Question 267 of 295

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:

A$θ < cos - 1 μ sin A + sin - 1 1 μ$B$\theta \gt {\sin }^{-1}⁡\left[\mu \sin ⁡\left(A-{\sin }^{-1}⁡\left(\dfrac{1}{\mu }\right)\right)\right]$C$θ < sin - 1 μ sin A - sin - 1 1 μ$D$\theta \gt {\cos }^{-1}⁡\left[\mu \sin ⁡\left(A+{\sin }^{-1}⁡\left(\dfrac{1}{\mu }\right)\right)\right]$

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Answer: (B) $\theta \gt {\sin }^{-1}⁡\left[\mu \sin ⁡\left(A-{\sin }^{-1}⁡\left(\dfrac{1}{\mu }\right)\right)\right]$

Central Idea: The ray will get transmitted through face $AC$ if ${i}_{AC}\lt {i}_{C}$
Consider the ray diagram is shown below.

A ray of light incident on face $AB$ at an angle $\theta$
${r}_{1}=$ Angle of refraction on face $AB$
${r}_{2}=$ Angle of incidence at face $AC$
For transmission of light through face $AC$
${i}_{AC}\lt {i}_{C}$
or $A-{r}_{1}\lt {i}_{C}$
or $\sin \left(A-{r}_{1}\right)\lt \sin {i}_{C}$
or $\sin \left(A-{r}_{1}\right)\lt \dfrac{1}{\mu }$
$A-{r}_{1}\lt {\sin }^{-1}\left(\dfrac{1}{\mu }\right)$
or $\sin {r}_{1}\gt \sin \left[A-{\sin }^{-1}\left(\dfrac{1}{\mu }\right)\right]$
Now, applying Snell's law at the face $AB$
$1\times \sin \theta =\mu \sin {r}_{1}$
or $\sin {r}_{1}=\dfrac{\sin \theta }{\mu }$
$\Rightarrow \dfrac{\sin \theta }{\mu }\gt \sin \left[A-{\sin }^{-1}\left(\dfrac{1}{\mu }\right)\right]$
or $\theta \gt {\sin }^{-1}\left[\mu \sin \left\{A-{\sin }^{-1}\left(\dfrac{1}{\mu }\right)\right\}\right]$