White light is incident on the interface of glass and air as shown in the figure. If green light is

Q: Why does green light graze the surface?

Because the angle of incidence is exactly equal to the critical angle for green light. At the critical angle, the angle of refraction is $90^\circ$.

Q: How does wavelength affect the critical angle?

According to Cauchy's formula, a longer wavelength results in a smaller refractive index. Since $\sin \theta_c = 1/\mu$, a smaller refractive index means a larger critical angle.

Q: Why don't violet, indigo, and blue emerge into the air?

These colors have shorter wavelengths than green, meaning they have a higher refractive index and a smaller critical angle. Since the angle of incidence is greater than their critical angles, they suffer Total Internal Reflection.

Q: What happens to the yellow, orange, and red light?

They have longer wavelengths than green, so their critical angles are larger than the angle of incidence. Thus, they undergo normal refraction and emerge into the air.

Q: Does the angle of incidence change for different colors?

No, the white light is incident as a single beam, so all its constituent colors strike the interface at the exact same angle of incidence. What differs is their critical angle!

Question

White light is incident on the interface of glass and air as shown in the figure. If green light is just totally internally reflected then the emerging ray in air contains.

Options:

(A) yellow, orange, red

(B) violet, indigo, blue

(C) all colours

(D) all colours except green

Sigma Percentile · Accepted Answer

### The Setup: A Colorful Boundary Imagine a beam of white light traveling through a dense glass medium and striking the boundary where it meets the air. White light is a beautiful mixture of all the colors of the rainbow—Violet, Indigo, Blue, Green, Yellow, Orange, and Red (VIBGYOR). The problem gives us a fascinating clue: the **green** component of this white light just grazes the glass-air interface. In the language of physics, this means the angle of incidence $i$ is exactly equal to the critical angle for green light, denoted as $ heta_{cG}$. $$i = heta_{cG}$$ But what happens to the other colors? Do they escape into the air, or are they trapped inside the glass? To answer this, we need to understand how the critical angle behaves for different colors. ### The Master Equation: Cauchy's Formula The critical angle $ heta_c$ for any medium is given by the relation: $$\sin heta_c = \frac{1}{\mu}$$ where $\mu$ is the refractive index of the medium. But here is the catch: the refractive index is not a constant! It depends on the wavelength $\lambda$ of the light. This relationship is beautifully captured by **Cauchy's formula**: $$\mu \approx A + \frac{B}{\lambda^2}$$ From this formula, we can draw a powerful conclusion: as the wavelength $\lambda$ increases, the refractive index $\mu$ decreases. And if $\mu$ decreases, the value of $\sin heta_c$ increases, which means the critical angle $ heta_c$ itself increases. In short: **Longer wavelength $\implies$ Smaller refractive index $\implies$ Larger critical angle.** ### Analyzing the Spectrum Now, let's look at our VIBGYOR spectrum. As we move from violet to red, the wavelength strictly increases: $$\lambda_V < \lambda_I < \lambda_B < \lambda_G < \lambda_Y < \lambda_O < \lambda_R$$ This means that Yellow, Orange, and Red have longer wavelengths than Green. Conversely, Violet, Indigo, and Blue have shorter wavelengths than Green. ### The Great Escape: Yellow, Orange, and Red Let's analyze the colors with longer wavelengths: Yellow, Orange, and Red. Since their wavelengths are greater than that of green ($\lambda > \lambda_G$), their critical angles must be greater than the critical angle of green ($ heta_c > heta_{cG}$). But remember, the angle of incidence $i$ for the entire white light beam is fixed at $ heta_{cG}$. Therefore, for Yellow, Orange, and Red, the angle of incidence is strictly less than their respective critical angles: $$i < heta_c$$ Because the angle of incidence is less than the critical angle, these colors will not suffer Total Internal Reflection. Instead, they will successfully refract and **emerge into the air**. ### The Trapped Colors: Violet, Indigo, and Blue What about Violet, Indigo, and Blue? Their wavelengths are shorter than that of green ($\lambda < \lambda_G$). Consequently, their critical angles are smaller than the critical angle of green ($ heta_c < heta_{cG}$). For these colors, our fixed angle of incidence $i$ is now strictly greater than their critical angles: $$i > heta_c$$ When the angle of incidence exceeds the critical angle, the light cannot escape. Violet, Indigo, and Blue will undergo **Total Internal Reflection (TIR)** and bounce back into the glass. ### Final Conclusion The rays that successfully cross the boundary and emerge into the air are the ones with longer wavelengths. Therefore, the emerging ray in the air contains **yellow, orange, and red**. This is a stunning example of how dispersion and total internal reflection work hand-in-hand!

Animated Solution for Physics - Optics: White light is incident on the interface of glass and air as shown in the figure. If green light is just totally internally reflected then the emerging ray in air contains.

Select Answer:

Visualized Solution

$i = θ_{c G}$

$θ_{c} = f (λ)$

$λ_{V} < \dots < λ_{R}$

$λ > λ_{G}$

$λ < λ_{G}$

$Final Answer$

The Sigma Insight: Refraction and Total Internal Reflection

The Setup

The Master Equation

Analyzing the Spectrum

The Great Escape

The Trapped Colors

Final Conclusion

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