The photoelectric effect occurs when light shines on a material surface and ejects electrons from that surface.
This phenomenon played a crucial role in the development of quantum theory because it could not be explained correctly by a purely classical wave model of light.
The key discovery is that light transfers energy in discrete packets, now called photons.
Each photon carries an energy proportional to its frequency:
\[
E_{\gamma}=hf,
\]
where \(h\) is Planck’s constant and \(f\) is the light frequency.
In electron-volt units, this is especially convenient because
\(h \approx 4.1357\times10^{-15}\,\text{eV}\cdot\text{s}\).
A material does not emit electrons unless each incoming photon has at least enough energy to overcome the material’s work function
\(\phi\).
The work function is the minimum energy needed to remove an electron from the surface.
This immediately gives the threshold condition
\[
hf \ge \phi.
\]
If the photon energy is smaller than the work function, then no electrons are emitted, no matter how intense the light is.
This is one of the most important conceptual points of the photoelectric effect:
intensity changes how many photons arrive, but the frequency determines whether each photon has enough energy to eject an electron.
The threshold frequency is obtained by setting the photon energy equal to the work function:
\[
f_{\min}=\frac{\phi}{h}.
\]
Any frequency below this value is too low to produce photoemission.
Since frequency and wavelength are related by
\(c=f\lambda\),
the corresponding threshold wavelength is
\[
\lambda_{\max}=\frac{c}{f_{\min}}=\frac{hc}{\phi}.
\]
This is the longest wavelength that can still eject electrons.
Longer wavelengths correspond to lower photon energies, so they fail to overcome the work function.
Once the photon energy exceeds the work function, the remaining energy appears as the maximum kinetic energy of the emitted electron.
Einstein’s photoelectric equation is therefore
\[
K_{\max}=hf-\phi.
\]
This equation is one of the central formulas of modern physics.
It explains why increasing frequency increases the electron kinetic energy, while increasing intensity mainly increases the number of emitted electrons, provided the light is already above threshold.
The stopping voltage
\(V_s\)
is the reverse potential needed to reduce the photocurrent to zero.
Since one electron-volt per electron corresponds numerically to one volt, the stopping voltage is related to the maximum kinetic energy by
\[
V_s=\frac{K_{\max}}{e}.
\]
Numerically, if
\(K_{\max}\)
is expressed in eV, then
\(V_s\)
has the same numerical value in volts.
For the sample case of sodium with
\(\phi=2.28\,\text{eV}\)
and incident frequency
\(f=8\times10^{14}\,\text{Hz}\),
the threshold frequency is
\[
f_{\min}=\frac{2.28}{4.1357\times10^{-15}}
\approx 5.51\times10^{14}\,\text{Hz}.
\]
The photon energy is
\[
hf=(4.1357\times10^{-15})(8\times10^{14})
\approx 3.31\,\text{eV}.
\]
Therefore the maximum kinetic energy is
\[
K_{\max}=3.31-2.28\approx 1.03\,\text{eV}.
\]
The stopping voltage is then approximately
\(1.03\,\text{V}\).
This matches the usual photoelectric-effect interpretation: the light frequency is above threshold, so electrons are emitted with nonzero kinetic energy.
At a more advanced university level, one can extend the discussion to surface states, realistic electron distributions, quantum efficiency, and multi-photon photoemission.
But the core physics is already contained in a few simple relations:
photon energy depends on frequency,
the work function sets the threshold,
and any excess energy appears as electron kinetic energy.
\[
f_{\min}=\frac{\phi}{h},
\qquad
K_{\max}=hf-\phi,
\qquad
V_s=\frac{K_{\max}}{e}.
\]
These formulas explain the threshold behavior, the frequency dependence of emitted-electron energy, and why the photoelectric effect provided such strong evidence that light is quantized.
