Question

When a ray of light is refracted from one medium to another, then the wavelength changes from 6000 Å to 4000 Å. The critical angle for the interface will be:

• A. \( \cos^{-1} \left( \frac{2}{3} \right) \)
• B. \( \sin^{-1} \left( \frac{2}{\sqrt{3}} \right) \)
• C. \( \cos^{-1} \left( \frac{2}{\sqrt{3}} \right) \)
• D. \( \sin^{-1} \left( \frac{2}{3} \right) \)

Hint:

The critical angle is calculated using the refractive index of the two media involved. It is the angle beyond which total internal reflection occurs.

Answer 1

The Correct Option is D

Step 1: Using the Wavelength and Refractive Index.
The refractive index \( n \) of the two mediums is related to the wavelengths \( \lambda_1 \) and \( \lambda_2 \) in the respective mediums by: \[ n = \frac{\lambda_1}{\lambda_2} \] Here, \( \lambda_1 = 6000 \, \text{Å} \) and \( \lambda_2 = 4000 \, \text{Å} \). Thus, \[ n = \frac{6000}{4000} = 1.5 \] Step 2: Critical Angle Formula.
The critical angle \( \theta_c \) is given by: \[ \sin \theta_c = \frac{1}{n} \] Substituting the value of \( n \): \[ \sin \theta_c = \frac{1}{1.5} = \frac{2}{3} \] Thus, \( \theta_c = \sin^{-1} \left( \frac{2}{3} \right) \).