Since \(c=\lambda f\), there is also a threshold wavelength. Longer wavelengths correspond to lower photon energies, so
only wavelengths shorter than or equal to the threshold wavelength can produce the photoelectric effect.
Threshold wavelength.
\[
\begin{aligned}
\lambda_{\text{threshold}} &= \frac{c}{f_{\min}} \\
&= \frac{h c}{\phi}
\end{aligned}
\]
Einstein’s photoelectric equation
If the photon energy is larger than the work function, the extra energy does not disappear. It becomes the kinetic
energy of the emitted electron. Einstein summarized this with the photoelectric equation
Photoelectric equation.
\[
\begin{aligned}
K_{\max} &= E_{\gamma} - \phi \\
&= h f - \phi
\end{aligned}
\]
The symbol \(K_{\max}\) means the maximum kinetic energy of the emitted photoelectrons. In real materials,
not every electron emerges with exactly the same energy, but the most energetic electrons satisfy this ideal relation.
If \(E_{\gamma}=\phi\), then the electrons are just barely freed and \(K_{\max}=0\).
Stopping potential
In laboratory measurements, the emitted electrons can be slowed and stopped by applying an opposing electric potential.
The smallest retarding voltage that stops even the fastest photoelectrons is called the stopping potential,
written as \(V_s\).
Stopping-potential relation.
\[
\begin{aligned}
e V_s &= K_{\max} \\
V_s &= \frac{K_{\max}}{e}
\end{aligned}
\]
This is especially convenient when kinetic energy is measured in electronvolts. A kinetic energy of
\(0.82\,\mathrm{eV}\) corresponds numerically to a stopping potential of \(0.82\,\mathrm{V}\).
Sample interpretation
Suppose the metal is sodium with work function \(\phi=2.28\,\mathrm{eV}\), and the incident wavelength is
\(400\,\mathrm{nm}\). First compute the photon energy:
Photon energy for the sample.
\[
\begin{aligned}
E_{\gamma} &= \frac{h c}{\lambda} \\
&= \frac{1239.84}{400} \\
&\approx 3.10\,\mathrm{eV}
\end{aligned}
\]
Since \(3.10\,\mathrm{eV} > 2.28\,\mathrm{eV}\), photoelectrons are emitted. The fastest of them have maximum kinetic
energy
Maximum kinetic energy for the sample.
\[
\begin{aligned}
K_{\max} &= E_{\gamma} - \phi \\
&\approx 3.10 - 2.28 \\
&\approx 0.82\,\mathrm{eV}
\end{aligned}
\]
The threshold frequency for sodium is
Threshold frequency for sodium.
\[
\begin{aligned}
f_{\min} &= \frac{\phi}{h} \\
&= \frac{2.28}{4.135667696\times 10^{-15}} \\
&\approx 5.51\times 10^{14}\,\mathrm{Hz}
\end{aligned}
\]
This result captures the main lesson of the photoelectric effect: frequency matters more fundamentally than
intensity for determining whether emission can occur. Bright low-frequency light may fail completely, while dim
higher-frequency light can succeed because each photon has enough energy.
Why this experiment was important
The photoelectric effect was crucial evidence for quantum theory. Classical wave ideas alone predicted that electrons
should slowly accumulate energy from the wave and eventually escape, but experiments showed something different. Below the
threshold frequency, no electrons were emitted at all, no matter how intense the light became. Above the threshold,
emission occurred essentially without delay. Einstein explained this by treating light as photons, and that explanation
became one of the foundations of modern quantum physics.
| Concept |
Main relation |
Meaning |
| Photon energy |
\(E_{\gamma}=hf=hc/\lambda\) |
Energy carried by one photon |
| Work function |
\(\phi\) |
Minimum energy required to free an electron |
| Threshold frequency |
\(f_{\min}=\phi/h\) |
Minimum frequency needed for emission |
| Threshold wavelength |
\(\lambda_{\text{threshold}}=hc/\phi\) |
Largest wavelength that still ejects electrons |
| Maximum kinetic energy |
\(K_{\max}=hf-\phi\) |
Excess photon energy after escape |
| Stopping potential |
\(V_s=K_{\max}/e\) |
Voltage needed to stop the fastest electrons |