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Question
State Einstein’s photoelectric equation. Explain all characteristics of the photoelectric effect, on the basis of Einstein’s photoelectric equation.
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Solution
Einstein’s photoelectric equation: K.E.max = (hν – `phi_0`)
Characteristics of photoelectric effect:
- The photoelectric work function `phi_0` is constant for a given emitter. Hence if the frequency ‘ν’ of the incident radiation is decreased, the maximum kinetic energy of the emitted photoelectrons decreases, till it becomes zero for a certain frequency ν0. Therefore, from Einstein’s equation,
0 = hv0 - `phi_0`
∴ `phi_0` = hv0 .......….(1)
This shows that the threshold frequency is related to the work function of the metal and hence it has different values for different metals. - Using equation (1), Einstein’s equation can be written as
`1/2"mv"_"max"^2 = "hv" - "hv"_0`
∴ `1/2"mv"_"max"^2 = "h"("v" - "v"_0)`
This equation shows that:
a. If ν < ν0, then K.E is negative, which is not possible. In this case, photoelectric emission is not possible.
b. If ν > ν0, then photoelectrons move with some velocity.
∴ K.E > 0, which is possible. Hence, photoelectrons are emitted.
c. If ν = ν0, the photoelectrons are just emitted. In this case, K.E = 0. - The photoelectric equation is,
`1/2"mv"_"max"^2 = "hv" - phi_0` .....….(2)
where, hν = energy of the photon of incident radiation.
`phi_0 = "hv"_0` = photoelectric work function of the metal.
Thus, both the terms on the R.H.S of equation (2) depends on the frequency and not on the intensity of radiation. Hence the maximum kinetic energy with which photoelectrons are emitted is independent of the intensity of radiation. However, since `phi_0` and h are constants, the maximum kinetic energy of the photoelectrons is directly proportional to the frequency. - According to the quantum theory, when the intensity of the radiation increases, there is a proportional increase in the number of photons incident per second on the surface. One photon can cause the emission of one photoelectron. Therefore, with the increase in the intensity of radiation, there will be an increase in the photoelectron interactions and the rate of emission of electrons.
- The emission of a photoelectron is the result of a collision between a photon and an electron. As soon as the radiation is incident on the photosensitive surface, the entire energy of the photon is absorbed by the electron at once. Therefore, the electrons are emitted at a moment when light is incident on the metal surface. This explains why photoelectric emission is instantaneous.
- If the electron, with which the incident photon collides, is situated on the emitting surface, the electron will be ejected with maximum K.E as given by Einstein’s equation.
- If, however, the electron is situated in the interior of the emitting material, it will lose some energy in coming to the surface. This explains why the photoelectrons are emitted with different kinetic energies.
Thus, all the features of the photoelectric effect are explained.
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