Question

A beam of light of wavelength \[\lambda \] and with illumination L falls on a clean surface of sodium. If N photoelectrons are emitted each with kinetic energy E, then
A. \[N \propto L{\rm{ \,and\, E}} \propto L\]
B. \[N \propto L{\rm{ \,and\, E}} \propto \dfrac{1}{\lambda }\]
C. \[N \propto \lambda {\rm{ \,and \,E}} \propto L\]
D. \[N \propto \dfrac{1}{\lambda }{\rm{\,and\, E}} \propto \dfrac{1}{L}\]

Answer 1

Hint: As the total energy of a photon is equal to the sum of the energy utilized to eject an electron and the maximum kinetic energy of electrons. If the light of frequency v is incident on a certain photoelectric substance with threshold frequency \[{v_0}\]. The work function for the substance is \[h{\upsilon _0}\]. Einstein gives the phenomenon of the photoelectric effect on the basis of Plank’s theory. In the photoelectric effect, the kinetic energy of photo-electrons emitted from the metal surface E and \[\phi \]is the work function of the metal.

Formula used:
Kinetic energy of photoelectrons is given as:
\[K{E_{\max }} = E - \phi \]
Where E is the energy and \[\phi \] is the work function.
Also, the energy of the photon is given by the equation is given as:
\[E = h\upsilon = \dfrac{{hc}}{\lambda }\].
Where, \[h\] is the Plank constant, \[\upsilon \] is the frequency of incident light, c is the speed of light and \[\lambda \] is the wavelength.
Work function is given as:
\[\phi = h{\upsilon _0}\]
Where h is Planck's constant and \[{\upsilon _0}\] is the threshold frequency.

Complete step by step solution:
As we know the kinetic energy(E) of photoelectrons is
\[E = \dfrac{{hc}}{\lambda } - \phi \]
Here we see that \[E \propto \dfrac{1}{\lambda }\].
It shows Kinetic energy E is inversely proportional to the wavelength \[\lambda \].
Also, \[I \propto \dfrac{n}{t}\]
It shows that the intensity is directly proportional to the number of photoelectrons generated by the source. i.e., \[N \propto L{\rm{ }}\]. Therefore, if N photoelectrons are emitted each with kinetic energy E, then \[N \propto L{\rm{ \,and \,E}} \propto \dfrac{1}{\lambda }\]

Hence option B is the correct answer.

Note: The photoelectric effect is defined as the phenomenon in which electrically charged particles are released from a material after it absorbs electromagnetic radiation. This photoelectric effect is also defined as the emission of electrons from a metal surface when light falls on it. The total energy of a photon is equal to the sum of the energy utilized to eject an electron and the maximum kinetic energy of electrons.