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A light beam travels at \(1.94 \times 10^8\) m/s in quartz. The wavelength of the light in quartz is 355 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
The index of refraction of quartz is 1.546, and the wavelength of light in air is 548.83 nm.

Step by step solution

01

Understand the Problem

We want to find the index of refraction for quartz and the wavelength of light in air given the speed and wavelength of light in quartz.
02

Use the Formula for Index of Refraction

The index of refraction, \( n \), is given by the formula \( n = \frac{c}{v} \), where \( c \) is the speed of light in vacuum \((3.00 \times 10^8\ m/s)\), and \( v \) is the speed of light in the medium (quartz). Use these values to calculate \( n \).
03

Calculate Index of Refraction

Substitute the given values into the formula: \( n = \frac{3.00 \times 10^8}{1.94 \times 10^8} \). This gives \( n = 1.546 \).
04

Use the Formula for Wavelength in Different Medium

The wavelength of light changes when it moves between two mediums. In air, the wavelength \( \lambda_{air} \) is given by \( \lambda_{air} = \frac{\lambda_{quartz} \times n}{n_{air}} \). The index of refraction of air \( n_{air} \) is approximately 1.
05

Calculate Wavelength in Air

Substitute the values into the formula: \( \lambda_{air} = \frac{355 \times 1.546}{1} \). This simplifies to \( \lambda_{air} = 548.83 \ nm \).
06

Verification

Verify the units and calculations to ensure accuracy: 355 nm and the calculated refractive index together yield a wavelength in air that is longer than in quartz, which makes sense because light travels faster in air.

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Key Concepts

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

Speed of Light in Different Media
The speed of light is famously approximately \(3 imes 10^8\, ext{m/s}\) in a vacuum, which is often rounded to 300,000 kilometers per second. However, when light travels through any other optical medium, like glass, water, or quartz, its speed decreases due to interactions with the atoms and molecules in that medium.

This reduction in speed is quantified by the medium's index of refraction, denoted as \(n\). The index of refraction is a dimensionless number defined by \(n = \frac{c}{v}\), where \(c\) is the speed of light in vacuum, and \(v\) is the speed of light in the medium.

For example, when light travels through quartz, with a calculated speed of \(1.94 \times 10^8\, ext{m/s}\), it moves slower than it would in a vacuum. This decrease in speed is why we calculated an index of refraction for quartz of 1.546.
Wavelength and Its Change in Different Media
Light's wavelength is another critical aspect of its behavior in different media. Wavelength refers to the distance between two consecutive peaks of a wave, which we generally measure in nanometers.

When light travels from one medium to another — for example, from quartz into air — its wavelength changes. This is because the speed of light alters due to the different optical properties of the media.

Using the formula \(\lambda_{air} = \frac{\lambda_{quartz} \times n}{n_{air}}\), where \(\lambda_{quartz}\) is the initial wavelength in quartz, and \(n\) and \(n_{air}\) are the refractive indices of quartz and air respectively, we deduced that the wavelength of light in air is 548.83 nm, longer than in quartz, where it was 355 nm.
Optical Medium and Its Influence on Light
An optical medium is any material through which light can propagate. Common examples include air, water, glass, and quartz. Each optical medium affects the speed and behavior of light differently.

Two fundamental properties influenced by the optical medium are the speed of light and its wavelength. In general, the denser the medium, the slower the speed of light within it, and the shorter the wavelength becomes.

For instance, in our exercise, we observed that light traveling in quartz moved slower compared to air. As a direct consequence, its wavelength in air increased. This behavior is due to quartz having a higher index of refraction compared to air, indicating that it is denser and reduces the speed of light more significantly.

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Most popular questions from this chapter

The indexes of refraction for violet light \((\lambda = 400 \, \mathrm{nm})\) and red light \((\lambda = 700 \, \mathrm{nm})\) in diamond are 2.46 and 2.41, respectively. A ray of light traveling through air strikes the diamond surface at an angle of 53.5\(^\circ\) to the normal. Calculate the angular separation between these two colors of light in the refracted ray.

In a physics lab, light with wavelength 490 nm travels in air from a laser to a photocell in 17.0 ns. When a slab of glass 0.840 m thick is placed in the light beam, with the beam incident along the normal to the parallel faces of the slab, it takes the light 21.2 ns to travel from the laser to the photocell. What is the wavelength of the light in the glass?

A ray of light is incident on a plane surface separating two sheets of glass with refractive indexes 1.70 and 1.58. The angle of incidence is 62.0\(^\circ\), and the ray originates in the glass with \(n\) = 1.70. Compute the angle of refraction.

A ray of light is traveling in a glass cube that is totally immersed in water. You find that if the ray is incident on the glass-water interface at an angle to the normal larger than 48.7\(^\circ\), no light is refracted into the water. What is the refractive index of the glass?

A beam of polarized light passes through a polarizing filter. When the angle between the polarizing axis of the filter and the direction of polarization of the light is \(\theta\), the intensity of the emerging beam is \(I\). If you now want the intensity to be \(I/2\), what should be the angle (in terms of \(\theta\)) between the polarizing angle of the filter and the original direction of polarization of the light?

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