Question

A ray of light enters a glass sphere, and after three total internal reflections it travels into original direction as shown in the figure. If $\mu$ is the refractive index of sphere, then the relation between $\mu$ and $\beta$ is expressed ${\sin }^{2}x\beta +$ ${\mu }^2{\sin }^2\beta =1$. Find the value of $x$.

Answer 1

Answer: 4

$2(\theta -\beta )+3\left(180^{\circ }-2\beta \right)=360^{\circ }$
$4\beta =\theta +90^{\circ }$
$\sin 4\beta =\cos \theta$.....(i)
$\mu \sin \beta =\sin \theta$....(ii)
Squaring and adding (i). and (ii), we get
$1={\mu }^2{\sin }^2\beta +{\sin }^{2}4\beta$