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I know that intensity determines the no. of electrons and the current produced. But if the intensity is constant and the frequency changes, the kinetic energy of the photoelectrons changes, so their velocity changes. So if the frequency increases, the kinetic energy of the photons increases so they move faster. In that case, the number of photoelectrons moving through a space in unit time increases because their velocity is more so the electrons are traveling faster. That is my reasoning behind this. But why does frequency not affect the current?

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You are right, current passing through the wire is equal to the charge passing through it per unit time, that is fact, BUT!

Photocurrent is produced because the electrons that were removed from the metallic plate and have enough kinetic energy jump to the other plate and interacts with that plate, which makes the metal from which electron was removed positively charged and the metallic plate to which electron jumps negatively charged.

The answer to your question is that, when the electron jumps to another plate, no matter how much kinetic energy it gained (of course I am not talking about energy that is high enough to make the electron pass through the metallic plate), it collides to the other plate (interacts) and looses its energy which increases the temperature of that plate (or metal), but charge difference rate will not change (dq/dt through the wire that is connecting the plates is constant), so current stays the same.

So the excess kinetic energy of the electrons turns into heat or other different forms of energy, which does not significantly effect thee speed of electrons that are moving through the wire that is connecting the plates.

After all, number of photons striking the metal per unit time is constant, which is directly proportional to the produced current => current must be constant!

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No. Each photon of high enough frequency to induce a photoelectric effect releases one electron. Therefore, high-energy photons per second are equal in number to new flowing electrons per second, which is the definition of current. What frequency determines is the per-electron kinetic energy, which is basically voltage.

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