quantum mechanics - Photoelectric Effect: How Can Number of Electrons Emitted be Independent of Frequency

I am reading about the photoelectric effect. Two classical results are:

1) Number of electrons emitted increases with intensity (above the threshold frequency)

2) Number of electrons emitted is independent of frequency

However if I want to increase frequency but keep intensity constant doesn't this mean that I have to send in fewer photons which would decrease the number of emitted electrons?

Note 1: Source (for assumptions and image)

Note 2: I found questions with the same title but they seemed to be asking something different!

Answer

I would hazard that the source you have linked is incorrect. Most often when I read about the photoelectric effect the main result is that:

1) The kinetic energy of the emitted electrons depends only on the frequency of the incident light. In particular, there are no emitted electrons until the light is at least a threshold frequency, $\nu_0$ and then after that the kinetic energy of the emitted electrons increases linearly with frequency of the incident light.

$$\text{KE} = hf - \phi$$

($\text{KE}$ is the kinetic energy of the emitted electrons, $h$ and $f$ are Planck's constant and the frequency of the incident light and $\phi$ is the work function of the material under consideration, the minimum amount of energy needed to eject an electron.)

2) (For fixed incident frequency) the number of emitted electrons increases linearly with the intensity of the incident light.

$$N_{\text{electron}} = N_{\text{photon}} = \frac{IA}{h f} = \frac{P}{hf} $$

($P$ is the power in the beam, $A$ is the area of the beam so the intensity, $I$ times $A$ gives the power. It is in fact power $P$ which equals $N_{\text{photon}} hf$ where $N_{\text{photon}}$ is the number of photons impinging upon the surface per second)

These two facts are consistent with the particle or photon description of light. That is, light comes in discrete packets where each packet carries a quanta of energy. The size of this quanta of energy is linearly proportional to the frequency of light and the intensity of the light is proportional to the number flux.

I don't know if I have seen many sources that claim the photocurrent is independent of the frequency of the incident field. As you have correctly noted (and shown in my 2nd equation above) the flux of photoelectrons depends directly on the flux of photons but, because of the relationship between intensity and flux, this in turn depends on intensity and frequency.

In short, I would expect that, at constant intensity, increasing the frequency of incident light would decrease the photocurrent. Someone should please correct me if I'm mistaken.

I think it would be extremely misleading to claim the intensity, in the context of the photoelectric effect, refers only to the number of photons in the beam and not the actual energy flux per unit area of the incident beam. Particularly because the experimenters at the time of the discovery would have been quantifying the experimentally accessible quantity of real intensity.