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Chapter 23: Problem 31

The speed of light with a wavelength of \(656 \mathrm{nm}\) in heavy flint glass is \(1.82 \times 10^{8} \mathrm{~m} / \mathrm{s}\). What is the index of refraction of the glass at this wavelength?

Short Answer

Expert verified
The index of refraction of the glass is approximately 1.65.

Step by step solution

01

Understanding the Formula for Index of Refraction

The index of refraction, denoted as \( n \), is calculated using the formula \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum (approximately \(3.00 \times 10^8 \text{ m/s}\)) and \( v \) is the speed of light in the medium (in this case, the glass). We need to solve for \( n \) using these parameters.
02

Plug in the Values

We know from the problem that the speed of light in vacuum, \( c \), is approximately \(3.00 \times 10^8 \text{ m/s}\) and the speed of light in heavy flint glass at the given wavelength, \( v \), is \(1.82 \times 10^8 \text{ m/s}\). Substitute these values into the formula: \( n = \frac{3.00 \times 10^8}{1.82 \times 10^8} \).
03

Calculate the Index of Refraction

Now that we have our values in place within the formula \( n = \frac{3.00 \times 10^8}{1.82 \times 10^8} \), perform the division to find \( n \). This gives \( n \approx 1.65 \).

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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, commonly denoted as _c_, is a fundamental constant of nature. It represents the speed at which light travels through a vacuum. The value of the speed of light in a vacuum is approximately \(3.00 \times 10^8 \text{ m/s}\). This constant plays a crucial role in many areas of physics, such as optics, electromagnetism, and relativity.

  • It is the maximum speed at which all energy, matter, and information in the universe can travel.
  • The speed of light is essential for calculating the index of refraction in different media.
  • It is also used to measure astronomical distances, though in a slightly different form, known as a light-year.
Understanding the speed of light is a key component in studying how light interacts with various materials and in solving related physics problems.
Light Wavelength
Wavelength is a fundamental property of a wave and for light, it refers to the distance between consecutive peaks of the electromagnetic wave. Different wavelengths are perceived as different colors.

  • Light with different wavelengths is refracted differently by materials.
  • A smaller wavelength corresponds to higher frequency and more energy.
  • Wavelengths measured in nanometers (nm) are commonly used to discuss visible light.
The exercise mentions a wavelength of \(656 \text{ nm}\), which is a specific wavelength of light within the red part of the visible spectrum. Understanding this helps us determine how that specific light behaves when entering different materials, like heavy flint glass.
Heavy Flint Glass
Heavy flint glass is a type of glass that has a high refractive index. It is known for its ability to bend light significantly due to its composition, which often includes lead. This makes it particularly useful in optics, like in lenses or prisms, where controlling the path of light is crucial.

  • Heavy flint glass has a higher refractive index compared to other types of glass.
  • It is dense, leading to its extensive use in optical instruments.
  • Such glass causes more refraction because of its chemical composition.
In the context of this exercise, the heavy flint glass affects the speed of light traveling through it, which subsequently alters the light's wavelength and direction.
Physics Problem-Solving
Solving physics problems often involves logical thinking and stepping through the underlying principles methodically. In this case, calculating the index of refraction is a classic example.

  • Identify the known values: speed of light in vacuum and in the medium.
  • Use the right formula: the index of refraction is \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum and \( v \) is the speed in the medium.
  • Input the given values and perform the calculation.
The key to, solving such problems is understanding how each variable interacts with the others. Precision in calculation and clarity in understanding how light's speed and wavelength change in different mediums are crucial for solving similar physics questions.

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Most popular questions from this chapter

We can reasonably model a \(75 \mathrm{~W}\) incandescent lightbulb as a sphere \(6.0 \mathrm{~cm}\) in diameter. Typically, only about \(5 \%\) of the energy goes to visible light; the rest goes largely to nonvisible infrared radiation. (a) What is the visible light intensity (in \(\mathrm{W} / \mathrm{m}^{2}\) ) at the surface of the bulb? (b) What are the amplitudes of the electric and magnetic fields at this surface, for a sinusoidal wave with this intensity?

A small helium-neon laser emits red visible light with a power of \(3.20 \mathrm{~mW}\) in a beam that has a diameter of \(2.50 \mathrm{~mm}\). (a) What are the amplitudes of the electric and magnetic fields of the light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a \(1.00 \mathrm{~m}\) length of the beam?

There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from \(320 \mathrm{nm}\) to \(400 \mathrm{nm}\). It is not so harmful to the skin and is necessary for the production of vitamin D. UVB, with a wavelength between \(280 \mathrm{nm}\) and \(320 \mathrm{nm},\) is much more dangerous because it causes skin cancer. (a) Find the frequency ranges of UVA and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?

Write the wave equation for the electric field of an electromagnetic wave that is traveling in the \(+x\) direction with a wavelength of \(2.0 \mathrm{~m}\) and an amplitude of \(100 \mathrm{~N} / \mathrm{C}\). Give the wave equation in terms of its angular frequency and wave number.

Laser surgery. Very short pulses of high-intensity laser beams are used to repair detached portions of the retina of the eye. The brief pulses of energy absorbed by the retina weld the detached portion back into place. In one such procedure, a laser beam has a wavelength of \(810 \mathrm{nm}\) and delivers \(250 \mathrm{~mW}\) of power spread over a circular spot \(510 \mu \mathrm{m}\) in diameter. The vitreous humor (the transparent fluid that fills most of the eye) has an index of refraction of \(1.34 .\) (a) If the laser pulses are each \(1.50 \mathrm{~ms}\) long, how much energy is delivered to the retina with each pulse? (b) What average pressure does the pulse of the laser beam exert on the retina as it is fully absorbed by the circular spot? (c) What are the wavelength and frequency of the laser light inside the vitreous humor of the eye? (d) What are the maximum values of the electric and magnetic fields in the laser beam?
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