To solve the given problem, let's break it down step-by-step.

Given Data:

1. Incident angle: $\alpha = 50^\circ$
2. Wavelength $\lambda_1$: $540 \, \text{nm}$ (green light)
3. Order of interference: $m = 1$
4. Refractive index $n$: $1.33$

Part (a): Thickness $d$ of the liquid film

When light reflects from a thin film, interference conditions apply. The reflected light undergoes a phase change, and constructive interference occurs when:

$2d n \cos(\alpha) = m \lambda_1$

Plugging in the given values:

• $\alpha = 50^\circ$
• $m = 1$
• $\lambda_1 = 540 \, \text{nm} = 540 \times 10^{-9} \, \text{m}$
• $n = 1.33$

Calculating $d$:

$2d \cdot 1.33 \cdot \cos(50^\circ) = 540 \times 10^{-9}$

We solve for $d$:

$d = \frac{540 \times 10^{-9}}{2 \cdot 1.33 \cdot \cos(50^\circ)}$

Now, let's compute it:

$\cos(50^\circ) \approx 0.6428$

$d = \frac{540 \times 10^{-9}}{2 \cdot 1.33 \cdot 0.6428}$

$d \approx \frac{540 \times 10^{-9}}{1.709}$

$d \approx 316 \, \text{nm}$

Thus, the thickness of the liquid film is approximately 316 nm.

Part (b): Wavelength $\lambda_2$ and color for normal incidence

For normal incidence ($\alpha = 0^\circ$), the interference condition simplifies to:

$2d n = m \lambda_2$

Using the thickness $d = 316 \, \text{nm}$ and $m = 1$:

$2 \cdot 316 \times 10^{-9} \cdot 1.33 = \lambda_2$

Calculating $\lambda_2$:

$\lambda_2 = 2 \cdot 316 \times 10^{-9} \cdot 1.33$

$\lambda_2 = 841 \, \text{nm}$

This wavelength ($\lambda_2 = 841 \, \text{nm}$) corresponds to infrared light, which is not visible to the human eye.

Summary of Answers

1. (a) The thickness $d$ of the liquid film is 316 nm.
2. (b) The wavelength of the reflected light at normal incidence is 841 nm (infrared light).

Do you need further details or clarifications?

Related Questions:

1. How does the interference condition change if the refractive index changes?
2. What happens if the angle $\alpha$ increases beyond $50^\circ$?
3. How does the order $m$ affect the thickness calculation?
4. Can destructive interference be used to determine the thickness?
5. What other colors can appear due to thin-film interference?

Tip:

In thin-film interference, constructive and destructive interference depend on both the film's thickness and the incident angle. Adjusting either can produce different colors.