Question

A beam of light is incident from air on the surface of a liquid. The angle of incidence is \(\theta\) and the angle of refraction is \(\alpha\). If the critical angle for the liquid when surrounded by air is \(\theta_c\), then \(\sin\theta_c\) is

• A. \(\dfrac{\sin\alpha}{\sin\theta}\)
• B. \(\sin\alpha \times \sin\theta\)
• C. \(\dfrac{\sin\theta}{\sin\alpha}\)
• D. \(\dfrac{\sin\alpha}{\cos\theta}\)

Hint:

Remember the two important relations: \[ n_1\sin i=n_2\sin r \] (Snell's law) and \[ \sin C=\frac{1}{n} \] (for a denser medium to air). Combining these relations often helps in critical angle problems.

Answer 1

The Correct Option is A

Step 1: Apply Snell's law at the air-liquid surface.
\( \sin\theta = n_l\sin\alpha \Rightarrow n_l = \frac{\sin\theta}{\sin\alpha} \).

Step 2: Use the critical angle definition.
At total internal reflection (liquid to air): \( n_l\sin\theta_c = 1 \Rightarrow \sin\theta_c = \frac{1}{n_l} = \frac{\sin\alpha}{\sin\theta} \). \[ \boxed{\sin\theta_c = \dfrac{\sin\alpha}{\sin\theta}} \]