Question

A glass rod of rectangular cross-section is bent into the shape shows in the Fig. (1). A parallel beam of light falls perpendicularly on the flat surface A. Determine the minimum value of the ratio R/d for which all light entering the glass through surface A will emerge from the glass through surface B. The index of refraction of the glass is 1.5.

• A. = 2
• B. = 4
• C. = 5
• D. = 7

Answer 1

Answer: (A)

= 2

Answer 2

Consider the representative rays shown in Fig. (2). A ray entering the glass through surface A and passing along the inner side of the rod will be reflected by the outer side with the smallest angle a, at which the reflected ray is tangent to the inner side. We have to consider the conditions under which the ray will undergo total internal reflection before reaching B.

Fig. (2)

If a > q_c, the critical angle, at which total internal reflection

occurs, all the incident beam will emerge through the surface

B. Hence we require

sin a > .

The geometry gives sin a =.

Therefore $\frac { \mathrm { R } } { \mathrm { R } + \mathrm { d } }$ ³ ,

or $\left( \frac { \mathrm { R } } { \mathrm { d } } \right) _ { \min }$ = $\frac { 1 } { n - 1 }$ = $\frac { 1 } { 1.5 - 1 }$ = 2.