Question

The angle of a prism made of a material of refractive index \( \sqrt{2} \) is \( 90^\circ \). The angle of incidence for a light ray on the first face of the prism such that the light ray suffers total internal reflection at the second face is

• A. \( 0^\circ \)
• B. \( 90^\circ \)
• C. \( 60^\circ \)
• D. \( 45^\circ \)

Hint:

Start by calculating the critical angle. Work backwards from the second face to the first face to find the required incidence angle.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

We need to find the condition for total internal reflection (TIR) at the second face of the prism. The limiting case for TIR is grazing emergence, where the angle of refraction at the second face is 90°. Equivalently, the angle of incidence at the second face r2 must be greater than or equal to the critical angle C.

Step 2: Key Formula or Approach:

1. Critical angle:
sin C = 1 / μ

2. Prism relation:
A = r1 + r2

3. Snell's law at first face:
sin i = μ sin r1

Step 3: Detailed Explanation:

Given:
μ = √2
A = 90°

Calculate the critical angle C:
sin C = 1 / √2
C = 45°

For TIR at the second face, we need:
r2 ≥ C

The limiting condition is:
r2 = 45°

Using the prism relation:
A = r1 + r2
90° = r1 + 45°
r1 = 45°

Now apply Snell's law at the first face:
sin i = μ sin r1
sin i = √2 × sin 45°
sin i = √2 × (1 / √2)
sin i = 1
i = 90°

Thus, at an incidence angle of 90° (grazing incidence), the ray strikes the second face at the critical angle.
For i < 90°, r1 would be less than 45°, making r2 > 45°, which ensures total internal reflection.

So, 90° is the limiting or boundary value asked in the problem.

Step 4: Final Answer:

The angle of incidence is 90°.