19. A light ray falls on a rectangular glass slab as shown. The index of refraction of the glass, if total internal reflection is to occur at the vertical face, is ?

19. A light ray falls on a rectangular glass slab as shown. The index of refraction of the glass, if total internal reflection is to occur at the vertical face, is ?

(a) \( \sqrt{\frac{3}{2}} \)  

(b) \( \frac{\sqrt{3} + 1}{2} \)  

(c) \( \frac{\sqrt{2} + 1}{2} \)  

(d) \( \frac{\sqrt{5}}{2} \)

Answer :  (a) \( \sqrt{\frac{3}{2}} \)

\(\frac{\sin 45^\circ}{\sin r} = \mu\)

 

\(\frac{1}{\sqrt{2} \sin r} = \mu\)

 

\(\sin i_c = \frac{1}{\mu}\)

 

\(\sin (90^\circ - r) = \frac{1}{\mu}\)

 

\( \cos r = \frac{1}{\mu}\)

 

\(\cos r = \sqrt{1 - \sin^2 r}\)

 

\(\frac{1}{\mu} = \sqrt{1 - \left( \frac{1}{\sqrt{2} \mu} \right)^2}\)

 

\(\frac{1}{\mu} = \sqrt{\frac{2\mu^2 - 1}{2\mu^2}}\)

 

\(\sqrt{2} = \sqrt{2\mu^2 - 1}\)

 

\(2 = 2\mu^2 - 1\)

 

\(\mu = \sqrt{\frac{3}{2}}\)