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Chapter 23: Problem 32

\(\bullet\) Light with a frequency of \(5.80 \times 10^{14}\) 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
In the vacuum, the wavelength is \(5.17 \times 10^{-7}\) m; in the glass, it is \(3.40 \times 10^{-7}\) m.

Step by step solution

01

Understand the Problem

We need to find two wavelengths, one in a vacuum and one in the glass. We are given the frequency of light and the refractive index of glass. We'll use the speed of light in a medium and the relationship between speed, frequency, and wavelength to find the solutions.
02

Calculate the Wavelength in Vacuum

The wavelength in a vacuum can be calculated using the formula \( \lambda = \frac{c}{f} \), where \( c = 3.00 \times 10^8 \) m/s (speed of light in a vacuum) and \( f = 5.80 \times 10^{14} \) Hz (frequency of light). Substitute the known values to get \( \lambda_{\text{vacuum}} = \frac{3.00 \times 10^8}{5.80 \times 10^{14}} = 5.17 \times 10^{-7} \) meters.
03

Calculate the Wavelength in Glass

The wavelength in the glass is found using the index of refraction \( n \) and the formula \( \lambda_{\text{glass}} = \frac{\lambda_{\text{vacuum}}}{n} \). Using \( n = 1.52 \), we substitute to find \( \lambda_{\text{glass}} = \frac{5.17 \times 10^{-7}}{1.52} = 3.40 \times 10^{-7} \) meters.

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Key Concepts

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

Index of Refraction
The index of refraction, often symbolized as "n," is a measure of how much a medium slows down light compared to its speed in a vacuum. It is defined by the equation \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium.
In simpler terms, the index of refraction tells us how much the light bends, or refracts, as it enters a new medium. The higher the index, the more the light slows down, resulting in a greater change in direction.
  • If the index of refraction is 1, the medium does not affect the speed of light (this is the case for a vacuum).
  • If the index is greater than 1, like in glass, the light travels slower in that medium than in a vacuum.
Speed of Light
The speed of light is a fundamental constant. In a vacuum, it travels at roughly \( 3.00 \times 10^{8} \) meters per second, which is considered the maximum speed at which all energy, matter, and information in the universe can travel.
When light enters a medium such as glass or water, it slows down. The speed of light in a medium is determined by its index of refraction, using the formula \( v = \frac{c}{n} \). This is crucial for understanding light behavior in different mediums.
  • The speed of light remains constant in a vacuum.
  • In mediums like glass or air, light travels at a speed determined by the material's refractive index.
Frequency of Light
The frequency of light, denoted as \( f \), is the number of waves that pass a fixed point in a given unit of time. In essence, it measures how "often" the wave oscillates per second. The unit for frequency is Hertz (Hz).
In the context of light, frequency is related to both the speed and wavelength of light by the formula \( c = \lambda f \), where \( \lambda \) is the wavelength. Importantly, the frequency of light does not change when it travels from one medium to another, unlike its speed or wavelength.
  • Frequency is determined by the source of the light, not the medium it travels through.
  • For a given light wave, if frequency is known, variations in speed or wavelength can be calculated.
Vacuum
A vacuum is a space devoid of matter, where light travels without interference at its maximum speed (\( 3.00 \times 10^{8} \) meters per second). It serves as the baseline for measuring the speed of light in other mediums.
In a vacuum, the behavior of light is not influenced by any material, thus refractive index is considered to be 1. This ideal condition allows us to use simple calculations to determine the fundamental properties of light, such as wavelength and frequency in a vacuum.
  • Light's speed in a vacuum is a universal constant.
  • As a reference, light's wavelength and speed calculations are often first considered in a vacuum to serve as a comparison to other media.
Glass
Glass is a common transparent material with an index of refraction greater than 1. It alters the speed and wavelength of light passing through it. For instance, the refractive index of glass is typically around 1.52, meaning light travels slower in glass compared to a vacuum.
In practical applications, this property of glass is used in lenses and optical instruments that rely on bending light to focus or disperse it (refraction). With known indices, precise calculations can be made to determine changes in light's wavelength as it passes through glass.
  • Glass is used in many applications due to its ability to change the path and properties of light.
  • Optical devices rely on glass to make precise alterations to the trajectory and focal points of light waves.

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Most popular questions from this chapter

\(\bullet$$\bullet\) 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?

\(\bullet$$\bullet\) (a) Light passes through three parallel slabs of different thicknesses and refractive indexes. The light is incident in the first slab and finally refracts into the third slab. Show that the middle slab has no effect on the final direction of the light. That is, show that the direction of the light in the third slab is the same as if the light had passed directly from the first slab into the third slab. (b) Generalize this result to a stack of \(N\) slabs. What determines the final direction of the light in the last slab?

\(\bullet\) Medical rays. Medical xays are taken with electromagnetic waves having a wavelength around 0.10 nm. What are the frequency, period, and wave number of such waves?

\(\bullet$$\bullet\) Using a fast-pulsed laser and electronic timing circuitry, you find that light travels 2.50 \(\mathrm{m}\) within a plastic rod in 11.5 \(\mathrm{ns} .\) What is the refractive index of the plastic?

If the entire slab (without the oil) is submerged in a fluid with an index of refraction of \(1.5,\) what will be the effect? A. The slab will appear to change color. B. Light striking the slab could be totally reflected. C. The slab will be very difficult to see. D. Light exiting the slab could be totally reflected.
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