A rod made of glass `(mu = 1.5)` and of square cross-section is equal is bent into the shape shown in figure. A parallel beam of light falls perpendicurly on the plane flat surface . A . Referring to the diagram, d is the width of a side & R is radius of inner semicircle. find the maximum value of ratio `(d)/(R)` so that all light entering the glass through surface A emerge from the glass through surface B.

Options

A. `1//2`
B. `2//3`
C. `3//2`
D. `2//1`

Answer

A

Explanation

As shown in the figure, angle of incidence will be minimum for the ray travelling along the inner edge. So the whole incident light entering the glass through surface A will emerge from the glass through surface B.

`theta gt I_(C )`
`sin theta gt sin I_(C )`
But as, `sin I_(C ) = (1)/(mu) =(2)/(3)`
And from the geometry of the figure,
`sin theta = (OC)/( OE ) =(R )/(OD+DE ) =(R )/(R+d)`
So, `(R )/(R+d) gt (2)/(3)`, I.e, `R gt 2d`
i.e., `(d)/(R ) lt (1)/(2)` so `((d)/(R ))=(1)/(2)`