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 liquid when surrounded by air is $\theta_{c}$ then $\sin\theta_{c}$ is

• A. $\frac{\sin\alpha}{\sin\theta}$
• B. $\sin\alpha \times \sin\theta$
• C. $\frac{\sin\theta}{\sin\alpha}$
• D. $\frac{\cos\theta}{\cos\alpha}$

Hint:

Use Snell’s law: \( n_1 \sin\theta = n_2 \sin\alpha \).
Critical angle occurs when the angle of refraction is \(90^\circ\).
Always place air’s refractive index as 1 when it’s the surrounding medium.

Answer 1

The Correct Option is C

Apply Snell’s law for refraction: \[ n_{\text{air}} \sin\theta = n_{\text{liquid}} \sin\alpha \Rightarrow \frac{n_{\text{liquid}}}{n_{\text{air}}} = \frac{\sin\theta}{\sin\alpha} \] Now, the critical angle is given by: \[ \sin\theta_c = \frac{n_{\text{air}}}{n_{\text{liquid}}} \Rightarrow \sin\theta_c = \frac{1}{\frac{n_{\text{liquid}}}{n_{\text{air}}}} = \frac{\sin\theta}{\sin\alpha} \]