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Chapter 33: Problem 4

Light with a frequency of \(5.80 \times 10^{14} \mathrm{~Hz}\) travels in a block of glass that has an index of refraction of \(1.52 .\) What is the wavelength of the light (a) in vacuum and (b) in the glass?

Short Answer

Expert verified
The wavelength of the light (a) in vacuum is approximately \(517 \mathrm{~nm}\) and (b) in the glass is approximately \(340 \mathrm{~nm}\).

Step by step solution

01

Calculate the wavelength in vacuum

First, to calculate the wavelength in vacuum, use the formula for the speed of light, \(c = f * \lambda\), where \(c = 3.00 \times 10^{8} \mathrm{~m/s}\) is the speed of light, \(f = 5.80 \times 10^{14} \mathrm{~Hz}\) is the frequency, and \(\lambda\) is the wavelength. Solve the equation for \(\lambda\) to obtain \(\lambda = c / f\).
02

Calculate the wavelength in the glass

The speed of light in a medium \(v = c / n\), where \(n = 1.52\) is the refractive index of the glass. Substituting \(v\) into the formula for the speed of light \(v = f * \lambda_g\) (where \(\lambda_g\) is the wavelength in glass), one gets \(c / n = f * \lambda_g\). Solve the equation for \(\lambda_g\) to obtain \(\lambda_g = c / (n * f)\).

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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 of nature, typically denoted by the symbol c. It represents the fastest speed at which energy, information, or matter can travel through the vacuum of space. In a vacuum, light travels at approximately \(3.00 \times 10^8 \text{ m/s}\). This velocity is crucial for a vast range of scientific calculations, including those concerning the wavelength of light, which we'll discuss next.

Understanding the interplay between the speed of light, wavelength, and frequency is pivotal in optics and physics. It's best described by the equation \(c = f \times \lambda\), where f stands for the frequency of light, and \lambda is the wavelength in a vacuum. When light enters different media, its speed decreases due to interaction with the material, though its frequency remains the same.
Refractive Index
The refractive index of a material, often symbolized as n, measures how much the speed of light is reduced inside that material compared to its speed in a vacuum. Essentially, it provides a way to quantify how much a light ray bends, or refracts, when transitioning from one medium to another.

The index of refraction is calculated as the ratio of the speed of light in a vacuum (\(c\)) to the speed of light in the material (\(v\)): \(n = \frac{c}{v}\). A refractive index greater than 1 indicates the light slows down inside the material. For example, the given exercise mentions glass with a refractive index of 1.52, implying light moves slower in glass than in a vacuum. As the refractive index changes, the wavelength of light changes accordingly, while the frequency remains fixed.
Frequency of Light
Frequency, represented by f, is the number of oscillations (or cycles) that a wave undergoes per unit of time, typically measured in hertz (Hz). For light, this is the number of times the electromagnetic wave oscillates through a certain point each second.

In the context of our exercise, the light frequency is given as \(5.80 \times 10^{14} \mathrm{ Hz}\). Frequency is an intrinsic property of light and remains constant regardless of the medium it travels through. When light enters a new medium and its speed changes, the frequency does not change, but its wavelength does. This change in wavelength is directly linked to the change in speed caused by the refractive index of the medium; a higher index will decrease the wavelength, as illustrated in the exercise's steps to determine the wavelength in glass.

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Most popular questions from this chapter

A beam of light has a wavelength of \(650 \mathrm{nm}\) in vacuum. (a) What is the speed of this light in a liquid whose index of refraction at this wavelength is \(1.47 ?\) (b) What is the wavelength of these waves in the liquid?

BIO Heart Sonogram. Physicians use high-frequency \((f=1-5 \mathrm{MHz})\) sound waves, called ultrasound, to image internal organs. The speed of these ultrasound waves is \(1480 \mathrm{~m} / \mathrm{s}\) in muscle and \(344 \mathrm{~m} / \mathrm{s}\) in air. We define the index of refraction of a material for sound waves to be the ratio of the speed of sound in air to the speed of sound in the material. Snell's law then applies to the refraction of sound waves. (a) At what angle from the normal does an ultrasound beam enter the heart if it leaves the lungs at an angle of \(9.73^{\circ}\) from the normal to the heart wall? (Assume that the speed of sound in the lungs is \(344 \mathrm{~m} / \mathrm{s}\).) (b) What is the critical angle for sound waves in air incident on muscle?

Unpolarized light with intensity \(I_{0}\) is incident on two polarizing filters. The axis of the first filter makes an angle of \(60.0^{\circ}\) with the vertical, and the axis of the second filter is horizontal. What is the intensity of the light after it has passed through the second filter?

A layer of liquid sits on top of the horizontal surface of a transparent solid. For a ray traveling in the solid and incident on the interface of the two materials, the critical angle is \(38.7^{\circ}\). (a) For a ray traveling in the solid and reflecting at the interface with the liquid, for what incident angle with respect to the normal is the reflected ray \(100 \%\) polarized? (b) What is the polarizing angle if the ray is traveling in the liquid?

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?
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