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  • An infinitely long cylinder of radius R is made of an unusual exotic material with refractive index –1 (Fig. 9.7). The cylinder is placed between two planes whose normal are along the y direction.

An infinitely long cylinder of radius R is made of an unusual exotic material with refractive index –1 (Fig. 9.7). The cylinder is placed between two planes whose normal are along the y direction. The center of the cylinder O lies along the y-axis. A narrow laser beam is directed along the y direction from the lower plate. The laser source is at a horizontal distance x from the diameter in the y direction. Find the range of x such that light emitted from the lower plane does not reach the upper plane.




 

Answers (1)

Explanation:-

Here the material has a refractive index -1,\theta _{r} is negative and \theta _{r}^{'} positive.

Now, \left | \theta _{i} \right |=\left | \theta _{r} \right |=\left | \theta _{r}^{'} \right | The total deviation of the outcoming ray from the

incoming ray is 4\theta _{r}.Rays shall not reach the receiving plate if

\frac{\pi }{2}\leq 4\theta _{i}\leq \frac{3\pi }{2}

[angles measured clockwise from the y-axis]

On solving,          

   \frac{\pi }{8}\leq \theta _{i}\leq \frac{3\pi }{8}

Now,

  \sin \theta _{i}=\frac{x}{R}

\frac{\pi}{8}\leq \sin^{-1}\frac{x}{R}\leq \frac{3\pi}{8}

or \frac{\pi}{8}\leq \frac{x}{R}\leq \frac{3\pi}{8}

Thus, for light emitted from the source shall not reach the receiving plate.

If \frac{R\pi}{8}\leq x\leq \frac{3R\pi }{8}

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