Question

A ray of yellow light undergoes total internal reflection when it is incident at the interface of two media. This ray is successively replaced by the ray of blue, green and red lights. Which of the following statements is true if the angle of incidence is same in all cases ?

• A. Total internal reflection occurs for blue light only.
• B. Total internal reflection occurs for red light only.
• C. Total internal reflection occurs for both green and blue lights.
• D. None of them will undergo total internal reflection.

Hint:

A quick rule of thumb for TIR color replacement: If a specific color undergoes TIR, all colors with a shorter wavelength (appearing to its left in the VIBGYOR sequence) will also undergo TIR at that same incident angle.

Answer 1

The Correct Option is C

Concept:
• Total Internal Reflection (TIR) occurs strictly when a light ray traveling in a denser medium strikes a rarer medium boundary at an angle of incidence \( i \) that is greater than the critical angle \( i_c \).

• The critical angle \( i_c \) is mathematically related to the refractive index \( \mu \) of the denser medium by \( \sin(i_c) = \frac{1}{\mu} \).

• According to Cauchy's dispersion formula (\( \mu = A + \frac{B}{\lambda^2} \)), the refractive index of a given material inversely depends on the wavelength \( \lambda \) of the light passing through it.

• Consequently, light colors with smaller wavelengths experience higher refractive indices, which in turn correspond to smaller critical angles.

Step 1: Establish the order of wavelengths and critical angles
The visible light spectrum ordered by increasing wavelength is Violet, Indigo, Blue, Green, Yellow, Orange, Red (VIBGYOR).
From this, we extract the order for the given colors:
\[ \lambda_{\text{blue}} < \lambda_{\text{green}} < \lambda_{\text{yellow}} < \lambda_{\text{red}} \]
Because refractive index is inversely related to wavelength:
\[ \mu_{\text{blue}} > \mu_{\text{green}} > \mu_{\text{yellow}} > \mu_{\text{red}} \]
Since a larger refractive index yields a smaller critical angle (\( \sin i_c = 1/\mu \)):
\[ i_{c,\text{blue}} < i_{c,\text{green}} < i_{c,\text{yellow}} < i_{c,\text{red}} \]

Step 2: Apply the condition for Total Internal Reflection
The problem explicitly states that the yellow ray successfully undergoes TIR at a specific angle of incidence \( i \).
This strictly means that the angle of incidence \( i \) must be greater than the critical angle for yellow light:
\[ i > i_{c,\text{yellow}} \]
Looking at our inequality chain from Step 1, both the critical angles for blue and green light are strictly smaller than the critical angle for yellow light.
Therefore, by mathematical transitivity:
\[ i > i_{c,\text{yellow}} > i_{c,\text{green}} > i_{c,\text{blue}} \]
This proves that the fixed angle of incidence \( i \) is simultaneously greater than \( i_{c,\text{green}} \) and \( i_{c,\text{blue}} \).
Consequently, both green and blue rays will easily satisfy the condition for TIR.
However, \( i_{c,\text{red}} \) is larger than \( i_{c,\text{yellow}} \), so we cannot guarantee that \( i > i_{c,\text{red}} \).

Step 3: Conclusion
Both green and blue light rays will absolutely undergo total internal reflection under the identical incident conditions.
This logic points directly to option (C).