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Chapter 23: Problem 30

\(\bullet\) A light beam travels at \(1.94 \times 10^{8} \mathrm{m} / \mathrm{s}\) in quartz. The wavelength of the light in quartz is 355 \(\mathrm{nm}\) . (a) What is the index of refraction of quartz at this wavelength? (b) If this same light travels through air, what is its wavelength there?

Short Answer

Expert verified
(a) The index of refraction of quartz is 1.55. (b) The wavelength in air is approximately 550 nm.

Step by step solution

01

Determine the speed of light in vacuum

The speed of light in a vacuum is denoted by \( c \) and is approximately \( 3.00 \times 10^8 \, \text{m/s} \). This value is crucial for solving the rest of the problem.
02

Calculate the index of refraction of quartz

The index of refraction \( n \) can be calculated using the formula \( n = \frac{c}{v} \), where \( v \) is the speed of light in the medium. Plug in \( c = 3.00 \times 10^8 \, \text{m/s} \) and \( v = 1.94 \times 10^8 \, \text{m/s} \) to get: \[ n = \frac{3.00 \times 10^8}{1.94 \times 10^8} \approximately 1.55 \] So, the index of refraction of quartz at this wavelength is 1.55.
03

Relate the vacuum wavelength and speed

The speed of light \( c \) and its wavelength in a medium \( \lambda \) are related by \( c = f \lambda \), where \( f \) is the frequency. This relationship holds regardless of the medium. Therefore, in vacuum or air, \( \lambda_0 = \frac{c}{f} \).
04

Determine the wavelength in air

The frequency \( f \) of light does not change between different media. Since the speed and index in the new medium (air) are \( c \) and approximately 1 (for air), the wavelength \( \lambda_{\text{air}} \) can be found by \( \lambda_{\text{air}} = n \lambda_{\text{quartz}} \), where \( \lambda_{\text{quartz}} = 355 \, \text{nm} \). Hence, \[ \lambda_{\text{air}} = (1.55) \times (355 \, \text{nm}) \approx 550.25 \, \text{nm} \]

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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 in a vacuum is a cornerstone of physics and is denoted by the symbol \( c \). This speed is approximately \( 3.00 \times 10^8 \) meters per second (m/s).Understanding this constant is vital because it serves as the maximum speed at which all energy, matter, and information in the universe can travel.
  • This speed is constant in a vacuum, meaning it does not change regardless of the observer's frame of reference.
  • However, the speed of light changes when it passes through different media, such as water, glass, or quartz.
In any medium other than vacuum, light slows down due to interactions with the matter it passes through, making it crucial to understand how different substances affect its speed.
Wavelength in Medium
When light enters a medium other than vacuum, its speed decreases, but its frequency remains the same.This has an impact on its wavelength, which is the distance over which the wave's shape repeats.To understand wavelength in a new medium, it's essential to note:
  • Wavelength in a medium, denoted as \( \lambda \), shortens when light slows down.
  • The relationship between speed, wavelength, and frequency is given by \( v = f \lambda \).
  • For light traveling through a medium like quartz, the wavelength is calculated using its new speed \( v \), which, in this case, is \( 1.94 \times 10^8 \) m/s.
  • By keeping the frequency constant, the change in speed directly affects the wavelength.
Thus, when light goes from vacuum into quartz, the wavelength becomes shorter due to the change in speed.
Frequency of Light
The frequency of light refers to how often the peaks of the wave pass a point in a second and is measured in hertz (Hz).It is a fundamental property of the light wave determined by the source of the light and remains unchanged as light moves between different media.
  • The frequency \( f \) is related to both the speed \( v \) and wavelength \( \lambda \) by the equation \( v = f \lambda \).
  • In the process of moving from one medium to another, while the speed and wavelength may change, the frequency remains constant.
  • For example, as light transitions from quartz to air, its frequency does not vary, ensuring the color and type of light remain the same.
This concept helps us understand why the pitch of sound changes when waves change medium, unlike light, where only speed and wavelength adjust.
Light Propagation in Different Media
When light travels through different media, its speed and wavelength are altered while frequency remains constant.This alteration affects how we perceive the light, as well as how it interacts with the medium.
  • In a medium like quartz, light travels slower compared to air due to its higher index of refraction.
  • The formula \( n = \frac{c}{v} \) helps us determine the index of refraction, where \( n \) is the index, \( c \) the speed of light in vacuum, and \( v \) the speed of light in the medium.
  • The higher the index, the slower the light travels in that medium, affecting the wavelength but not the frequency.
  • For air, which has an index of approximately 1, light propagates very near to its speed in vacuum.
  • This property is crucial in designing lenses and other optical devices that depend on bending light to focus or disperse.
Understanding light propagation through different media allows us to better design and utilize optical technologies in everyday life.

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Most popular questions from this chapter

\(\bullet\) The electric field of a sinusoidal electromagnetic wave obeys the equation \(E=-(375 \mathrm{V} / \mathrm{m}) \sin [(5.97 \times\) \(10^{15} \operatorname{rad} / \mathrm{s} ) t+\left(1.99 \times 10^{7} \mathrm{rad} / \mathrm{m}\right) x ] .\) (a) What are the amplitudes of the electric and magnetic fields of this wave? (b) What are the frequency, wavelength, and period of the wave? Is this light visible to humans? (c) What is the speed of the wave?

\(\bullet\) 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?

\(\bullet\) At the floor of a room, the intensity of light from bright overhead lights is 8.00 \(\mathrm{W} / \mathrm{m}^{2} .\) Find the radiation pressure on a totally absorbing section of the floor.

\(\bullet\) Ultraviolet radiation. There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 \(\mathrm{nm}\) to 400 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?

\(\bullet\) \(\cdot\) High-energy cancer treatment. Scientists are working on a new technique to kill cancer cells by zapping them with ultrahigh-energy (in the range of \(10^{12}\) W) pulses of light that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk 5.0\(\mu \mathrm{m}\) in diameter, with the pulse lasting for 4.0 \(\mathrm{ns}\) with an average power of \(2.0 \times 10^{12} \mathrm{W}\) . We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in \(\mathrm{W} / \mathrm{m}^{2} )\) delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?
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