Physics · Atomic and quantum physics · Grade 11

The photoelectric effect

Why can strong red light not knock out an electron, while weak ultraviolet light can?

The situation

In night-vision goggles, weak light first falls on a thin metal layer — the photocathode — and knocks electrons out of it; the electrons are amplified and form an image on a screen. Light sensors and old photocells are also based on this phenomenon. (Solar panels, on the other hand, work by the internal photoelectric effect: the electron does not leave the metal but becomes free inside a semiconductor.) Why can strong red light not release an electron, while weak ultraviolet light can?

Topic

The photoelectric effect. Photocell and a night-vision device

Key ideas

Photon energy
Work function
Einstein's equation

Formulas

E = h ν = \frac{h c}{λ}

The energy of a single photon depends on the frequency, not on the intensity.

h ν = A + \frac{m v_{m a x}^{2}}{2}

Einstein's equation: the photon's energy is spent on the work function and on the electron's kinetic energy.

e U_{t} = \frac{m v_{m a x}^{2}}{2} = K_{m a x}

The stopping (retarding) voltage prevents even the fastest electron from reaching the anode.

ν_{0} = \frac{A}{h}, λ_{m a x} = \frac{h c}{A}

The red limit of the photoelectric effect: at a wavelength longer than this no electron is emitted.

I_{t} = e η N_{f}

The saturation current is proportional to the number $N_{f}$ of photons arriving per second, that is, to the intensity.

Theory

Light — a stream of photons

Light energy arrives “in pieces” — as photons. A single photon gives all its energy to a single electron. Increasing the intensity increases the number of photons, but the energy $h ν$ of each photon does not change. A photon has no mass, but it does have momentum: $p = h / λ = h ν / c$ — this is the origin of the pressure of light (in detail in the “Photon momentum and the Compton effect” experiment).

Why does frequency decide?

To free an electron from a metal, an energy of at least $A$ (the work function) is needed. If $h ν < A$ , however many photons there are, none of them can free an electron — energy is not accumulated bit by bit. That is why ultraviolet light, not red, knocks electrons out of zinc.

Current–voltage characteristic

If a positive voltage is applied to the anode, almost all the electrons are collected — the current saturates. At a negative (retarding) voltage only sufficiently fast electrons get through; at $U = - U_{t}$ the current falls to zero. From $U_{t}$ we find $K_{m a x} = e U_{t}$ .

Measuring Planck's constant

If we measure $U_{t}$ at different frequencies and plot $e U_{t}$ against $ν$ , a straight line results: its slope gives $h$ and its intercept gives $A$ . Robert Millikan confirmed Einstein's equation in 1916 by exactly this method. Model: the electron energy is uniformly distributed over the range $[0; K_{m a x}]$ , the direction follows Lambert's law, the quantum yield is $η = 0.1 %$ , the electrodes are flat, and the gap is 20 mm. The contact potential difference between anode and cathode is not taken into account: in a real apparatus it adds to the measured $U_{t}$ , so the intercept of the straight line actually gives the work function of the anode material, while its slope ( $h$ ) does not change.

The laws of the photoelectric effect and photocells

In 1888–1890 A. G. Stoletov experimented with an illuminated metal plate (the cathode) and a metal mesh in front of it (the anode), and established that the saturation current is proportional to the light flux (Stoletov's law) and that the current appears the moment the plate is illuminated — the photoelectric effect is instantaneous. P. Lenard (1902) found that the maximum energy of photoelectrons does not depend on the intensity and increases with frequency; for every substance there is a red limit $ν_{0} = A / h$ . Classical wave theory could not explain the last two laws — in 1905 Einstein explained them with the idea of the photon. Applications: vacuum photocells (reading the sound track in old film projectors, photorelays); the photomultiplier tube (PMT) — each electron released from the photocathode hits a series of dynodes and knocks out secondary electrons, and with 10–12 dynodes the current is amplified 10⁶–10⁷ times, so a PMT can detect even a single photon; night-vision devices accelerate the electrons released from the photocathode and direct them onto a luminescent screen. In the internal photoelectric effect the electron does not leave the material but becomes free inside a semiconductor and raises its conductivity — the photoresistor, photodiode and solar cell (in the “Semiconductor physics” experiment).

The chemical action of light and photography

Light heats a substance, knocks out electrons, exerts pressure and also drives chemical reactions: photosynthesis, the change of the rhodopsin molecule in the retina of the eye, the tanning of skin in the sun, the fading of dyes. Here too one photon changes one molecule, so, as in the photoelectric effect, it is the photon energy that decides, not the strength of the light: ultraviolet light burns, while red light does not burn even if it falls for hours. In photography, in crystals of silver bromide (AgBr) a photon frees an electron, and the electron turns a silver ion into an atom — centres of the “latent image” form; the developer turns them into grains of black silver. Ordinary black-and-white photographic paper is sensitive only to blue-violet and ultraviolet light — which is why in a darkroom you can work under a red lamp.

Did you know?

Albert Einstein received the Nobel Prize for 1921 not for the theory of relativity, but precisely for explaining the law of the photoelectric effect.
Since 2019 Planck's constant has an exactly defined value: $h = 6.62607015 \cdot 10^{- 34}$ J·s. The kilogram is now defined through this very constant.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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