Question

The critical angle for glass is \( \theta_1 \) and that for water is \( \theta_2 \). The critical angle for the glass-water surface would be (given \( \mu_g = 1.5 \), \( \mu_w = 1.33 \)):

• A. \( \) less than \( \theta_2 \)
• B. \( \) between \( \theta_1 \) and \( \theta_2 \)
• C. \( \) greater than \( \theta_2 \)
• D. \( \) less than \( \theta_1 \)

Hint:

The critical angle for a boundary between two media depends on the refractive indices of the two media. The critical angle for the boundary between glass and water lies between the critical angles for each medium individually.

Answer 1

The Correct Option is C

Given Data
Refractive index of glass, \( \mu_g = 1.5 \)
Refractive index of water, \( \mu_w = 1.33 \)
Critical angle for glass, \( \theta_1 \)
Critical angle for water, \( \theta_2 \) Critical Angle Formula
The critical angle \( \theta_c \) for a boundary between two media is given by: \[ \sin \theta_c = \frac{\mu_2}{\mu_1} \] where \( \mu_1 \) is the refractive index of the denser medium (glass) and \( \mu_2 \) is the refractive index of the less dense medium (water). Critical Angle for Glass-Water Interface
For the glass-water interface:
\[ \sin \theta_c = \frac{\mu_w}{\mu_g} = \frac{1.33}{1.5} \approx 0.8867 \] Therefore:
\[ \theta_c = \sin^{-1}(0.8867) \approx 62.5^\circ \] Comparison with Given Critical Angles

The critical angle for glass (\( \theta_1 \)) is: \[ \sin \theta_1 = \frac{1}{\mu_g} = \frac{1}{1.5} \approx 0.6667 \Rightarrow \theta_1 \approx 41.8^\circ \]
The critical angle for water (\( \theta_2 \)) is:
\[ \sin \theta_2 = \frac{1}{\mu_w} = \frac{1}{1.33} \approx 0.7519 \Rightarrow \theta_2 \approx 48.8^\circ \] The critical angle for the glass-water interface (\( \theta_c \approx 62.5^\circ \)) is greater than both \( \theta_1 \) and \( \theta_2 \). Conclusion
The critical angle for the glass-water interface is greater than the critical angle for water (\( \theta_2 \)). Final Answer \begin{center} \boxed{\text{(C) greater than } \theta_2} \end{center}