The photoelectric current in an ideal photocell depends on how many photons strike the surface each second and how
efficiently those photons produce emitted electrons. In the simplest model, the current is not determined directly by
the excess kinetic energy of the photoelectrons, but by the emission rate of electrons that are successfully ejected
and collected.
The first step is to compute the optical power reaching the illuminated area:
Optical power on the surface.
\[
P_{\text{light}} = I_{\text{light}}A
\]
Here, \(I_{\text{light}}\) is the light intensity and \(A\) is the illuminated area. Once the power is known, it can
be converted into a photon arrival rate by dividing by the energy of one photon.
Photon energy.
\[
E_{\gamma} = hf = \frac{hc}{\lambda}
\]
In this expression, \(h\) is Planck’s constant, \(f\) is the light frequency, \(c\) is the speed of light, and
\(\lambda\) is the wavelength. This is the same photon-energy relation used throughout quantum physics. A shorter
wavelength means a higher frequency and therefore a larger photon energy.
Photons per second.
\[
N_{\gamma} = \frac{P_{\text{light}}}{E_{\gamma}}
\]
This gives the number of photons incident per second. However, not every photon necessarily produces a photoelectron.
The fraction that succeeds is described by the quantum efficiency \(\eta\).
Emitted-electron rate.
\[
N_e = \eta N_{\gamma}
\]
If \(\eta = 0.1\), for example, then about 10% of the incident photons produce collected photoelectrons in the ideal
model. Once the number of emitted electrons per second is known, the current follows directly from charge per unit
time.
Photoelectric current.
\[
I = eN_e
\]
Here, \(e\) is the elementary charge. If all emitted electrons are collected, this is also the ideal saturation
current.
Threshold condition
The above counting argument only applies if the incident photons have enough energy to overcome the material’s work
function \(\phi\). The threshold condition is
Emission criterion.
\[
E_{\gamma} \ge \phi
\]
If \(E_{\gamma} < \phi\), no electrons are emitted in the ideal single-photon picture, even if the light intensity is
large. This is one of the hallmark results of the photoelectric effect: increasing intensity below threshold increases
the number of photons, but not the energy per photon, so emission still does not occur.
The corresponding threshold frequency and threshold wavelength are
\[
f_{\min} = \frac{\phi}{h},
\qquad
\lambda_{\max} = \frac{hc}{\phi}.
\]
Sample ideal calculation
Suppose the light intensity is \(10\ \mathrm{W/m^2}\), the illuminated area is \(1\ \mathrm{cm^2} = 10^{-4}\ \mathrm{m^2}\),
the quantum efficiency is \(\eta = 0.1\), and the wavelength is \(400\ \mathrm{nm}\), with photon energy above the work
function. Then the power incident on the surface is
\[
P_{\text{light}} = 10 \times 10^{-4} = 10^{-3}\ \mathrm{W}.
\]
The photon energy is
\[
E_{\gamma} = \frac{hc}{\lambda} \approx 4.97\times10^{-19}\ \mathrm{J}.
\]
Therefore, the photon rate is
\[
N_{\gamma} = \frac{10^{-3}}{4.97\times10^{-19}} \approx 2.01\times10^{15}\ \mathrm{s^{-1}}.
\]
With quantum efficiency \(0.1\), the emitted-electron rate is
\[
N_e \approx 2.01\times10^{14}\ \mathrm{s^{-1}}.
\]
The current becomes
\[
I = eN_e \approx 3.22\times10^{-5}\ \mathrm{A} = 32.2\ \mathrm{\mu A}.
\]
This is the ideal saturation current for this simple model. Real photocathodes may show losses, nonuniform work
function, reflection effects, space-charge effects, and collection inefficiencies, so experimental currents can differ
from the ideal estimate.
Physical interpretation
Above threshold, the current is proportional to intensity because more optical power means more photons per second. At
fixed intensity and area, the current is also proportional to quantum efficiency. But below threshold, the ideal
current is zero regardless of intensity, because no single photon has enough energy to release an electron.
| Concept |
Main relation |
Meaning |
| Optical power |
\(P_{\text{light}} = I_{\text{light}}A\) |
Power reaching the illuminated surface |
| Photon energy |
\(E_{\gamma}=hf=hc/\lambda\) |
Energy of one photon |
| Photon rate |
\(N_{\gamma}=P_{\text{light}}/E_{\gamma}\) |
Photons arriving each second |
| Electron rate |
\(N_e=\eta N_{\gamma}\) |
Electrons emitted each second |
| Photoelectric current |
\(I=eN_e\) |
Ideal collected current |
| Threshold condition |
\(E_{\gamma}\ge\phi\) |
Emission occurs only if photons exceed the work function |