons are emitted, no matter how bright the light — a single low-energy photon still can't free an electron, and increasing intensity just supplies more (still too-weak) photons.

Einstein's photoelectric equation

• Any photon energy above the work function goes into the ejected electron's kinetic energy. Electrons ejected from right at the surface, with no energy lost on the way out, leave with the maximum possible kinetic energy, $E_{k(max)}$:

$hf = W_0 + E_{k(max)}$

• Each variable, with sub-bullets:
  • $hf$ — the energy delivered by one absorbed photon (J)
  • $W_0$ — the work function of the metal (J)
  • $E_{k(max)}$ — the maximum kinetic energy of an emitted electron (J)
• Increasing the light's intensity (brightness) at a fixed frequency increases the number of photons per second, so it increases the number of electrons emitted per second (the photoelectric current) — but not $E_{k(max)}$, since each photon still only carries $hf$.
• Increasing the light's frequency increases the energy per photon, so it increases $E_{k(max)}$ of the emitted electrons.
• These two results are exactly what a wave model of light cannot explain: a wave model predicts intensity (not frequency) should control the electrons' energy, and predicts a delay before emission at low intensity — neither is observed.

Stopping voltage

• $E_{k(max)}$ can be measured directly using a stopping voltage, $V_s$: the minimum retarding voltage that stops even the fastest photoelectrons from reaching a collector.
• All of the electron's kinetic energy is converted to electric potential energy at the point it is stopped:

$eV_s = E_{k(max)}$

• $e$ — the electron charge, $1.60\times10^{-19}$ C
• $V_s$ — the stopping voltage (V)
• Combining with Einstein's equation: $eV_s = hf - W_0$, so a graph of $V_s$ against $f$ is a straight line of gradient $h/e$ and $y$-intercept $-W_0/e$ — this is how $h$ was first measured experimentally.

Tips

• "No electrons below threshold frequency" is about frequency, not intensity. A very bright light below $f_0$ still emits nothing — this is the single most-tested conceptual point on this topic. Never say "if it's bright enough, electrons will eventually be emitted."
• When asked to explain why the photon model fits the evidence but the wave model doesn't, name both observations: (1) no emission below $f_0$ regardless of intensity, and (2) $E_{k(max)}$ depends on frequency, not intensity. One alone rarely earns full marks.