question_answer

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency Ω such that \[\frac{4\pi M\Omega }{h}\,=\,{{10}^{24}}{{m}^{-2}}\]  with h as h Planck's constant. N photons of wavelength \[\lambda \,=\,8\pi \,\times {{10}^{-6}}\] m strike the mirror simultaneously at normal incidence   such that the mirror gets displaced by 1 \[\mu m.\] If the value  of N is \[x\times {{10}^{12}},\] then the value  of x is ___ [Consider the spring   as massless]

A) 1

B) 7

C) 4

D) 5

Correct Answer: A

Solution :

\[m{{\text{V}}_{\text{max}}}=\left( \frac{2h}{\lambda } \right)\text{N}\]

\[m.\omega \text{A}=\frac{2h}{\lambda }\text{N}\]

\[\text{N}=\frac{m\omega \lambda \Alpha }{2h}\]

\[\text{N}=\frac{{{10}^{24}}}{4\pi }\times \frac{8\pi \times {{10}^{-6}}}{2}({{10}^{-6}})\]

\[\text{N}={{10}^{12}}\]

\[\therefore \]      \[X=1\]