This calculator combines two closely related ideas from photoelectric measurements: estimating the material work function from stopping-voltage data, and estimating quantum efficiency from the ratio of emitted electrons to incident photons.
Both quantities are important in detector physics, photoemission studies, and device characterization.
The work function tells you how much energy an electron must gain to escape the surface, while the quantum efficiency tells you how successfully the incoming light is converted into emitted charge carriers.
The work-function part comes directly from Einstein’s photoelectric equation. For a photon of frequency
\(f\),
the photon energy is
\[
hf.
\]
If the emitted electrons are stopped by a reverse potential
\(V_s\),
then the maximum kinetic energy is
\[
K_{\max}=eV_s.
\]
Einstein’s relation says
\[
hf=\phi+K_{\max},
\]
so the work function is
\[
\phi=hf-eV_s.
\]
In the calculator, the photon energy is expressed in electron-volts, so the numerical relation becomes especially convenient:
if
\(hf\)
is in eV and
\(V_s\)
is in volts, then
\(\phi\)
comes out directly in eV because one electron accelerated through one volt gains one electron-volt of energy.
For the sample frequency
\(f=8\times10^{14}\,\text{Hz}\),
the photon energy is
\[
hf=(4.1357\times10^{-15}\,\text{eV}\cdot\text{s})(8\times10^{14}\,\text{Hz})
\approx 3.31\,\text{eV}.
\]
If the stopping voltage is
\(V_s=1.0\,\text{V}\),
then
\[
\phi \approx 3.31-1.00=2.31\,\text{eV}.
\]
That matches the sample work-function estimate in the prompt.
Quantum efficiency, usually written as
\(\eta\),
is a different quantity. It measures how many electrons are emitted per incoming photon:
\[
\eta=\frac{\text{number of electrons}}{\text{number of photons}}.
\]
To estimate it from laboratory-style data, you need more than stopping voltage and frequency alone.
You also need a photon-rate estimate and an electron-rate estimate.
In this calculator, the electron rate is obtained from the photocurrent:
\[
\dot N_e=\frac{I}{e},
\]
while the photon rate is obtained from the optical power:
\[
\dot N_\gamma=\frac{P}{hf}.
\]
Combining these gives
\[
\eta=\frac{\dot N_e}{\dot N_\gamma}
=\frac{I/e}{P/(hf)}
=\frac{Ihf}{eP}.
\]
For the example values
\(P=1.0\,\text{mW}\)
and
\(I=30.2\,\mu\text{A}\),
together with
\(f=8\times10^{14}\,\text{Hz}\),
the calculator gives
\(\eta\approx 0.10\),
meaning about ten percent of the incident photons produce one emitted electron in this simplified interpretation.
The threshold frequency also follows from the estimated work function:
\[
f_0=\frac{\phi}{h}.
\]
This is the minimum frequency needed to eject electrons at all.
Below threshold, the stopping-voltage picture no longer produces a positive kinetic-energy interpretation.
At a more advanced university level, one studies spectral response, surface contamination, recombination losses, escape probability, semiconductor band structure, and multi-photon or nonlinear photoemission.
In real detectors and solar cells, the measured efficiency may also be external rather than internal, and reflection or transport losses matter.
This calculator intentionally keeps the model simple so the two central ideas remain clear:
stopping voltage gives access to the work function, and current/power data allow an efficiency estimate.
\[
\phi=hf-eV_s,
\qquad
\eta=\frac{Ihf}{eP},
\qquad
f_0=\frac{\phi}{h}.
\]
These formulas connect measurable quantities to the basic energetics and yield of the photoelectric process.
