If current I is given by I = nAev, where n is the number of electrons per unit volume, A is the area, e is the charge of an electron and v is the velocity of the electron, it means that current increases with increase in velocity of the electron which increases with the frequency of light incident on the metal emitter. Why then doesn't saturation current increase with increase in frequency?
Answer
In the context of the photoelectric effect it is probably not a good idea to use the equation $ I = nAev$ because the speed of the electrons is not constant across the gap.
It is better to use $I=Ne$ where $N$ is the number of photoelectrons collected per second by the plate remote from the plate emitting the photoelectrons and $I$ is the current.
The saturation current occurs when all the emitted photoelectrons are collected.
The saturation current can change with changing frequency and so a lot of the graphs in textbooks are not correct.
The intensity of light is proportional to the number of photons arriving per second times the frequency of a photon.
If you increase the frequency of the light keeping the intensity the same then the number of photons hitting the surface per second decreases which might well have an effect on the number of photoelectrons being collected per second (photoelectric current).
If you have all the incoming photons of high enough energy for each of them to release a photoelectron and then increase the frequency whilst keeping the intensity the same the saturation current will decrease.
The Phet photoelectric emission simulation can be used to show such behaviour.
If the frequency is kept constant then the saturation current does increase as the intensity is increased.
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