Question 403: Ray Optics

Question

For the given incident ray as shown in figure, the condition of total internal refraction of this ray the required refractive index of prism will be

Accepted Answer

Applying SnellÃ¢ÂÂs law of refraction at A,
$$\mu  = {{\sin 45^\circ } \over {\sin r}}$$
$$ \Rightarrow $$ r = $${\sin ^{ - 1}}\left( {{1 \over {\sqrt 2 \mu }}} \right)$$ ....(1)
Applying the condition of total internal reflection at B,
i
c
= $${\sin ^{ - 1}}\left( {{1 \over \mu }} \right)$$ ..........(2)
where i
c
is the critical angle.
From figure, r + i
c
= $${\pi  \over 2}$$
$$ \Rightarrow $$ $${\sin ^{ - 1}}\left( {{1 \over {\sqrt 2 \mu }}} \right) = {\pi  \over 2} - {\sin ^{ - 1}}\left( {{1 \over \mu }} \right)$$
$$ \Rightarrow $$ $${\sin ^{ - 1}}\left( {{1 \over {\sqrt 2 \mu }}} \right) = {\cos ^{ - 1}}\left( {{1 \over \mu }} \right)$$ = $${\sin ^{ - 1}}\left( {{{\sqrt {{\mu ^2} - 1} } \over \mu }} \right)$$
$$ \Rightarrow $$ $${{1 \over {\sqrt 2 \mu }} = {{\sqrt {{\mu ^2} - 1} } \over \mu }}$$
$$ \Rightarrow $$ $${\mu ^2} = 1 + {1 \over 2}$$
$$ \Rightarrow $$ $$\mu  = \sqrt {{3 \over 2}} $$

Answer

$${{\sqrt 3  + 1} \over 2}$$

Answer

$${{\sqrt 2  + 1} \over 2}$$

Answer

$$\sqrt {{3 \over 2}} $$

Answer

$$\sqrt {{7 \over 6}} $$

Ray Optics

For the given incident ray as shown in figure, the condition of total internal refraction of this ray the required refractive index of prism will be

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