An infinitely long cylinder of radius R is made of an unusual exotic material with refractive index-1 (see figure). The cyliner is palced between two planes whose normals 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.

Explanation

`rArr` For a material with negative refractive index, from of Snell.s law, is,
`- n =(sin theta_i)/(sin theta_r)`
`therefore - (-1) = (sin theta_i)/(sin theta_r)`
`therefore sin theta_i = sin theta_r`
`therefore theta_i = sin theta_r`
`rArr` Since two interior angles made by a chord of a circle with its centre are equal, we can write for above figure,
`theta_r = theta_r`
`rArr` Now, applying Snell.s law at point C, we get
`theta._r = theta_e`
`rArr` Thus, in magnitudes `theta_i = theta_r= theta._r = theta_e`
`rArr` Here, incident ray `vec(AB)`, first gets deviated by angle `2 theta_i` at first point of incidence B and then again it gets deviated by another angle `2 theta_i` at second point of incidence C. Thus, it undergoes total deviation by angle `4 theta_i` (Here `sin theta_i= sin theta_r)` and so we can consider as if incident light ray undergoes reflection instead of refraction.
`rArr` Now, angle `4 theta_i` measured clockwise with respect to Y - axis is such that (in radian),
`(pi)/(2) lt 4 theta_i le (3pi)/(2)......(1)`
then emergent light does not reach to upper plate. Now, for the right angled `Delta OEB,` since angle `theta_i` is extremely small.
`sin theta_i ~~ theta_i = (x)/(R) .....(2)`
`rArr` From equation (1),
`pi/8 lt theta_i le (3pi)/(8)`
`therefore (pi)/(8) lt x/R le (3pi)/(8)`
`rArr` Multiplying by R,
`(piR)/(8) le x le (3pi R)/(8)` ......(3)
`rArr` Above result is required result.