Problem 45 - College Physics

Question

A beam of light with red and blue components of wavelengths \(670 \mathrm{nm}\) and \(425 \mathrm{nm}\), respectively, strikes a slab of fused quartz at an incident angle of \(30^{\circ} .\) On refraction, the different components are separated by an angle of 0.00131 rad. If the index of refraction of the red light is \(1.4925,\) what is the index of refraction of the blue light?

Step-by-Step Solution

Answer

The index of refraction for blue light is approximately 1.4929.

Step 1: Understand the Problem

We are given two components of light with different wavelengths. The incident angle and the index of refraction for red light are provided. We need to find the index of refraction for blue light given the angle of separation upon refraction.

Step 2: Use Snell's Law

Snell's Law relates the angles of incidence and refraction to the indices of refraction for each component of light. The law is given by \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \), where \( n_1 \) and \( n_2 \) are the indices of refraction of the incident medium and refracted medium, and \( \theta_1 \) and \( \theta_2 \) are their respective angles.

Step 3: Calculate Refraction Angle for Red Light

Using Snell's Law for the red light component:\[\sin(\theta_{r,\text{red}}) = \frac{n_1}{n_2, \text{red}} \sin(\theta_1)\]Substitute \( n_1 = 1 \) for air, \( \theta_1 = 30^{\circ} \), and \( n_2, \text{red} = 1.4925 \):\[\sin(\theta_{r,\text{red}}) = \frac{1}{1.4925} \sin(30^{\circ})= \frac{0.5}{1.4925}\approx 0.335 \] \( \theta_{r,\text{red}} = \sin^{-1}(0.335) \approx 19.5^{\circ} \).

Step 4: Determine Separation Angle for Blue Light

We know the separation angle between the red and blue components is 0.00131 rad. Convert this to degrees: \( 0.00131 \) rad \( \times \frac{180}{\pi} \approx 0.075^{\circ} \). The angle for the blue component is thus \( \theta_{r,\text{blue}} = 19.5^{\circ} + 0.075^{\circ} = 19.575^{\circ} \).

Step 5: Calculate the Index of Refraction for Blue Light

Using Snell's Law again for the blue light:\[\sin(\theta_{r,\text{blue}}) = \frac{n_1}{n_2, \text{blue}} \sin(\theta_1)\]Solve for \( n_2, \text{blue} \):\[n_2, \text{blue} = \frac{\sin(30^{\circ})}{\sin(19.575^{\circ})}= \frac{0.5}{0.3354} \approx 1.4929\]

Step 6: Conclusion

Using Snell's law, we've found that the index of refraction for blue light, considering the separation upon refraction, is approximately 1.4929.

Key Concepts

Index of Refraction

The index of refraction is a measure of how much light bends when it enters a new medium. Every material has a unique index of refraction, symbolized by "\( n \)". This index depends on the speed of light in that medium compared to the speed of light in a vacuum. Here's what you need to know:

• If the index is high, like glass or water, light slows down more and bends significantly.
• If the index is low, like air, light barely bends.

The formula for the index of refraction is \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum and \( v \) is the speed of light in the medium.

In the problem, you're given the index of refraction for red light in fused quartz as 1.4925. This tells you how much slower light travels through quartz compared to air.

When dealing with light components like red and blue, these each have different indices of refraction because they travel at slightly different speeds when entering a new medium due to their different frequencies.

Incident Angle

The incident angle is the angle at which a light beam hits a surface. It's measured between the incoming light and the perpendicular (normal) to the surface.

In our context, the incident angle influences how light refracts after hitting a boundary like air to quartz. This phenomenon follows Snell's Law which is crucial for solving many optics problems. Snell's Law states:

• \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \)

Here, \( \theta_1 \) is the incident angle, while \( \theta_2 \) is the refraction angle. The problem gives \( \theta_1 = 30^{\circ} \). This was used in calculations to find out how much the red and blue light gets bent in the quartz medium.

Remember, the incident angle is about incoming light. How it refracts or scatter depends heavily on this angle, including how the different colors of light separate due to varying refraction indices.

Wavelengths

Wavelengths are distances between consecutive peaks of a wave. They define the color of light and are measured in nanometers (nm). Visible light has wavelengths ranging from about 400 nm to 700 nm.

• Red light often has longer wavelengths, like 670 nm, which bends differently when entering a medium.
• Blue light has a shorter wavelength, such as 425 nm, affecting how it refracts.

The exercise involves these wavelengths to show how different colors of light behave differently in materials like quartz. The difference in behavior leads to the phenomenon of dispersion, causing the separation into different colors.

In simpler terms, though both red and blue light enter quartz at the same incident angle, their differences in wavelength cause them to refract at slightly different angles. That's why we observe them spreading apart by a small separation angle inside the medium.