Photoelectric effect — Einstein's explanation, work function: Visual walkthrough

Step 1 — A photon is a coin of light

WHAT. Forget waves for a moment. Picture light arriving as a stream of tiny identical coins. Each coin is called a photon. Every coin of a given colour carries exactly the same amount of energy — never a little more, never a little less.

WHY. Experiments (see the parent's table) showed that light's effect on electrons depends on the colour, not on the amount of light. A stream of identical coins captures this perfectly: the value of one coin is fixed by colour; brightness is just how many coins per second. This is the idea inherited from Planck's quantum hypothesis & blackbody radiation.

PICTURE. In the figure, colour of light changes left → right (red to blue). The number written on each coin — its energy — grows as we go bluer.

Figure — Photoelectric effect — Einstein's explanation, work function

So: bluer light ⇒ bigger $ν$ ⇒ each coin is worth more.

Step 2 — The electron is glued to the metal

WHAT. Inside the metal sit electrons. The one nearest the surface is held by the weakest glue. To rip that easiest electron out into empty space costs a fixed amount of energy we call the ==work function $ϕ$ == (Greek "phi").

WHY. An electron does not float free — something binds it. Before any energy can turn into motion, the escape toll must be paid. Naming that toll $ϕ$ lets us do honest bookkeeping.

PICTURE. Think of a ball at the bottom of a well of depth $ϕ$ . The ball cannot roll away across the ground until it has been lifted out of the well.

Step 3 — One coin, one electron, and the receipt

WHAT. A single photon is absorbed by a single electron. All of the coin's energy $h ν$ goes into that one electron. That energy is then spent on two things, in order:

First, pay the escape toll $ϕ$ .
Whatever is left over becomes the electron's kinetic energy (its speed).

WHY. This is Energy conservation — the receipt must balance. Money in ( $h ν$ ) equals money spent (toll $ϕ$ ) plus change (leftover motion). One coin buys one escape; you cannot pool two weak coins, because a second coin arrives too late — the electron already relaxed. That is why a below-toll coin achieves nothing, ever.

PICTURE. A jar of energy $h ν$ pours out. The first portion fills the toll box $ϕ$ ; the overflow spills into the "speed" cup.

Why the subscript "max"? We wrote the receipt for the surface electron — the cheapest one to free. A deeper electron pays a bigger toll, so it keeps less change. Therefore the surface electron ends up fastest. Its kinetic energy is the maximum any ejected electron can have — hence $K E_{m a x}$ .

Step 4 — Rearrange into Einstein's equation

WHAT. Move the toll to the other side of the receipt.

WHY. We want a formula that answers the practical question "given the colour, how fast is the fastest electron?" So we isolate $K E_{m a x}$ .

PICTURE. The receipt tips like a see-saw: energy-in on one side, toll subtracted, motion left over.

Step 5 — The threshold and the "nothing happens" case

WHAT. Ask: what is the weakest coin that still frees an electron? Answer: the one whose energy exactly equals the toll, leaving zero change. That colour has frequency ==threshold frequency $ν_{0}$ ==.

WHY. This is the degenerate, break-even case — the boundary between "works" and "does nothing". Below it the coin cannot even pay the toll, so $K E_{m a x}$ would come out negative, which is impossible. A negative answer is nature's way of saying no electron leaves at all.

PICTURE. Three coins hit the well:

too weak ( $h ν < ϕ$ ): coin can't fill the toll box → electron stays (crossed out).
exactly threshold ( $h ν = ϕ$ ): toll box just full, zero overflow → electron barely escapes, speed = 0.
strong ( $h ν > ϕ$ ): overflow spills → electron flies out.

Step 6 — Weigh the electron with a voltage (stopping potential)

WHAT. We cannot read $K E_{m a x}$ off a ruler. Instead we make the escaped electron climb an electrical hill and find the exact hill height $V_{0}$ that stops even the fastest one. That $V_{0}$ is the stopping potential.

WHY. $K E_{m a x}$ is invisible; voltage is easy to dial and read. Pushing a charge $e$ (the electron's charge) "uphill" through a voltage $V_{0}$ costs energy $e V_{0}$ . When the hill is just tall enough to freeze the fastest electron, its entire kinetic energy has been spent climbing. So $e V_{0}$ equals $K E_{m a x}$ — a measurable stand-in for an invisible quantity.

PICTURE. The fastest electron rolls up a ramp of height $V_{0}$ and stops right at the top; all its motion-energy is now "climb-energy".

Step 7 — The straight line that measured Planck's constant

WHAT. Plot the measured $V_{0}$ (up) against frequency $ν$ (across). You get a perfectly straight line.

WHY. Because $V_{0} = \frac{h}{e} ν - \frac{ϕ}{e}$ has the shape (value) = (slope)(input) − (offset). The magic: the slope $h / e$ contains only universal constants — so every metal gives a line of the same steepness. Different metals just slide the line sideways (different toll $ϕ$ ⇒ different $ν_{0}$ ). Measuring that slope hands you $h$ — exactly how Millikan confirmed Einstein (see Millikan oil drop & measurement of h).

PICTURE. Two metals, two lines, same slope, different x-intercepts $ν_{0}$ . The steepness is Planck's constant made visible.

Recall Where does each graph feature live?

Slope of $V_{0}$ vs $ν$ ::: $h / e$ — identical for every metal. x-intercept of the line ::: threshold frequency $ν_{0} = ϕ / h$ . y-intercept of the line ::: $- ϕ / e$ . Effect of a bigger $ϕ$ ::: line shifts right (larger $ν_{0}$ ), same slope.

The one-picture summary

Everything above compressed into a single image: the coin ( $h ν$ ) enters, the toll ( $ϕ$ ) is subtracted, the change ( $K E_{m a x}$ ) becomes speed, the voltage ramp ( $e V_{0}$ ) measures it, and the plot of many colours traces the straight line of slope $h / e$ .

Recall Feynman retelling of the whole walkthrough

Light comes as coins. Each coin's value is set by its colour: bluer coins are worth more ( $E = h ν$ ). An electron is stuck in the metal by glue; the cheapest one to peel off costs a fixed toll $ϕ$ . When a coin hits, one electron grabs one coin. It pays the toll first; the change it keeps turns into speed ( $K E_{m a x} = h ν - ϕ$ ). If the coin is worth less than the toll (colour redder than $ν_{0}$ ), nobody escapes — pooling weak coins doesn't help, because each electron only ever grabs one. To find how fast the fastest escapee is, we make it climb an electrical hill until it stops; the hill height $V_{0}$ times the charge $e$ equals its energy ( $e V_{0} = h ν - ϕ$ ). Plot that hill height against colour and you get a straight line whose steepness is $h / e$ — the same for every metal on Earth. From that steepness Millikan literally read off Planck's constant.

Connections

Planck's quantum hypothesis & blackbody radiation — where $E = h ν$ (Step 1) is born.
Wave-particle duality of light — the "coin" is the particle face of light.
Energy conservation — the receipt in Step 3.
Millikan oil drop & measurement of h — reading $h$ off the Step 7 line.
Compton effect · de Broglie wavelength — where the particle picture goes next.