Question

Consider an equilateral prism (refractive index \( \sqrt{2} \)). A ray of light is incident on its one surface at a certain angle \( i \). If the emergent ray is found to graze along the other surface, then the angle of refraction at the incident surface is close to

• A. \(15^\circ\)
• B. \(40^\circ\)
• C. \(20^\circ\)
• D. \(30^\circ\)

Hint:

When a ray grazes the surface during emergence, the angle of refraction at that surface equals the critical angle.

Answer 1

The Correct Option is A

For an equilateral prism, the angle of the prism is \[ A = 60^\circ. \]
Step 1: Condition for grazing emergence.
If the emergent ray grazes along the surface, the angle of emergence is \[ e = 90^\circ. \] Hence, the angle of refraction at the second surface equals the critical angle.
Step 2: Find the critical angle.
Given refractive index \[ \mu = \sqrt{2}. \] The critical angle is given by \[ \sin C = \frac{1}{\mu} = \frac{1}{\sqrt{2}}. \] Thus, \[ C = 45^\circ. \]
Step 3: Use prism geometry.
For a prism, \[ r_1 + r_2 = A. \] Here, \[ r_2 = 45^\circ, \] so \[ r_1 = 60^\circ - 45^\circ = 15^\circ. \]
Final Answer: \[ \boxed{15^\circ} \]