A monochromatic beam of light enters a square enclosure with mirrored interior surface at an angle of incidence `theta I (ne 0)` (see the figure below). For some value (s) of `theta i`, the beam is reflected by every mirrored wall (other than the one with the opening ) exactly once and exits the enclosure through the same hole. which of the following statements about this beam is correct?

Options

A. The beam will not come out the enclosure for any value of `thetai.`
B. The beam will not come out for more than two values of `thetai. `
C. The beam will not come out only at `thetai=45^@`
D. The beam will come out for exactly two values of `thetai`.

Answer

C

Explanation

From geometry, `alpha+theta_i=pi/2rArralpha=pi/2-theta`
`beta+alpha=pi/2rArrbeta=theta_i`
`beta+gamma=pi/2rArrgamma=pi/2-theta`
Also AD II BC & AB II CD
`therefore` ABCD is a parallelogram & AB = CD
Also, `DeltaABQ=DeltaCDS `
`therefore` From trignometry
`(AQ)/(QB)=(CR)/(BR)`
Let length of each side of square by l, AQ = x, & QB = y
`thereforex/y=(l-x)/(l-y)rArrx=y`
`thereforetheta_i=pi/4`