AB and CD are two slabs. The medium between the slabs has refractive index 2. Find the minimum angle of incidence (in degrees) at Q so that the ray is totally reflected by both the slabs.

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Solution

Let the minimum angle be $α$
For first surface to completely reflect, $2 S i n α ⩾ \sqrt{2} S i n 90$ => $s i n α ⩾ 1 / \sqrt{2}$
=> $α ⩾ 45^{\circ}$

For second surface to completely reflect, $2 S i n α ⩾ \sqrt{3} S i n 90$ => $s i n α ⩾ \sqrt{3} / 2$
=> $\alpha \geqslant 60^\circ$

So $\alpha \geqslant 60^\circ$ satisfies both condition

Answer. Minimum angle of incidence so as to totally reflect from both surface is $α = 60^{\circ}$

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