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Question

Light is incident at an angle $$\alpha$$ on one planer end of a transparent cylindrical rod of refractive index $$n$$. Determine the least value of $$n$$ so that the light entering the rod does not emerge from the curved surface of the rod irrespective of the value of $$\alpha$$
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Solution

The light entering the rod does nor emerge from the curved surface of rod when angle $$90-r$$ is greater than critical angle.
$$\mu \le \dfrac{1}{\sin c}$$ ,c is critical angle.
Here, $$c=90-r$$
$$\mu \le \dfrac{1}{\sin(90-1)}$$
$$\mu \le \dfrac{1}{\cos r}$$
As a limiting case , $$\mu=\dfrac{1}{\cos r}$$
By Snell's law at A,
$$\mu=\dfrac{\sin alpha}{\sin r}$$
$$\sin r=\dfrac{\sin \alpha}{\mu}$$
Smallest angle of incident on curved surface is when $$\alpha=\dfrac{\pi}{2}$$
$$\sin r=\dfrac{\sin(\pi/2)}{\mu}$$
$$\mu=\dfrac{1}{\sin r}$$
$$\sin r=\cos r$$
$$\mu=\dfrac{1}{\cos 45}=\dfrac{1/1}{1 / \sqrt2}$$
$$\mu=\sqrt2$$

Physics

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