Question

A perfectly reflecting mirror of mass $M$ mounted on a spring constitutes a spring-mass system of angular frequency $\Omega$ such that $\frac{4\pi M\Omega }{h}=10^{24}{m}^{-2}$ with $h$ as Planck's constant. $N$ photons of wave- length $\lambda =8\pi \times 10^{-6}m$ 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

Answer 1

Answer: 1

Total Momentum transferred on mirror $=\frac{2Nh}{\lambda }$ (for perfect reflection)
In an $SHM$
$V_{max}=A.\Omega$
Where $A=$ amplitude and $\Omega =$ angular frequency
$V_{(\text{mean position })}=\Omega A$ (where $A=1\mu m)$
$\Rightarrow \frac{2Nh}{\lambda }=M\Omega A($ where $\lambda =8\pi \times 10^{-6})$
$\therefore N=\frac{\lambda M\Omega A}{2h}$
$\Rightarrow N=\frac{M\Omega 10^{-6}\lambda }{2h}=\frac{M\Omega 8\pi \times 10^{-6}\times 10^{-6}}{2h}$
$N=\frac{4\pi M\Omega }{h}\times 10^{-12}$
$=10^{24}\times 10^{-12}$
$N=1\times 10^{12}$
$x=1$