Spherical refracting surface's

Question

A semi-cylinder made of transparent plastic has a refractive index n = $\sqrt{2}$ and a radius of R. There is a narrow incident light ray perpendicular to the flat side of the semi cylinder at d distance from the axis of symmetry.

Moderate

Question

What can the maximum value of d be so that the light ray can still leave the other side of the semi cylinder?

Solution

$\begin{array}{l}\mathrm{sin}{\mathrm{\theta}}_{\mathrm{C}}=\frac{1}{\sqrt{2}}\\ \Rightarrow {\mathrm{\theta}}_{\mathrm{C}}={45}^{\circ}\phantom{\rule{1em}{0ex}}\Rightarrow \mathrm{d}=\frac{\mathrm{R}}{\sqrt{2}}\end{array}$

Question

When the value of d chosen is such that TIR just takes place then time for which light remains inside the cylinder is:

Solution

Total distance travel in the semi cylinder $=2\sqrt{2}R$

Optical path travel$=\left(2\sqrt{2}\mathrm{R}\right)\times \sqrt{2}=4R\phantom{\rule{0ex}{0ex}}$

$\text{time}=\frac{4\mathrm{R}}{\mathrm{c}}$

Question

Distance d is now varied, so that the ray always emerges from the other side of the semi cylinder. What is the range of OB?

Solution

$\text{Range}\mathrm{R}\sqrt{2}\text{to}\mathrm{\infty}$

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