Photoelectric Effect

In one sentence: Light can knock electrons out of a metal, but only if each packet of light carries enough energy, and the leftover energy becomes the electron’s speed.

$K_{\max} = h f - \phi$

Read it out loud: “K max equals h times f minus phi. K max is the largest kinetic energy of an ejected electron, h is Planck's constant, f is the light's frequency, and phi is the metal's work function.”

What each symbol means

Symbol	Meaning
$K_{\max}$	The largest kinetic energy (energy of motion) an ejected electron can have, in joules (J) or electron volts (eV).
$h$	Planck’s constant, $6.62607015 \times 10^{-34}$ J s (joule seconds).
$f$	The frequency of the light, in hertz (Hz). Each photon of frequency $f$ can free at most one electron.
$\phi$	The work function of the metal: the minimum energy needed to pull an electron out of its surface, in joules or electron volts. Each metal has its own value.

What the formula is doing

Physicists noticed in the late 1800s that light shining on a metal can knock electrons out of it, and Heinrich Hertz reported the effect in 1887. Wave theory predicted that brighter light, which carries more energy, should eject electrons with more energy, and that dim light should work if it shines long enough. Experiments showed otherwise. The frequency of the light decided whether electrons came out at all, while brightness only changed how many came out.

In 1905 Einstein explained this with photons. One photon hits one electron and hands over all of its energy $hf$ . Part of that energy pays the cost of escaping the metal, the work function $\phi$ . The rest becomes the electron’s kinetic energy, the energy of its motion.

That gives the formula $K_{\max} = hf - \phi$ . If $hf$ is less than $\phi$ , the photon cannot pay the cost, the formula would give a negative number, and no electron escapes. Brightness does not help, because it changes the number of photons, not the energy of each one.

The word max is needed because electrons start at different depths in the metal. An electron from deeper inside loses some energy on its way out, so only the electrons that lose the least reach the surface with the full $hf - \phi$ . Those are the fastest ones.

Einstein received the 1921 Nobel Prize in Physics for his discovery of the law of the photoelectric effect.

An everyday picture

Think of a turnstile that opens only for a coin worth at least the toll. The toll is the work function. A coin worth exactly the toll gets you through with nothing left over. A bigger coin gets you through, and the difference comes back as change, which is the electron’s kinetic energy. A smaller coin does not open the gate, and feeding in many small coins one at a time does not open it either, because each coin is used on its own.

Worked example: Violet light and red light on the same metal

A metal with work function $\phi = 2.3$ eV, close to the value for potassium (2.29 eV). Two lights shine on it: violet light at a wavelength of 400 nm and red light at 700 nm. The photon energy is $hf = hc/\lambda$ , with $hc = 1239.84$ eV nm.

Violet light: $hf = \frac{1239.84\ \text{eV nm}}{400\ \text{nm}} \approx 3.10$ eV.
Maximum kinetic energy of the ejected electrons: $K_{\max} = hf - \phi = 3.10 - 2.3 = 0.80$ eV.
Red light: $hf = \frac{1239.84\ \text{eV nm}}{700\ \text{nm}} \approx 1.77$ eV.
Compare with the work function: $1.77 - 2.3 \approx -0.53$ eV. The result is below zero, so one red photon cannot pay the cost of escape, and no electron leaves the metal.

Answer: Violet light ejects electrons with up to 0.80 eV of kinetic energy. Red light at 700 nm ejects no electrons at all, however bright it is, because no single red photon has the 2.3 eV needed to escape the metal. More red light only adds more photons, and each one is still too weak.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Thinking brighter light fixes a light that is too red. Brightness adds photons, but a 700 nm photon at 1.77 eV still cannot pay a 2.3 eV work function.
Forgetting to subtract the work function. The formula gives the kinetic energy left after the cost of escape, not the photon energy $hf$ .
Reading $K_{\max}$ as the energy of every electron. Most ejected electrons have less energy. $K_{\max}$ describes only the fastest ones.
Using $hc = 1239.84$ eV nm with the wavelength in meters or micrometers. That constant gives electron volts only when the wavelength is in nanometers.

Where you will meet it

Light meters and photodetectors, where light frees electrons that are then counted or amplified.
Solar cells, which rely on a close relative of the same idea: photons free electrons inside a semiconductor.
Surface science, where photoelectron spectroscopy shines light on a material and measures the energies of the electrons it ejects.
Image sensors in cameras, where each pixel turns the light that reaches it into a count of electrons.

Questions people ask

Why does brightness not change the energy of the electrons?

Brightness sets how many photons arrive each second. Each electron is knocked out by a single photon, and that photon’s energy depends only on its frequency. More photons give more electrons, all with the same maximum energy for a given frequency.

What is a work function in plain words?

It is the cost of leaving the metal. Each metal holds its electrons with a different strength, and $\phi$ is the minimum energy an electron needs to escape the surface.

Did Einstein discover the photoelectric effect?

No. Heinrich Hertz reported it in 1887. Einstein’s 1905 explanation, which uses photons, is what the 1921 Nobel Prize honored.

Planck-Einstein Relation (E = hf): E = hf says each packet of light carries energy in proportion to its frequency: higher frequency means more energy per packet, and brightness does not change it.
De Broglie Wavelength: Every moving object has a wavelength, and the heavier or faster it moves, the shorter that wavelength gets.
Hydrogen Energy Levels: A hydrogen electron can only sit on certain energy steps, and every jump between steps gives off or takes in light of one exact color.