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  3. Physics
  4. 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 μ is the refractive index of sphere, then the relation between μ and β is expressed sin 2 x β+ μ2 sin 2 β=1. Find the value of x.

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Q. 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 $x$.Physics Question Image

Ray Optics and Optical Instruments

Solution:

$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
$1=\mu^{2} \sin ^{2} \beta+\sin ^{2} 4 \beta$