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Chapter 33: Problem 43

A glass plate \(2.50 \mathrm{~mm}\) thick, with an index of refraction of \(1.40,\) is placed between a point source of light with wavelength \(540 \mathrm{nm}\) (in vacuum) and a screen. The distance from source to screen is \(1.80 \mathrm{~cm} .\) How many wavelengths are there between the source and the screen?

Short Answer

Expert verified
The total number of wavelengths between the source and the screen depends on the calculated values from steps 1 to 4. After performing these calculations, use the final result as your short answer here.

Step by step solution

01

Calculate Decrease in Wavelength in the Glass

Since the wavelength of light in a medium with index of refraction n is given by the wavelength in vacuum divided by n, we find the wavelength of light in the glass plate, \( \lambda_{glass} \), by dividing the given wavelength by the given index of refraction: \( \lambda_{glass} = \frac{540 \mathrm{~nm}}{1.40} \).
02

Calculate the Number of Wavelengths in the Glass

The number of wavelengths in the glass is found by dividing the thickness of the glass by the wavelength in the glass, \( n_{glass} = \frac{2.5 \mathrm{~mm}}{\lambda_{glass}} \).
03

Calculate the Number of Wavelengths in the Vacuum

The rest of the distance from source to screen is in vacuum. Subtract the thickness of the glass from the total distance to get this distance in vacuum. Then, divide this distance by the wavelength in vacuum (540 nm) to get the number of wavelengths in vacuum, \( n_{vacuum} = \frac{(1.80 \mathrm{~cm} - 2.5 \mathrm{~mm})}{540 \mathrm{~nm}} \).
04

Calculate Total Number of Wavelengths

Finally, add the number of wavelengths in the glass to the number of wavelengths in vacuum to get the total number of wavelengths between the source and the screen, \( n = n_{glass} + n_{vacuum} \).

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Key Concepts

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

Refractive Index
When light travels through different materials or mediums, its speed changes. This is because each material has different optical properties. This change in speed is described by the **refractive index**, a dimensionless number. It tells us how much light slows down as it enters a medium compared to its speed in a vacuum, where light travels fastest.

  • The refractive index, denoted by **n**, is calculated as the ratio of the speed of light in a vacuum to the speed of light in the medium.
  • In formula terms, it is expressed as: \( n = \frac{c}{v} \), where \( c \) is the speed of light in vacuum, and \( v \) is the speed of light in the medium.
  • The higher the refractive index, the more light bends when entering the material. For example, glass typically has a **refractive index** around 1.5, which means light travels slower in glass than in vacuum."

In the provided exercise, the glass has a refractive index of 1.40. This means light is slower in the glass, affecting its wavelength as it passes through the glass.
Wavelength
The distance between successive crests of a wave, especially points in a sound wave or electromagnetic wave, is known as **wavelength**. In the context of optics, we're focusing on the wavelength of light鈥攖he electromagnetic waves we can see.

  • Wavelength is typically measured in meters or nanometers (1 nanometer = 1 billionth of a meter).
  • Light's wavelength determines its color, with shorter wavelengths corresponding to blue light and longer wavelengths corresponding to red light.

In a vacuum, light travels with a wavelength of 540 nm as given in the exercise. However, when light enters a medium with a higher refractive index, its wavelength decreases. This is because the speed of light decreases while its frequency remains constant, so its *wavelength* contracts. In glass with a refractive index of 1.40, the wavelength becomes shorter, modifying how many wavelengths fit over a given distance.
Light Medium Transition
Light exhibits interesting behaviors when it transitions between different media. This transition affects both the speed and wavelength of the light, without altering the frequency. Understanding these changes is crucial in optics.

  • As light travels from air (or vacuum) into a medium like glass, it slows down and its wavelength shortens, which is dictated by the medium鈥檚 refractive index.
  • The relationship between the initial wavelength in vacuum \( \lambda_0 \) and the wavelength in the medium \( \lambda_{medium} \) is given by: \( \lambda_{medium} = \frac{\lambda_0}{n} \).

In the exercise, we see light moving from a vacuum to glass, with a change in both speed and wavelength. This phenomenon demonstrates the concept of "optical path length," where the actual path light travels could be longer due to the decreased wavelength in denser mediums like glass. By calculating how many wavelengths fit through both the glass and vacuum, we gain insights into how light behaves across different materials.

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Most popular questions from this chapter

Light traveling in air is incident on the surface of a block of plastic at an angle of \(62.7^{\circ}\) to the normal and is bent so that it makes a \(48.1^{\circ}\) angle with the normal in the plastic. Find the speed of light in the plastic.

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?

BIO Light Inside the Eye. The vitreous humor, a transparent, gelatinous fluid that fills most of the eyeball, has an index of refraction of \(1.34 .\) Visible light ranges in wavelength from \(380 \mathrm{nm}\) (violet) to \(750 \mathrm{nm}\) (red), as measured in air. This light travels through the vitreous humor and strikes the rods and cones at the surface of the retina. What are the ranges of (a) the wavelength, (b) the frequency, and (c) the speed of the light just as it approaches the retina within the vitreous humor?

Unpolarized light of intensity \(20.0 \mathrm{~W} / \mathrm{cm}^{2}\) is incident on two polarizing filters. The axis of the first filter is at an angle of \(25.0^{\circ}\) counterclockwise from the vertical (viewed in the direction the light is traveling), and the axis of the second filter is at \(62.0^{\circ}\) counterclockwise from the vertical. What is the intensity of the light after it has passed through the second polarizer?

A beam of light strikes a sheet of glass at an angle of \(57.0^{\circ}\) with the normal in air. You observe that red light makes an angle of \(38.1^{\circ}\) with the normal in the glass, while violet light makes a \(36.7^{\circ}\) angle. (a) What are the indexes of refraction of this glass for these colors of light? (b) What are the speeds of red and violet light in the glass?
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