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

The Correct Option is D

 To solve this problem, let's understand the conditions for the transmission of light through a prism. When monochromatic light is incident on a glass prism, it undergoes refraction at both surfaces of the prism. We need to determine the condition required for successful transmission of this light through the second face of the prism.

Snell's Law:

According to Snell's Law, the relation between the angle of incidence \((\theta)\) and the angle of refraction \((r_1)\) at the first surface is given by:

\[\mu = \frac{\sin \theta}{\sin r_1}\]

Refraction at the Second Surface:

When the ray refracts at the second surface \((AC)\), the angle of incidence inside the prism is \((r_1)\), and the angle of refraction outside is \((r_2)\). The second application of Snell's Law gives us:

\[\mu \sin r_2 = \sin r_1\]

Condition for Transmission:

For successful transmission through the prism, the angle \((r_2)\) must be less than the critical angle \((\theta_c)\) which is given by:

\[r_2 < \sin^{-1}\left(\frac{1}{\mu}\right)\]

Using the angle of the prism \((A)\), we know:

\[r_1 + r_2 = A\]

Substituting for \((r_2)\) and simplifying, we find:

\[r_1 > A - \sin^{-1}\left(\frac{1}{\mu}\right)\]

Final Condition on \((\theta)\):

Transforming \((r_1)\) into terms of \((\theta)\) using Snell's Law, we have:

\[\theta > \sin^{-1}\left[\mu \sin \left(A - \sin^{-1}\left(\frac{1}{\mu}\right)\right)\right]\]

Thus, the correct option is:

\[\theta > \sin^{-1}\left[\mu \, \sin\left(A-\sin^{-1}\left(\frac{1}{\mu}\right)\right)\right]\]

This condition ensures that the light ray successfully emerges from the prism through the second face \((AC)\).