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A light beam travels at 1.94×108 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=cv, where c is the speed of light in vacuum (3.00×108 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=3.00×1081.94×108. 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 λair is given by λair=λquartz×nnair. The index of refraction of air nair is approximately 1.
05

Calculate Wavelength in Air

Substitute the values into the formula: λair=355×1.5461. This simplifies to λair=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 3imes108extm/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=cv, 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 1.94×108extm/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 λair=λquartz×nnair, where λquartz is the initial wavelength in quartz, and n and nair 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

Optical fibers are constructed with a cylindrical core surrounded by a sheath of cladding material. Common materials used are pure silica n2=1.4502 for the cladding and silica doped with germanium n1=1.4652 for the core. (a) What is the critical angle θcrit for light traveling in the core and reflecting at the interface with the cladding material? (b) The numerical aperture (NA) is defined as the angle of incidence thetai at the flat end of the cable for which light is incident on the core-cladding interface at angle θcrit (Fig. P33.46). Show that sin θi =n12n22 . (c) What is the value of θi for n1 = 1.465 and n2 = 1.450?

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(a) A tank containing methanol has walls 2.50 cm thick made of glass of refractive index 1.550. Light from the outside air strikes the glass at a 41.3 angle with the normal to the glass. Find the angle the light makes with the normal in the methanol. (b) The tank is emptied and refilled with an unknown liquid. If light incident at the same angle as in part (a) enters the liquid in the tank at an angle of 20.2 from the normal, what is the refractive index of the unknown liquid?

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?

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