Question

Two separate monochromatic light beams A and B of the same intensity are falling normally on a unit area of a metallic surface. Their wavelengths are \[{{\lambda }_{A}}\] and \[{{\lambda }_{B}}\] ​, respectively. Assuming that all the incident light is used in ejecting the photoelectrons, the ratio of the number of photoelectrons from beam A to that from B is:
A. \[\dfrac{{{\lambda }_{A}}}{{{\lambda }_{B}}}\]
B. \[\dfrac{{{\lambda }_{B}}}{{{\lambda }_{A}}}\]
C. \[{{\left( \dfrac{{{\lambda }_{B}}}{{{\lambda }_{A}}} \right)}^{2}}\]
D. \[{{\left( \dfrac{{{\lambda }_{A}}}{{{\lambda }_{B}}} \right)}^{2}}\]

Answer 1

Hint: In this question we have been asked to calculate the ratio of number of photoelectrons ejected from beam A to beam B. It is given that all the light is used to eject photoelectrons. Therefore, the intensity of the light is equal for both. We know that the formula for energy or Planck’s equation depends inversely on the wavelength. Therefore, using these two conditions, we shall calculate our answer.
Formula Used:
\[E=n\dfrac{hc}{\lambda }\]
Where, n is the number of photoelectrons.

Complete answer:
From the Planck’s equation we know that
\[E=n\dfrac{hc}{\lambda }\]
Therefore, energy of beam A can be given by,
\[{{E}_{A}}={{n}_{A}}\dfrac{hc}{{{\lambda }_{A}}}\] ………… (1)
Similarly, the energy of beam B,
\[{{E}_{B}}={{n}_{B}}\dfrac{hc}{{{\lambda }_{B}}}\] …………….. (2)
Now, we know that the intensity of both beams is equal. We also know that, intensity of the beam can be calculated by calculating the energy density i.e. energy per unit volume. However, the volume in this case will be constant. Therefore, energy becomes directly proportional to energy
Therefore,
\[{{E}_{A}}={{E}_{B}}\]
From (1) and (2)
\[{{n}_{A}}\dfrac{hc}{{{\lambda }_{A}}}={{n}_{B}}\dfrac{hc}{{{\lambda }_{B}}}\]
On solving,
\[\dfrac{{{n}_{A}}}{{{n}_{B}}}=\dfrac{{{\lambda }_{A}}}{{{\lambda }_{B}}}\]

Therefore, the correct answer is option A.

Note:
The monochromatic light is the electromagnetic radiation obtained from a single photon emission from atoms. The colour of the light depends on the wavelength of the light and the energy level determines the frequency. The name mono-chrome means having only one colour. Sunlight is white but it consists of 7 different colours. Therefore, the light from the sun is not monochromatic as it does not have purely one colour.