A ray of light falling on a glass sphere of refractive index $η = \sqrt{3}$ such that the directions of the incident ray and emergent ray when produced meet the surface at the same point. The value of angle of incidence is $θ$ degrees. Calculate the value of $\frac{θ}{30^{o}}$ .

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Detailed Solution

Ray diagram is shown

Question Image
$∠ A O B = π - 2 r$
Now $∠ A M B = π - \frac{1}{2} ∠ A O B (w h y ?)$
$∴ ∠ A M B = π - \frac{1}{2} (π - 2 r) = \frac{π}{2} + r$
$∴ I n Δ A M B$
$i - r + \frac{π}{2} + r + i + - r = π \Rightarrow 2 i - r = \frac{π}{2}$
Now $\frac{\sin i}{\sin r} = μ$
$\Rightarrow \sin i = \sqrt{3} \sin (2 i - \frac{π}{2}) \Rightarrow \sin i = - \frac{\sin i}{\sin r \cos 2 i} \Rightarrow 2 \sqrt{3} \sin^{2} i - \sin i - \sqrt{3} = 0 \Rightarrow \sin i = \frac{\sqrt{3}}{2}, - \frac{1}{\sqrt{3}}$
Rejecting the negative sign
we get i = 60o.