The Photoelectric Effect: Einstein's Quantum Revolution

The photoelectric effect is the experiment that launched quantum mechanics. When light shines on a metal surface, electrons can be ejected — but only if the light's frequency is high enough. Brighter light ejects more electrons but doesn't make them move faster. Only higher-frequency light increases their kinetic energy. This simple observation was completely inexplicable by classical wave theory and led Einstein, in 1905, to propose that light comes in discrete packets of energy called photons.

The classical prediction (and why it fails)

Classical physics treats light as a continuous wave. A wave carries energy proportional to its intensity (amplitude squared). So classically, you'd expect:

1. Brighter light → more energy → electrons ejected with more kinetic energy
2. Any frequency should work — dim red light would just need more time to build up enough energy
3. There should be a time delay at low intensities while the electron "absorbs" enough wave energy

Every single one of these predictions is wrong. Experiments show:

1. Brighter light ejects more electrons, but each individual electron's energy depends only on frequency
2. Below a certain threshold frequency, NO electrons are ejected — not even with blinding intensity
3. Electrons are ejected instantaneously, even at very low light levels

Einstein's quantum explanation

Einstein proposed that light energy comes in discrete quanta — photons — each carrying energy:

$E = hf$

where $h = 6.626 \times 10^{-34}$ J·s is Planck's constant and $f$ is the frequency.

When a single photon hits a metal surface, it can transfer ALL its energy to a single electron. The electron needs a minimum energy $\phi$ (the work function) to escape the metal surface. If $hf \geq \phi$, the electron escapes; any leftover energy becomes kinetic energy:

$KE_{\max} = hf - \phi$

If $hf < \phi$, the photon doesn't have enough energy and the electron stays put — no matter how many photons (intensity) you throw at the surface.

Solving the problem

Cesium: $\phi = 2.1$ eV. First, the threshold frequency:

$f_{0} = \dfrac{\phi}{h} = \dfrac{2.1 \times 1.602 \times 10^{-19}\text{ J}}{6.626 \times 10^{-34}\text{ J·s}} = 5.08 \times 10^{14}\text{ Hz}$

This corresponds to a wavelength:

$\lambda_{0} = \dfrac{c}{f_{0}} = \dfrac{3 \times 10^{8}}{5.08 \times 10^{14}} = 590\text{ nm (yellow-orange light)}$

So yellow-orange and any color with longer wavelength (red, infrared) will NOT eject electrons from cesium, no matter how bright. Green, blue, violet, and ultraviolet will.

Example 1: Violet light, $f = 7.5 \times 10^{14}$ Hz ($\lambda = 400$ nm):

$KE_{\max} = hf - \phi = (6.626 \times 10^{-34})(7.5 \times 10^{14}) - 2.1 \times 1.602 \times 10^{-19}$ $= 4.97 \times 10^{-19} - 3.36 \times 10^{-19} = 1.61 \times 10^{-19}\text{ J} = 1.0\text{ eV}$

Example 2: UV light, $f = 1.0 \times 10^{15}$ Hz ($\lambda = 300$ nm):

$KE_{\max} = (6.626 \times 10^{-34})(10^{15}) - 3.36 \times 10^{-19} = 6.63 \times 10^{-19} - 3.36 \times 10^{-19} = 3.27 \times 10^{-19}\text{ J} = 2.04\text{ eV}$

The stopping voltage method

Experimentally, $KE_{\max}$ is measured by applying a reverse voltage $V_{s}$ that just barely stops the fastest electrons:

$eV_{s} = KE_{\max} = hf - \phi$

where $e$ is the electron charge. Plotting $V_{s}$ vs $f$ gives a straight line with slope $h/e$ and $x$-intercept at $f_{0}$. This is how Millikan experimentally confirmed Einstein's equation (ironically, Millikan initially set out to disprove it).

Work functions of common metals

• Cesium: 2.1 eV (lowest — used in photocells)
• Potassium: 2.3 eV
• Sodium: 2.75 eV
• Calcium: 2.9 eV
• Zinc: 4.3 eV
• Copper: 4.65 eV
• Platinum: 5.65 eV

Metals with lower work functions are easier to trigger. Cesium is used in photocathodes precisely because even visible light can eject its electrons.

Real-world applications

• Solar cells (photovoltaics): Photons with energy above the semiconductor's band gap create electron-hole pairs that generate current. Too-low-energy photons are wasted as heat; too-high-energy photons waste the excess above the band gap.
• Photomultiplier tubes: Use the photoelectric effect to detect individual photons. A photon ejects one electron, which is then amplified through a cascade of secondary emissions.
• Digital cameras (CCD/CMOS sensors): Each pixel is a photoelectric device — photons create charge carriers that are counted.
• Night vision: Uses the photoelectric effect with materials sensitive to infrared.

Common mistakes

• Thinking brighter light gives faster electrons. Intensity affects the number of photons per second (and thus the number of ejected electrons — the photocurrent), but not the energy per photon.
• Forgetting unit conversions. Work functions are often in eV; $h$ is in J·s. Convert: $1\text{ eV} = 1.602 \times 10^{-19}\text{ J}$.
• Using wavelength instead of frequency in $E = hf$. If given $\lambda$, convert first: $f = c/\lambda$.

Try it in the visualization

Drag the frequency slider from infrared to ultraviolet. Below the threshold (marked with a red line), no electrons are ejected — the photons just bounce off. Above the threshold, electrons fly out and the KE bar grows linearly with frequency. Switch metals to see how the threshold shifts. Toggle intensity — more photons per second means more electrons, but each electron's KE stays the same.