The Photoelectric Effect

Take a clean sheet of metal, connect it to a sensitive meter, and shine a light on it. Under exactly the right light, something startling happens: the metal starts spitting out electrons. They leap clean off the surface and, if you set up the circuit, flow round it as a tiny current. Light going in, electrons coming out — with nothing touching the metal at all. This is the photoelectric effect, and the electrons it knocks loose are called photoelectrons.

It sounds like a curiosity, but it broke physics wide open. Everything we knew from the electromagnetic spectrum said light was a wave — and a wave should behave one way. The photoelectric effect stubbornly does the opposite. Explaining it forced Einstein to a radical idea — that light comes in tiny lumps of energy — and won him the Nobel Prize. This page walks through the whole story: the puzzle, Einstein's fix, the equation you can compute with, and the straight-line graph that pins it all down.

The puzzle: a threshold that shouldn't be there

Here is the experiment that refused to make sense. Shine light of a chosen colour (that is, a chosen frequency) on the metal and watch for photoelectrons:

Below a certain frequency — called the threshold frequency f_0 — no electrons come off at all, no matter how bright the light or how long you wait. A blindingly intense red beam: nothing.
Above f_0, electrons come off the instant the light lands — even the faintest glimmer works. And the electrons come off with more energy the higher the frequency.
Turning the light brighter (more intense) above the threshold gives you more electrons per second, but each one still leaves with the same maximum energy.

Now here is why this is a scandal for wave theory. A wave carries energy smoothly and continuously; a brighter wave carries more energy. So a wave picture makes a firm prediction: shine a bright enough light of any colour for long enough, and an electron should eventually soak up enough energy to escape. There should be no threshold — just a delay while dim light "charges up" the electron. The experiment flatly disagrees: colour (frequency) is king, brightness is powerless below f_0, and there is no delay. The wave theory of light simply cannot explain this.

Einstein's fix: light comes in lumps (photons)

In 1905 Einstein made a daring proposal. Light is not a smooth, continuous wave delivering its energy in a steady trickle. Instead it arrives in tiny indivisible packets — photons (also called quanta). Each photon carries a fixed lump of energy set entirely by the light's frequency:

E = hf,

where h = 6.63 \times 10^{-34}\ \text{J s} is Planck's constant and f is the frequency of the light. A higher frequency (bluer light) means a more energetic photon; a lower frequency (redder light) means a feebler one. Since every photon in a beam of one colour has the same energy, making the beam brighter does not make the photons stronger — it just sends more of them.

Light of frequency f is carried by photons, each of energy E = hf.
Using c = f\lambda, this is also E = \dfrac{hc}{\lambda} in terms of wavelength.
The energy of one photon depends only on frequency, never on how bright the beam is. Intensity sets the number of photons per second, not their individual energy.

One photon, one electron — and the price of escape

The second half of Einstein's idea is just as important as the first: in the photoelectric effect one photon interacts with exactly one electron, and hands over all of its energy in a single all-or-nothing kick. An electron cannot save up two feeble photons for later — each photon acts alone.

But an electron sitting inside a metal is held there; it takes a minimum amount of energy to tear it free from the surface. That minimum is the metal's work function, \phi. So when a photon strikes an electron:

If the photon's energy hf is less than \phi, the electron cannot escape at all — nothing happens.
If hf is at least \phi, the electron escapes, spending \phi on the getaway and keeping the rest as kinetic energy.

Energy is simply conserved: photon in, escape cost plus leftover motion out. The electron that was least tightly held keeps the most leftover energy, so the photoelectrons emerge with a maximum kinetic energy E_{k,\max} given by the photoelectric equation:

hf = \phi + E_{k,\max}, \qquad\text{so}\qquad E_{k,\max} = hf - \phi.

Energy balance: hf = \phi + E_{k,\max} — the photon energy equals the work function plus the maximum kinetic energy of the freed electron.
Work function \phi: the minimum energy needed to remove an electron from the metal's surface. Different metals have different \phi.
Threshold frequency: emission only occurs when hf \ge \phi, i.e. above f_0 = \dfrac{\phi}{h}. At exactly f_0 the electron just escapes with zero kinetic energy.

Why frequency matters and intensity doesn't (for energy)

The photoelectric equation settles the whole puzzle in one line. Look at E_{k,\max} = hf - \phi:

The maximum kinetic energy depends on frequency f, not on brightness. Bluer light → more energetic photons → faster photoelectrons. This is exactly what the experiment shows.
Below f_0, every photon carries less than \phi. No single photon can free an electron, and since electrons take their energy one photon at a time, piling on more such photons (a brighter beam) changes nothing. Threshold explained.
Above f_0, a brighter beam means more photons per second, so more electrons freed per second — a larger current — but each electron still leaves with the same maximum E_{k,\max}. Intensity controls the number of electrons, never their maximum energy.

And because a photon gives its energy in one instant, emission starts the moment the light of the right frequency arrives — no delay. Every strange feature of the experiment falls straight out of "light is photons." This was decisive evidence that light has a particle nature as well as a wave nature — the idea we now call wave–particle duality. (The very electrons being ejected are the ones you met circling the nucleus in the atom.)

Worked examples

Use h = 6.63 \times 10^{-34}\ \text{J s} throughout.

Example 1 — photon energy. Blue light has frequency f = 6.0 \times 10^{14}\ \text{Hz}. What is the energy of one photon?

E = hf = (6.63\times10^{-34})(6.0\times10^{14}) = 3.98\times10^{-19}\ \text{J}.

Example 2 — maximum kinetic energy. A metal has work function \phi = 3.0\times10^{-19}\ \text{J}. Light of frequency 7.0\times10^{14}\ \text{Hz} shines on it. Find E_{k,\max}.

hf = (6.63\times10^{-34})(7.0\times10^{14}) = 4.64\times10^{-19}\ \text{J},

E_{k,\max} = hf - \phi = 4.64\times10^{-19} - 3.0\times10^{-19} = 1.64\times10^{-19}\ \text{J}.

Example 3 — work function from the threshold. A metal's threshold frequency is f_0 = 5.0\times10^{14}\ \text{Hz}. What is its work function?

\phi = h f_0 = (6.63\times10^{-34})(5.0\times10^{14}) = 3.32\times10^{-19}\ \text{J}.

Example 4 — threshold frequency from the work function. A metal has \phi = 4.0\times10^{-19}\ \text{J}. Below what frequency will no electrons be emitted?

f_0 = \frac{\phi}{h} = \frac{4.0\times10^{-19}}{6.63\times10^{-34}} = 6.03\times10^{14}\ \text{Hz}.

Any light dimmer in frequency than 6.03\times10^{14}\ \text{Hz} — no matter how intense — ejects nothing.

The straight-line graph

Rewrite the photoelectric equation as E_{k,\max} = hf - \phi and compare it with the equation of a straight line, y = mx + c. Plotting the maximum kinetic energy E_{k,\max} (the y-axis) against frequency f (the x-axis) gives a straight line whose features are packed with meaning:

the gradient is h — Planck's constant, the same for every metal;
the x-intercept (where E_{k,\max} = 0) is the threshold frequency f_0 = \phi/h;
the y-intercept (extending the line back to f = 0) is -\phi, so the work function is read straight off it.

Every metal gives a line of the same gradient h, just shifted along — a metal with a bigger work function needs a higher threshold frequency before any electrons appear. Drag the frequency slider below and watch E_{k,\max} climb along the line above the threshold, and stay firmly at nothing below it.

It is the FREQUENCY, not the brightness, that must beat the threshold. A dim blue light can eject electrons from a metal on which a blindingly bright red light does absolutely nothing. Below f_0, no amount of intensity helps — because each photon is individually too weak and electrons take energy one photon at a time.
Intensity changes the NUMBER of electrons, not their energy. Turning a suitable light brighter frees more photoelectrons per second (a bigger current), but each one still leaves with the same maximum kinetic energy E_{k,\max} = hf - \phi. To speed the electrons up you must raise the frequency.
One photon interacts with one electron. An electron cannot combine the energy of two sub-threshold photons to escape. This "one-to-one" rule is the whole reason a threshold exists — miss it and the effect looks impossible.
The work function is a property of the metal, not the light. \phi and the threshold f_0 = \phi/h belong to the surface you shine on; the frequency and intensity belong to the light. Don't mix them up.

It really was. When Albert Einstein was awarded the 1921 Nobel Prize in Physics, the citation was "for his discovery of the law of the photoelectric effect" — not for special or general relativity, the theories that made him a household name. Relativity was still thought too bold and untested; the photoelectric law, by contrast, was solid, testable, and had opened the door to quantum physics. The idea that light comes in photons is one of the founding stones of the whole quantum world.

And it is everywhere in your life. A solar cell turns sunlight into electricity by exactly this trick — photons knocking electrons loose to drive a current. The automatic light sensor that switches on a street lamp at dusk, the photodiode in a smoke alarm or a camera's light meter, the sensors in the checkout scanner and the automatic door — all lean on light kicking electrons free. Einstein's "curiosity" quietly runs a large chunk of modern technology.