Question 189: Ray Optics

Question

Monochromatic light is incident on a glass prism of angle $$A$$. If the refractive index of the material of the prism is $$\mu $$, a ray, incident at an angle $$	heta $$. on the face $$AB$$ would get transmitted through the face $$AC$$ of the prism provided :

Accepted Answer

When $${r_2} = C,\,\angle {N_2}Rc = {90^ \circ }$$
Where $$C = $$ critical angle
As $$\sin C = {1 \over v} = \sin {r_2}$$
Applying snell's law at $$'R'$$
$$\mu \sin {r_2} = 1\sin {90^ \circ }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i ight)$$
Applying snell's law at $$'Q'$$
$$1 	imes \sin 	heta  = \mu \sin {r_1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} ight)$$
But $${r_1} = A - {r_2}$$
So, $$\sin 	heta  = \mu \sin \left( {A - {r_2}} ight)$$
$$	heta  = \mu \sin A\cos {r_2} - \cos \,A\,\,\,\,\,...\left( {iii} ight)$$ [using $$(i)$$]
From $$(1)$$
$$\cos \,{r_2} = \sqrt {1 - {{\sin }^2}{r_2}}  = \sqrt {1 - {1 \over {{\mu ^2}}}} \,\,\,\,\,\,\,...\left( {iv} ight)$$
By eq.$$(iii)$$ and $$(iv)$$
$$\sin 	heta  = \mu \sin A\sqrt {1 - {1 \over {{\mu ^2}}}}  - \cos A$$
on further solving we can show for ray not to transmitted through face $$AC$$
$$	heta  = {\sin ^{ - 1}}\left[ {u\sin \left( {A - {{\sin }^{ - 1}}\left( {{1 \over \mu }} ight)} ight.} ight]$$
So, for transmission through face $$AC$$
$$	heta  > {\sin ^{ - 1}}\left[ {u\sin \left( {A - {{\sin }^{ - 1}}\left( {{1 \over \mu }} ight.} ight)} ight]$$

Answer

$$	heta  > {\cos ^{ - 1}}\left[ {\mu \,\sin \left( {A + {{\sin }^{ - 1}}} ight.\left( {{1 \over \mu }} ight)} ight]$$

Answer

$$	heta  < {\cos ^{ - 1}}\left[ {\mu \,\sin \left( {A + {{\sin }^{ - 1}}} ight.\left( {{1 \over \mu }} ight)} ight]$$

Answer

$$	heta  > si{n^{ - 1}}\left[ {\mu \,\sin \left( {A - {{\sin }^{ - 1}}} ight.\left( {{1 \over \mu }} ight)} ight]$$

Answer

$$	heta  < si{n^{ - 1}}\left[ {\mu \,\sin \left( {A - {{\sin }^{ - 1}}} ight.\left( {{1 \over \mu }} ight)} ight]$$

Ray Optics

Monochromatic light is incident on a glass prism of angle $A. If the refractive index of the material of the prism is \mu , a ray, incident at an angle \theta . on the face AB would get transmitted through the face AC$ of the prism provided :

Choose the correct answer: