Consider a configuration of n identical units, each consisting of three layers. The first layer is a column of air of height h = 1 3   cm , and the second and third layers are of equal thickness d = 3 - 1 2   cm , and refractive indices μ 1 = 3 2 and μ 2 = 3 , respectively. A…

Question

Consider a configuration of n identical units, each consisting of three layers. The first layer is a column of air of height h = 1 3 &#160; cm , and the second and third layers are of equal thickness d = 3 - 1 2 &#160; cm , and refractive indices &#956; 1 = 3 2 and &#956; 2 = 3 , respectively. A light source O is placed on the top of the first unit, as shown in the figure. A ray of light from O is incident on the second layer of the first unit at an angle of &#952; = 60 &#176; to the normal. For a specific value of n , the ray of light emerges from the bottom of the configuration at a distance l = 8 3 &#160; cm , as shown in the figure. The value of n is _______.

Aicharya · Accepted Answer

Considering a single unit with three layers. The figure below shows the refraction of light as it moves from one medium to others.
For the first interface,
$x_{1} = \frac{1}{3} \times \tan 60 ° = \frac{1}{\sqrt{3}} cm$
Using Snell's law,
$μ_{1} \sin θ_{1} = μ_{2} \sin θ_{2}$
$\Rightarrow 1 \times \frac{\sqrt{3}}{2} = \sqrt{\frac{3}{2}} \times \sin θ_{2}$
$\Rightarrow θ_{2} = 45 °$
For the second interface,
$\Rightarrow x_{2} = d \times \tan 45 ° = d$
And again using Snell's law,
$\sqrt{\frac{3}{2}} \times \sin 45 = \sqrt{3} \times \sin θ_{3}$
$\Rightarrow \sqrt{\frac{3}{2}} \times \frac{1}{\sqrt{2}} = \sqrt{3} \sin θ_{3}$
$\Rightarrow θ_{3} = 30 °$
$\Rightarrow x_{3} = d \times \tan 30 ° = \frac{d}{\sqrt{3}}$
$∴ x_{1} + x_{2} + x_{3} = \frac{1}{\sqrt{3}} + \frac{(\sqrt{3} - 1)}{2} (1 + \frac{1}{\sqrt{3}})$
$= \frac{2}{\sqrt{3}} cm$
$∴ n = \frac{1}{x_{1} + x_{2} + x_{3}} = \frac{\frac{8}{\sqrt{3}}}{\frac{2}{\sqrt{3}}} = 4$
Thus, the value of $n = 4$ .

Consider a configuration of n identical units, each consisting of three layers. The first layer is a column…

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