Question

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

• A. $\theta <co{s}^{-1}\left[\mu sin\right(A+si{n}^{-1}\left(\frac{1}{\mu }\right)\left)\right]$
• B. $\theta <si{n}^{-1}\left[\mu sin\right(A-si{n}^{-1}\left(\frac{1}{\mu }\right)\left)\right]$
• C. $\theta >co{s}^{-1}\left[\mu sin\right(A+si{n}^{-1}\left(\frac{1}{\mu }\right)\left)\right]$
• D. $\theta >si{n}^{-1}\left[\mu sin\right(A-si{n}^{-1}\left(\frac{1}{\mu }\right)\left)\right]$

Answer 1

Answer: (D)

$r_2<{\theta }_c$
$r_2<{\sin }^{-1}(1/\mu )$
$\sin r_2<1/\mu$
$\sin \theta =\mu \sin r_1$
$r_1={\sin }^{-1}(\sin \theta /\mu )$
$\sin \left(A-r_1\right)<1/\mu$
$\sin \left(A-\left({\sin }^{-1}\left(\frac{\sin \theta }{\mu }\right)\right)\right)<\frac{1}{\mu }$
$A-{\sin }^{-1}\left(\frac{\sin \theta }{\mu }\right)<{\sin }^{-1}\left(\frac{1}{\mu }\right)$
$A-{\sin }^{-1}\left(\frac{1}{\mu }\right)<{\sin }^{-1}\left(\frac{\sin \theta }{\mu }\right)$
$\left[\sin \left(A-{\sin }^{-1}\left(\frac{1}{m}\right)\right)\right]<\frac{\sin \theta }{\mu }$
$[{\sin }^{-1}\left[\mu \left[\sin \left(A-{\sin }^{-1}\left(\frac{1}{m}\right)\right)\right]\right]<\theta$