Section 02
The Idea, Step by Step
Imagine a vending machine that only releases a snack when you insert one coin of the right value. Ten tiny coins won't do, even though together they're worth more — the machine takes them one at a time, and each is too small on its own. Light hitting a metal behaves exactly like this. It arrives as a stream of energy packets called photons, and each electron is freed by a single photon or by none at all. Pour on more dim photons and nothing happens; what matters is whether each individual packet is big enough.
Putting numbers on it
Three quantities tell the whole story. The frequency $f$ of the light fixes the energy of each photon, $E = hf$, where $h$ is Planck's constant. The work function $\phi$ is the "entry fee" — the least energy needed to pull an electron out of that particular metal. Whatever energy is left over becomes the electron's motion, its maximum kinetic energy $K_{\max} = hf - \phi$. Take sodium, $\phi = 2.3$ eV, lit at $f = 6.0\times10^{14}$ Hz. Each photon carries $E = hf \approx 2.48$ eV, so $K_{\max} \approx 2.48 - 2.3 = 0.18$ eV — pay the 2.3 eV fee, keep 0.18 eV as speed. If a photon carries less than $\phi$, there's simply nothing to free the electron with, and turning up the brightness can't change that.
The precise picture
There is a sharp cutoff. Setting $K_{\max} = 0$ gives the threshold frequency $f_0 = \phi/h$; below it, no electrons escape however intense the beam. Above it, $K_{\max}$ climbs in a straight line with frequency. Experimenters read $K_{\max}$ off directly by applying a reverse stopping voltage $V_s$ that just halts the fastest electrons, so $K_{\max} = eV_s$ and $V_s = (h/e)(f - f_0)$ — a line whose slope is $h/e$, which is exactly how Millikan measured $h$. Intensity controls only how many electrons leave (the saturation current), never how fast. In the sim, the frequency slider drives $E_{\rm photon}$ and $K_{\max}$, the work-function slider slides $f_0$, and the intensity slider changes the electron count alone.
Try this in the sim above
In Setup, drag the frequency slider down until the electrons disappear — you've dropped below $f_0$; now push the intensity to maximum and watch nothing happen, the headline result of the whole experiment. Then raise the frequency back up and see $K_{\max}$ grow while $\phi$ stays fixed. Finally open the KE_max vs f graph and confirm the straight line that starts not at zero frequency but at $f_0$ — its slope is Planck's constant itself.
Section 03
Equations & Derivation
Photoelectric Equation (Einstein 1905)
$$hf=\phi+K_{\max},\quad K_{\max}=hf-\phi=eV_s$$
Threshold Frequency & Stopping Potential
$$f_0=\frac{\phi}{h},\quad V_s=\frac{K_{\max}}{e}=\frac{h}{e}(f-f_0)$$
Photon Energy
$$E=hf=\frac{hc}{\lambda},\quad h=6.626\times10^{-34}\;\text{J·s}=4.136\times10^{-15}\;\text{eV·s}$$
Current Saturation
$$I_{\rm sat}\propto\text{intensity},\quad I=0\text{ for }V<-V_s,\quad I=I_{\rm sat}\text{ for }V\gg V_s$$
Key Results
| Symbol | Quantity | Unit/Value |
|---|
| $h=4.136\times10^{-15}$ | Planck constant | eV·s |
| $\phi$ | Work function | eV (metal-dependent) |
| $f_0=\phi/h$ | Threshold frequency | Hz |
| $V_s=h(f-f_0)/e$ | Stopping potential | V |
| $K_{\max}=eV_s$ | Maximum electron KE | eV |
1
What classical physics predicted (wrong). Classical wave theory: brighter light = more energy = electrons ejected regardless of frequency; time needed to accumulate energy. Observed: (1) Threshold frequency exists — below $f_0$, no electrons no matter how bright; (2) $K_\max$ depends on $f$, not intensity; (3) electrons ejected instantly.
2
Einstein's quantum explanation (1905). Light comes in quanta (photons), each with energy $E=hf$. One photon ejects one electron. Intensity = number of photons, not energy per photon. Below $f_0$: $hf<\phi$, photon can't overcome work function. This is why Einstein won the Nobel Prize (1921), not for relativity.
3
Stopping potential measures $K_\max$. Reverse voltage $V_s$ decelerates electrons. Current drops to zero at $V_s=K_\max/e$. Plotting $V_s$ vs $f$: straight line with slope $h/e$. This gave first accurate measurement of $h$, confirming quantisation.
4
Work functions. $\phi$ is the minimum energy to remove an electron from the surface: Na 2.28 eV, Al 4.08 eV, Cu 4.65 eV, Pt 5.65 eV. Most metals have $f_0$ in UV range. Cs and Na have $f_0$ in visible — used in early photomultiplier tubes.
Ref: Halliday, Resnick & Walker 10th Ed., Ch. 38; Einstein (1905) Ann. Phys. 17, 132; Millikan (1916) Phys. Rev. 7, 355.
Section 05
Common Misconceptions
❌ Brighter light always causes more energetic electrons.
✅ Intensity (brightness) controls the number of emitted electrons (saturation current), not their energy. $K_\max$ depends only on frequency: $K_\max=hf-\phi$. A dim UV beam ejects fewer electrons but each has the same $K_\max$ as from a bright UV beam at the same frequency.
📖 HRW 10th Ed., §38-3.
❌ The photoelectric effect requires a metal surface.
✅ The photoelectric effect occurs on any material with a well-defined work function. Semiconductors exhibit it (photodetectors, solar cells), as do insulators (X-ray induced electron emission in crystals). The principle is the same: photon energy must exceed the binding energy of the electron.
📖 Kittel — Introduction to Solid State Physics, Ch. 8.
❌ A time delay is needed for electrons to accumulate enough energy to be ejected.
✅ Classical wave theory predicted a delay of minutes to hours for weak light. Experimentally, electrons are emitted within nanoseconds even for very dim light. In the quantum picture, there is no accumulation — a single photon instantaneously delivers all its energy $hf$ to a single electron.
📖 HRW 10th Ed., §38-3; Millikan (1916), Phys. Rev.
Misconception research: Arons — A Guide to Introductory Physics Teaching; Chabay & Sherwood — Matter & Interactions.