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 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]

Answer 1

Let momentum of one photon is p and after reflection velocity of the mirror is v. conservation of linear momentum
$Np\hat{i} = -Np\hat{i} + mv\hat{i}$
$mv\hat{i} = 2pN\hat{i}$
$mv = 2Np\quad\quad\quad....\left(1\right)$
since v is velocity of mirror (spring mass system) at mean position,
$v = A\Omega$
Where A is maximum deflection of mirror from mean position. (A = 1 μm) and $\Omega$ is angular frequency of mirror spring system,
momentum of 1 photon, $p = \frac{h}{\lambda}$
$mv = 2Np\quad\quad\quad .....\left(i\right)$
$mA\Omega = 2N \frac{h}{\lambda}$
$N = \frac{m\Omega}{h}\times\frac{\lambda A}{2}$
given, $\frac{m\Omega}{h} = \frac{10^{24}}{4\pi}m^{-2}$
$\lambda = 8\pi \times 10^{-6}\,m$
$N = \frac{10^{24}}{4\pi}\times\frac{8\pi\times10^{-6}\times10^{-6}}{2}$
$N = 10^{12} = x \times 10^{12}$