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Chapter 33: Problem 5

A light beam travels at \(1.94 \times 10^{8} \mathrm{m} / \mathrm{s}\) in quartz. The wave-length of the light in quartz is 355 \(\mathrm{nm}\) . (a) What is the index of refraction of quartz at this wavelength? (b) If this same light travels through air, what is its wavelength there?

Short Answer

Expert verified
(a) The index of refraction of quartz is approximately 1.546; (b) the wavelength in air is about 548.83 nm.

Step by step solution

01

Calculate the Speed of Light in Vacuum

The speed of light in a vacuum is known to be approximately \(c = 3.00 \times 10^8 \) meters per second. We'll use this constant to find the index of refraction of quartz.
02

Determine the Index of Refraction of Quartz

The index of refraction \(n\) is defined as the ratio of the speed of light in vacuum \(c\) to the speed of light in the medium \(v\). Therefore, for quartz, we have:\[ n = \frac{c}{v} = \frac{3.00 \times 10^8 \text{ m/s}}{1.94 \times 10^8 \text{ m/s}} \]Calculating this gives:\[ n \approx 1.546 \]
03

Understand the Relationship Between Wavelength and Index of Refraction

The wavelength of light in a medium is related to its wavelength in a vacuum by the index of refraction. Given that the wavelength of light in quartz is 355 nm, we need to find its wavelength in air.
04

Calculate the Wavelength in Air

Using the relation \( \lambda_{air} = n \times \lambda_{quartz} \), where:- \( \lambda_{air} \) is the wavelength in air,- \( n \) is the index of refraction of quartz, and- \( \lambda_{quartz} \) is the wavelength in quartz,we have:\[ \lambda_{air} = 1.546 \times 355 \text{ nm} \]This gives:\[ \lambda_{air} \approx 548.83 \text{ nm} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed of Light
The speed of light is a fundamental constant essential in physics and is often represented by the symbol \( c \). It measures how fast light travels through a vacuum 鈥� an environment without any matter. Typically, this speed is approximately \( 3.00 \times 10^8 \) meters per second. This speed is a critical part of many equations, including those involving the index of refraction of materials.

In different mediums, like air or quartz, light travels slower than in a vacuum due to interactions with the atoms in the medium. This difference in speed helps us understand the behavior of light as it passes through various substances, impacting phenomena like refraction and dispersion. Knowing the speed of light in different mediums is crucial in calculations to understand how light behaves in these environments.
Wavelength of Light
The wavelength of light is the distance between two consecutive peaks of a light wave. It's usually measured in nanometers (nm), where 1 nm equals a billionth of a meter. Different wavelengths correspond to different colors of light. For example, the visible spectrum ranges from about 400 nm (violet) to 700 nm (red).

In different mediums, the wavelength of light changes because its speed changes, even though the frequency remains constant. This relation is important for solving problems where light transitions between materials. To find the wavelength of light in a medium like quartz, you can use the index of refraction to compute how much the wavelength has shortened compared to its wavelength in vacuum.
Light in Quartz
Quartz is a transparent mineral known for its ability to refract light. When light enters quartz, it slows down compared to its speed in a vacuum. The speed of light within quartz is given as \( 1.94 \times 10^{8} \) meters per second in the exercise.

This slower speed results in a different index of refraction for quartz, calculated by dividing the speed of light in a vacuum by its speed in quartz. The index of refraction reveals how much light bends when entering the quartz. For the given speed, it's approximated to be 1.546. Moreover, the wavelength of light also shortens as it enters quartz, an aspect crucial when studying optical properties.
Light in Air
Air, like quartz, affects the speed and wavelength of light. However, the effect is much less pronounced than in denser materials like quartz or water.

When light travels through air, it moves almost as fast as it does in a vacuum, with its speed slightly reduced due to the air's molecular structure. A very thin medium, like air, has an index of refraction near 1, marking a minimal bending effect on the light path.
  • The index of refraction is used to compare the wavelengths of light in different environments.
  • For example, the wavelength of light entering air can be calculated using the herm index of refraction found earlier, aiding in determining its behavior post-transition.
Understanding these subtle changes helps in designing lenses and optical systems where precision is key.

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Most popular questions from this chapter

A horizontal, parallel-sided plate of glass having a refractive index of 1.52 is in contact with the surface of water in a tank. A ray coming from above in air makes an angle of incidence of \(35.0^{\circ}\) with the normal to the top surface of the glass. (a) What angle does the ray refracted into the water make with the normal to the surface? (b) What is the dependence of this angle on the refractive index of the glass?

A polarizer and an analyzer are oriented so that the maximum amount of light is transmitted. To what fraction of its maximum value is the intensity of the transmitted light reduced when the analyzer is rotated through (a) \(22.5^{\circ} ;(\mathrm{b}) 45.0^{\circ} ;(\mathrm{c}) 67.5^{\circ} ?\)

Bending Around Corners. Traveling particles do not bend around corners, but waves do. To see why, suppose that a plane wave front strikes the edge of a sharp object traveling perpendicular to the surface (Fig. 33.44\()\) . Use Huygens's principle to show that this wave will bend around the upper edge of the object. (Note: This effect, called diffraction, can easily be seen for water waves, but it also occurs for light, as you will see in Chapters 35 and 36 . However due to the very short wavelength of visible light, it is not so apparent in daily life.)

Light is incident along the normal on face \(A B\) of a glass prism of refractive index 1.52 , as shown in Fig. 33.41 . Find the largest value the angle \(\alpha\) can have without any light refracted out of the prism at face \(A C\) if (a) the prism is immersed in air and (b) the prism is immersed in water.

A beam of light has a wavelength of 650 nm in vacuum. (a) What is the speed of this light in a liquid whose index of refraction at this wavelength is 1.477 (b) What is the wavelength of these waves in the liquid?
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