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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

Light Pipe. Light enters a solid pipe made of plastic having an index of refraction of 1.60 . The light travels parallel to the upper part of the pipe (Fig. E33.15). You want to cut the face \(A B\) so that all the light will reflect back into the pipe after it first strikes that face. (a) What is the largest that \(\beta\) can be if the pipe is in air? (b) If the pipe is immersed in water of refractive index 1.33 , what is the largest that \(\beta\) can be?

BIO Seeing Polarized Light. Some insect eyes have two types of cells that are sensitive to the plane of polarization of light. In a simple model, one cell type (type \(\mathrm{H}\) ) is sensitive to horizontally polarized light only, and the other cell type (type \(\mathrm{V}\) ) is sensitive to vertically polarized light only. To study the responses of these cells, researchers fix the insect in a normal, upright position so that one eye is illuminated by a light source. Then several experiments are carried out. First, light with a plane of polarization at \(45^{\circ}\) to the horizontal shines on the insect. Which statement is true about the two types of cells? (a) Both types detect this light. (b) Neither type detects this light. (c) Only type \(\mathrm{H}\) detects the light. (d) Only type \(\mathrm{V}\) detects the light.

A ray of light traveling in a block of glass \((n=1.52)\) is incident on the top surface at an angle of \(57.2^{\circ}\) with respect to the normal in the glass. If a layer of oil is placed on the top surface of the glass, the ray is totally reflected. What is the maximum possible index of refraction of the oil?

A parallel beam of unpolarized light in air is incident at an angle of \(54.5^{\circ}\) (with respect to the normal) on a plane glass surface. The reflected beam is completely linearly polarized. (a) What is the refractive index of the glass? (b) What is the angle of refraction of the transmitted beam?

Light is incident in air at an angle \(\theta_{a}\) (Fig. \(\mathbf{P 3 3 . 5 2}\) ) on the upper surface of a transparent plate, the surfaces of the plate being plane and parallel to each other. (a) Prove that \(\theta_{a}=\theta_{a}^{\prime}\). (b) Show that this is true for any number of different parallel plates. (c) Prove that the lateral displacement \(d\) of the emergent beam is given by the relationship $$ d=t \frac{\sin \left(\theta_{a}-\theta_{b}^{\prime}\right)}{\cos \theta^{\prime}} $$ where \(t\) is the thickness of the plate. (d) A ray of light is incident at an angle of \(66.0^{\circ}\) on one surface of a glass plate \(2.40 \mathrm{~cm}\) thick with an index of refraction of \(1.80 .\) The medium on either side of the plate is air. Find the lateral displacement between the incident and emergent rays.
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