Question 51: Ray Optics

Question

A parallel beam of light is incident  from air at an anglea on the side PQ of  a right angled triangular prism of  refractive index n = $$\sqrt 2 $$. Light  undergoes total  internal reflection in  the prism at the face  PR when a has a  minimum value of  45°. The angle q of the  prism is :

Accepted Answer

Applying Snell's law at M,
$$n = {{\sin \alpha } \over {\sin {r_1}}} \Rightarrow \sqrt 2  = {{\sin 45^\circ } \over {\sin {r_1}}}$$
$$ \Rightarrow \sin {r_1} = {{\sin 45^\circ } \over {\sqrt 2 }} = {{1/\sqrt 2 } \over {\sqrt 2 }} = {1 \over 2}$$
$${r_1} = 30^\circ $$
$$\sin {	heta _c} = {1 \over n} = {1 \over {\sqrt 2 }} \Rightarrow {	heta _c} = 45^\circ $$
Let us take $${r_2} = {	heta _c} = 45^\circ $$ for just satisfying the condition of TIR.
In $$\Delta PNM$$,
$$	heta  + 90 + {r_1} + 90 - {r_2} = 180^\circ $$
or $$	heta  = {r_2} - {r_1} = 45^\circ  - 30^\circ  = 15^\circ $$
Note If $$\alpha$$ = 45$$^\circ $$ (the given value). Then, r
1
> 30$$^\circ $$ (the obtained value)
$${r_2} > {	heta _c}$$ (as r
2
$$-$$ r
1
= $$	heta$$ or r
2
= $$	heta$$ + r
1
)
or TIR will take place. So, for taking TIR under all conditions $$\alpha$$ should be greater than 45$$^\circ$$ or this is the minimum value of $$\alpha$$.

Answer

15o

Answer

22.5o

Answer

30o

Answer

45o

Ray Optics

A parallel beam of light is incident from air at an anglea on the side PQ of a right angled triangular prism of refractive index n = $\sqrt 2 $. Light undergoes total internal reflection in the prism at the face PR when a has a minimum value of 45°. The angle q of the prism is :

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