Question

The critical angle for diamond with respect to air is nearly

• A. $48.8^\circ$
• B. $41.1^\circ$
• C. $37.3^\circ$
• D. $24.4^\circ$

Hint:

Critical Angle Formula: \[ \theta_c = \arcsin\left(\fracn_\textairn_\textdense\right) \] Total internal reflection occurs for $\theta>\theta_c$.

Answer 1

The Correct Option is D

Using Snell’s law at the critical angle ($\theta_c$): \[ \sin\theta_c = \frac{n_2}{n_1} \] For diamond: $n_1 = 2.418$, for air: $n_2 = 1$. Then: \[ \sin\theta_c = \frac{1}{2.418} \approx 0.4136 \Rightarrow \theta_c = \arcsin(0.4136) \approx 24.4^\circ \] Thus, the correct option is (4).

Answer 2

Step 1: Understanding the Critical Angle
The critical angle is the minimum angle of incidence inside a denser medium at which light is totally internally reflected when it hits the boundary with a less dense medium. It is the angle beyond which light cannot pass into the less dense medium and is instead reflected back.

Step 2: Formula for Critical Angle
The critical angle (θ_c) can be calculated using Snell’s law:
sin θ_c = n₂ / n₁
where n₁ is the refractive index of the denser medium (diamond here), and n₂ is the refractive index of the less dense medium (air).

Step 3: Refractive Indices of Diamond and Air
- Refractive index of diamond, n₁ ≈ 2.42
- Refractive index of air, n₂ ≈ 1.00

Step 4: Calculate the Critical Angle
Using the formula:
sin θ_c = 1 / 2.42 ≈ 0.413
θ_c = sin⁻¹(0.413) ≈ 24.4°

Step 5: Conclusion
Hence, the critical angle for diamond with respect to air is approximately 24.4°, meaning light inside diamond hitting the surface at angles greater than this will undergo total internal reflection.