Question

The refractive index of a prism is \( \sqrt{2} \). What should be the angle of incidence for a light ray such that the emerging ray grazes out of the surface?

• A. \(90^\circ\)
• B. \(60^\circ\)
• C. \(30^\circ\)
• D. \(45^\circ\)

Hint:

For grazing emergence from a prism: \begin{itemize} \item Angle of incidence at the second face = critical angle \item First find the critical angle using \( \sin C = \frac{1}{\mu} \) \item Then use prism geometry and Snell’s law \end{itemize}

Answer 1

The Correct Option is D

To determine the angle of incidence for a light ray such that the emerging ray grazes out of the surface of a prism with a refractive index of \( \sqrt{2} \), we need to use the concept of critical angle and Snell's Law.

Step-by-step Solution: 

1. Critical Angle Calculation:
   • The critical angle (\( C \)) is defined by the equation \(\sin C = \frac{1}{\text{Refractive Index}}\).
   • Given the refractive index (\( n \)) of the prism is \( \sqrt{2} \), the critical angle is: \(\sin C = \frac{1}{\sqrt{2}}\).
   • Therefore, \(C = 45^\circ\).
2. Relation with the Prism's Geometry:
   • For the emerging ray to graze, it must be at the critical angle when exiting. This happens when the angle of incidence is such that the internal refraction leads to this scenario.
   • Given the geometry of the prism (equilateral right-angled triangle), if the angle of refraction at the second face is the critical angle, \( 45^\circ \), then the angle of incidence at the first surface must also be \( 45^\circ \).

Conclusion:

Hence, the angle of incidence for the light ray such that the emerging ray grazes out of the surface is \(45^\circ\).