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. $\theta < cos^{-1}\Bigg[\mu sin\Bigg(A+sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]$
• B. $\theta < sin^{-1}\Bigg[\mu\, sin\Bigg(A-sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]$
• C. $\theta > cos^{-1}\Bigg[\mu\, sin\Bigg(A+sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]$
• D. $\theta > sin^{-1}\Bigg[\mu\, sin\Bigg(A-sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]$

Answer 1

Answer: (D)

$r_{2}<\theta_{c}$
$r_{2}<\sin ^{-1}(1 / \mu)$
$\sin r_{2}<1 / \mu$
$\sin \theta=\mu \sin r_{1}$
$r_{1}=\sin ^{-1}(\sin \theta / \mu)$
$\sin \left(A-r_{1}\right)<1 / \mu$
$\sin \left(A-\left(\sin ^{-1}\left(\frac{\sin \theta}{\mu}\right)\right)\right)<\frac{1}{\mu}$
$A-\sin ^{-1}\left(\frac{\sin \theta}{\mu}\right)<\sin ^{-1}\left(\frac{1}{\mu}\right)$
$A-\sin ^{-1}\left(\frac{1}{\mu}\right)<\sin ^{-1}\left(\frac{\sin \theta}{\mu}\right)$
$\left[\sin \left(A-\sin ^{-1}\left(\frac{1}{m}\right)\right)\right]<\frac{\sin \theta}{\mu}$
$\left[\sin ^{-1}\left[\mu\left[\sin \left(A-\sin ^{-1}\left(\frac{1}{m}\right)\right)\right]\right]<\theta\right.$