The Photoelectric Effect Explained
The photoelectric effect is the emission of electrons from a metal surface when light of sufficiently high frequency shines on it, and it is described by Einstein’s equation KEmax = hν − φ, where φ is the work function of the metal. In brief: violet light of 400 nm releases electrons from sodium (φ = 2.3 eV) with a maximum kinetic energy of 0.80 eV, but red light of 700 nm does not release any electrons, however intense it is.
The photoelectric effect could not be explained by the wave theory of light, and its explanation by Einstein in 1905 was a main step in the development of quantum theory. It is also the basis of solar cells, light sensors and digital cameras. This article describes the observations, shows why the wave theory fails, derives the equation and works through examples including the threshold frequency and the speed of the emitted electrons.
What is observed?
When light falls on a clean metal surface in a vacuum, electrons (photoelectrons) may be ejected. The experiments show:
- There is a threshold frequency ν0 for each metal. Below it no electrons are emitted, however intense the light is.
- Above the threshold, the number of electrons emitted per second is proportional to the intensity of the light.
- The maximum kinetic energy of the electrons is independent of the intensity, and increases linearly with the frequency of the light.
- The emission starts almost instantaneously (within about 10−9 s), even for faint light.
Why does classical physics fail?
| Observation | Wave theory predicts | Photon theory explains |
|---|---|---|
| Threshold frequency | Any frequency should work if intense enough | One photon must have at least energy φ |
| Kinetic energy independent of intensity | More intense light should give faster electrons | Each electron absorbs one photon; intensity gives the number of photons |
| Kinetic energy rises with frequency | No dependence on frequency expected | KE = hν − φ |
| No time delay | Time needed to accumulate energy (seconds to minutes for dim light) | One photon transfers all its energy at once |
What is Einstein’s explanation?
Einstein proposed that light consists of particles, photons, each with an energy of hν. One photon is absorbed by one electron, and its energy is used to free the electron from the metal (the work function φ), and the rest becomes kinetic energy:
hν = φ + KEmax, or KEmax = hν − φ
The work function φ is the minimum energy required to remove an electron from the metal, and the threshold frequency is ν0 = φ/h, with the corresponding threshold wavelength λ0 = hc/φ. The kinetic energy is maximal for the electrons at the surface; deeper electrons lose some energy on their way out. In an experiment, the maximum kinetic energy is measured by the stopping potential V0, the retarding voltage that just stops the fastest electrons: eV0 = KEmax. A plot of V0 against ν is a straight line with a slope h/e, and this plot gives an experimental value of Planck’s constant.
Work functions of metals
| Metal | Work function φ (eV) | φ (10−19 J) | Threshold ν0 (1014 Hz) | Threshold λ0 (nm) |
|---|---|---|---|---|
| Cesium | 2.1 | 3.4 | 5.1 | 590 |
| Sodium | 2.3 | 3.7 | 5.6 | 540 |
| Zinc | 4.3 | 6.9 | 10.4 | 290 |
| Copper | 4.7 | 7.5 | 11.4 | 264 |
| Platinum | 5.6 | 9.0 | 13.5 | 222 |
Alkali metals have low work functions, and so respond to visible light, while most other metals need ultraviolet light. Values depend on the surface condition and crystal face, and differ slightly between sources.
Worked examples
Example 1: threshold frequency of sodium
φ = 2.3 eV = 2.3 × 1.602 × 10−19 J = 3.69 × 10−19 J. ν0 = φ / h = 3.69 × 10−19 / 6.626 × 10−34 = 5.56 × 1014 Hz. λ0 = c/ν0 = 3.00 × 108 / 5.56 × 1014 = 5.4 × 10−7 m = 540 nm. Light with a wavelength longer than 540 nm (orange, red) cannot emit electrons from sodium.
Example 2: kinetic energy and speed of the electrons
Light of 400 nm falls on sodium. The photon energy is E = hc/λ = 1.988 × 10−25 / 400 × 10−9 = 4.97 × 10−19 J = 3.10 eV.
- KEmax = 3.10 − 2.3 = 0.80 eV = 1.28 × 10−19 J.
- v = √(2 KE / m) = √(2 × 1.28 × 10−19 / 9.109 × 10−31) = √(2.81 × 1011) = 5.3 × 105 m/s.
- The stopping potential is V0 = KE/e = 0.80 V.
Example 3: no emission
Red light of 700 nm has a photon energy of 2.84 × 10−19 J = 1.77 eV, lower than 2.3 eV for sodium. No electrons are emitted, no matter how many photons arrive per second, because each electron can absorb only one photon at a time, which has insufficient energy.
Example 4: effect of doubling the intensity
If the intensity of the 400 nm light is doubled, the number of photons per second doubles, so twice as many electrons are emitted (the current doubles), but the maximum kinetic energy stays at 0.80 eV.
Example 5: finding the work function
Light of wavelength 250 nm ejects electrons from a metal with a maximum speed of 6.0 × 105 m/s. The photon energy is 7.95 × 10−19 J. KE = ½mv2 = 0.5 × 9.109 × 10−31 × (6.0 × 105)2 = 1.64 × 10−19 J. Then φ = 7.95 × 10−19 − 1.64 × 10−19 = 6.31 × 10−19 J = 3.9 eV.
Where is the effect used?
- Solar cells: photons promote electrons in a semiconductor, an internal version of the effect that generates a current.
- Photodiodes and light meters: detection of light intensity.
- Photomultiplier tubes: a photoelectron is multiplied by a cascade of electrodes, so single photons can be detected.
- Digital cameras: charge-coupled devices and CMOS sensors convert light into electric charge.
- Photoelectron spectroscopy: the kinetic energies of electrons ejected from atoms or molecules by UV or X-ray photons give their binding energies, and so the energy levels of the electrons.
Who discovered and tested it?
Heinrich Hertz noticed in 1887 that ultraviolet light helps to produce a spark between electrodes. Philipp Lenard showed in 1902 that the energy of the electrons does not depend on the intensity of the light. Albert Einstein explained the effect in 1905 with the hypothesis of light quanta, and received the Nobel Prize in Physics in 1921, specifically for the explanation of the photoelectric effect. Robert Millikan, who at first doubted Einstein’s theory, carried out precise measurements in 1914–1916 that confirmed the equation and gave a value of Planck’s constant to within about 0.5 %. He received the Nobel Prize in 1923.
Which mistakes are common?
- Thinking that a brighter light always ejects electrons: only if the frequency is above the threshold.
- Saying that intensity changes the electron energy: it changes only the number.
- Mixing up eV and J: 1 eV = 1.602 × 10−19 J.
- Forgetting that φ is a property of the metal: it does not depend on the light.
- Assuming that electrons accumulate energy from several photons: in the ordinary effect, one photon is absorbed by one electron.