Einstein’s Photoelectric Equation

In 1905, Albert Einstein proposed a revolutionary explanation for the photoelectric effect — the phenomenon where light shining on a metal surface ejects electrons. Classical wave theory predicted that any frequency of light, given enough intensity, should eventually knock electrons loose. Experiments told a completely different story: below a certain threshold frequency, no electrons were emitted no matter how bright the light. Einstein's insight was that light itself is quantised into discrete packets of energy called photons, and each photon interacts with exactly one electron in an all-or-nothing transaction. Einstein's photoelectric equation is a precise energy-balance statement. A single photon arrives carrying energy E = hf. A portion of that energy — called the work function φ — is spent simply breaking the electron free from the metal's surface. Whatever energy remains becomes the kinetic energy the electron carries away. If the photon's energy is less than the work function, the electron never escapes, regardless of how many photons arrive per second. This is why intensity (number of photons) cannot substitute for frequency (energy per photon). This equation earned Einstein the 1921 Nobel Prize in Physics and laid one of the cornerstones of quantum mechanics. It demonstrated that energy is not delivered continuously by waves but in discrete quanta, fundamentally changing how physicists understood the nature of light and matter. Today, Einstein's photoelectric equation underpins technologies ranging from solar cells and digital camera sensors to medical X-ray detectors and night-vision devices.

Key takeaways
  • Photon Energy: The Currency of the Transaction
  • The Work Function: The Entry Fee
  • Energy Conservation: What's Left Becomes Kinetic Energy
  • Threshold Frequency: The Minimum Coin Value
  • Stopping Voltage: Measuring KE_max Experimentally

Photon Energy: The Currency of the Transaction

Planck had already shown that oscillating charges emit energy in quanta of size hf, where h = 6.626 × 10⁻³⁴ J·s is Planck's constant and f is frequency. Einstein went further: he proposed that light itself travels as photons, each carrying exactly E_photon = hf. Think of each photon as a single coin with a value set by its colour (frequency). Blue light coins are worth more than red light coins.

The Work Function: The Entry Fee

Electrons inside a metal are bound by attractive forces from the lattice of positive ions. To escape the surface, an electron must do work against these forces. The minimum energy required to free an electron from the surface is called the work function φ (phi), measured in joules or electron-volts (eV). It is a property of the specific metal — for example, sodium has φ ≈ 2.3 eV while platinum has φ ≈ 5.7 eV. Think of φ as the 'cover charge' to get out of the metal.

Energy Conservation: What's Left Becomes Kinetic Energy

When a photon is absorbed by an electron at the surface, conservation of energy demands: (photon energy) = (energy to escape) + (kinetic energy of freed electron). The freed electron's kinetic energy is maximised when it starts right at the surface and loses the minimum possible energy escaping. Electrons deeper in the metal lose additional energy in collisions before reaching the surface, so they emerge with less KE. The maximum kinetic energy any electron can have is therefore KE_max = hf − φ. If hf < φ, the result is negative — physically impossible — so no electrons escape.

Threshold Frequency: The Minimum Coin Value

Emission only occurs when hf ≥ φ. Setting KE_max = 0 defines the threshold frequency f₀ = φ/h. Light below this frequency simply cannot liberate electrons, no matter how intense. Above f₀, increasing intensity means more photons per second, so more electrons are ejected per second — but each electron's maximum KE depends only on frequency, not intensity. This is the signature quantum result that wave theory cannot explain.

Stopping Voltage: Measuring KE_max Experimentally

Experimentally, KE_max is measured using a stopping voltage V_s: an opposing electric potential is applied until even the fastest electrons are just stopped. At this point, eV_s = KE_max, so KE_max = eV_s, where e = 1.602 × 10⁻¹⁹ C is the elementary charge. Millikan's precise measurements (1916) confirmed Einstein's linear prediction of KE_max vs. frequency and independently measured h with high accuracy.

The Key Equation

KEmax⁡=hf−φKE_{\max} = hf - \varphi
  • KE_{\max} — Maximum kinetic energy of ejected electron (Joules (J) or electron-volts (eV)): The largest kinetic energy a photoelectron can have; achieved by electrons that escape from the very surface without losing energy in collisions.
  • h — Planck's constant (J·s): The fundamental quantum of action, h = 6.626 × 10⁻³⁴ J·s. It sets the scale of all quantum phenomena.
  • f — Frequency of incident light (Hertz (Hz)): The number of wave cycles per second of the incoming photon. Higher frequency means higher photon energy. Related to wavelength by f = c/λ.
  • \varphi — Work function (Joules (J) or electron-volts (eV)): The minimum energy needed to liberate one electron from the metal surface. It is a constant characteristic of the material.

Worked Example

Problem. Ultraviolet light of wavelength 200 nm shines on a sodium surface whose work function is φ = 2.28 eV. (a) Calculate the energy of one photon in eV. (b) Find KE_max of the ejected electrons in eV. (c) Determine the stopping voltage needed to halt the fastest electrons.

Ephoton=6.20 eV,KEmax⁡=3.92 eV,Vs=3.92 VE_{photon} = 6.20\text{ eV},\quad KE_{\max} = 3.92\text{ eV},\quad V_s = 3.92\text{ V}

Answer. Photon energy = 6.20 eV; KE_max = 3.92 eV; stopping voltage = 3.92 V

Figure
Diagram
Einstein's Photoelectric Equation — Energy BalanceMETALe⁻ bound insideSurfacePhotonhf = 6.20 eVe⁻KE_max= 3.92 eVEnergy Barhf6.20 eVφ = 2.28 eVKE_max= 3.92 eVKE_max = hf − φ → 3.92 eV = 6.20 eV − 2.28 eV
Origins · 1 minAlbert Einstein and how this was discovered
AE
Albert Einstein
1879–1955 · German-American

In his 1905 'miracle year', Einstein published his paper 'On a Heuristic Point of View Concerning the Production and Transformation of Light', proposing that light consists of quantised packets (photons) and deriving the photoelectric equation KE_max = hf − φ. This was the work for which he was awarded the Nobel Prize in Physics in 1921, cited explicitly for 'his discovery of the law of the photoelectric effect'. Einstein received the Nobel Prize not for relativity — which was still considered controversial — but for the photoelectric effect. He was actually traveling by ship to Japan when the prize was announced and could not attend the ceremony in Stockholm.

  • Observations of the Photoelectric Effect — Understanding what was experimentally observed — threshold frequency, independence of KE from intensity — is essential context for why Einstein's equation was needed.
  • Work Function and Threshold Frequency — The work function φ and the threshold frequency f₀ = φ/h are key quantities in Einstein's equation and vary by material.
  • Planck's Quantum Hypothesis — Einstein built directly on Planck's idea that energy comes in quanta of size hf; understanding Planck provides the foundation for Einstein's photon model.
  • Wave-Particle Duality of Light — The photon model proposed by Einstein is a central example of light behaving as a particle, leading to the broader concept of wave-particle duality.