A planar structure of length and width is made of two different optical media of refractive indices and , as shown in the figure. If , a ray entering from end AB will emerge from end CD only if the total internal reflection condition is met inside the

Question

A planar structure of length $L$ and width $W$ is made of two different optical media of refractive indices $n_{1} = 1.5$ and $n_{2} = 1.44$ , as shown in the figure. If $L ≫ W$ , a ray entering from end AB will emerge from end CD only if the total internal reflection condition is met inside the structure. For $L = 9.6 m$ , if the incident angle $θ$ is varied, the maximum time taken by a ray to exit the plane CD is $t \times 10^{- 9} s$ , where $t$ is ______. [2019]

Smart Achievers · Accepted Answer

(50)

Let $W$ be the critical angle.

$1.5 \sin C = 1.44 \sin 90 °$

$\Rightarrow \sin C = \frac{1.44}{1.50}$

But, $\sin C = \frac{A D}{A B}$

$∴$ $\frac{A D}{A B} = \frac{1.44}{1.50}$ $∴$ $A B = \frac{1.50}{1.44} A D$

If we replace AD by 9.6, then the total length travelled by light is $\frac{1.50}{1.44} \times 9.6 = 10 m$

Maximum time taken by a ray to exit the plane is $t = \frac{d}{v_{2}} = \frac{d}{c / n_{2}}$

$∴$ $t = \frac{10}{3 \times 10^{8} / 1.5} = 5 \times 10^{- 8} s = 50 \times 10^{- 9} s$

Comparing it with $t \times 10^{- 9}$ s, we get $t = 50.00$

Solution

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