Question

When photons of energy $h$ fall on an aluminium plate (of work function $\phi$ ), photoelectrons of maximum kinetic energy of $K$ are ejected. If the frequency of the radiation is doubled then ejected photoelectrons will be.
$A.2K$
$B.0$
$C.K+h\nu$
$D.K$

Answer 1

We will apply the formula of maximum kinetic energy of photoelectrons. Concept of the work function will be used to find the correct answer. Work function is defined as the minimum energy which is required to remove an electron from the surface of a material. Work function is the property of the surface of the material.

Formula Used:
We are using the following equation to solve the correct answer:-
$K=E-\phi$.

Complete answer:
We know that the maximum kinetic energy of photoelectrons is given by the following equation:-
$K=E-\phi$…………………. $(i)$
Where, $K$ denotes the maximum kinetic energy, $E$ is the incident energy of the photon and $\phi$ is the work function of the aluminium metal.
We know that the incident energy of the photon is given with the help of the following formula:-
$E=h\nu$………………. $(ii)$
Using $(ii)$ in $(i)$ we get
$K=h\nu -\phi$
$\Rightarrow \phi =h\nu -K$………………. $(iii)$
Now, according to question the frequency of the incident photons is doubled then the incident energy will be given as follows:-
${{E}_{n}}=h(2\nu )$
${{E}_{n}}=2h\nu$………….. $(iv)$
Now, the new maximum kinetic energy is given as follows:-
${{K}_{n}}={{E}_{n}}-\phi$……………… $(v)$
Putting $(iii)$ and $(iv)$ in $(v)$ we get
${{K}_{n}}=2h\nu -(h\nu -K)$
$\Rightarrow {{K}_{n}}=2h\nu -h\nu +K$
$\Rightarrow {{K}_{n}}=h\nu +K$
Hence, ${{K}_{n}}=K+h\nu$ is the required answer.

Therefore, option $(C)$ is the correct one.

Note:
We should remember the fact that the maximum kinetic energy of photoelectrons is directly proportional to the frequency of the incident light. It should also be noted that the maximum kinetic energy of the photoelectrons does not depend on the intensity of the incident light. It should also be noted that if the energy of the photons is less than the work function of the metal surface then electrons will never be ejected from the metal surface regardless of frequency of the incident light.