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Chapter 3: Problem 38

A plane, harmonic. linearly polarized light wave has an electric field intensity given by $$E_{z}=E_{0} \cos \pi 10^{15}\left(t-\frac{x}{0.65 c}\right)$$ while traveling in a piecc of glass. Find (a) The frequency of the light. (b) Its wavelength. (c) The index of refraction of the glass.

Short Answer

Expert verified
(a) Frequency is \(5 \times 10^{14}\) Hz. (b) Wavelength is approximately \(3.9 \times 10^{-7}\) m. (c) Index of refraction is approximately 1.54.

Step by step solution

01

Identify the wave equation

The electric field intensity of the light wave is given by \(E_z = E_0 \cos \left(\pi \times 10^{15} \left(t - \frac{x}{0.65c}\right)\right)\). This equation helps in identifying the components needed for frequency, wavelength, and index of refraction.
02

Determine the angular frequency

The general form of a wave equation is \(E = E_0 \cos(\omega t - kx)\), where \(\omega\) is the angular frequency. From the given equation, \(\omega = \pi \times 10^{15}\) rad/s.
03

Calculate the frequency (a)

The frequency \(f\) is related to the angular frequency \(\omega\) by \(\omega = 2\pi f\). Solving for \(f\), we have: \[ f = \frac{\omega}{2\pi} = \frac{\pi \times 10^{15}}{2\pi} = 5 \times 10^{14} \text{ Hz} \].
04

Calculate the wavelength (b)

The wavelength \(\lambda\) in a medium is related to the speed of light \(v\) and frequency \(f\) by \(v = f\lambda\). The speed of the wave in the medium is \(0.65c\). Thus, \(\lambda = \frac{v}{f} = \frac{0.65c}{5 \times 10^{14}}\). Substituting \(c = 3 \times 10^8\) m/s, we get: \[ \lambda = \frac{0.65 \times 3 \times 10^8}{5 \times 10^{14}} \approx 3.9 \times 10^{-7} \text{ m} \].
05

Calculate the index of refraction (c)

The index of refraction \(n\) is given by \(n = \frac{c}{v}\), where \(v = 0.65c\) is the speed of light in glass. Therefore: \[ n = \frac{c}{0.65c} \approx \frac{1}{0.65} \approx 1.54 \].

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Key Concepts

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

Wave Equation
A wave equation is a mathematical description of a wave as it propagates through a medium. In optics, this is essential to understand how light waves behave. The given equation for electric field intensity is \( E_z = E_0 \cos \left(\pi \times 10^{15} \left(t - \frac{x}{0.65c}\right)\right) \). It indicates a wave moving in space and time. Key components of any wave equation include:
  • Amplitude \(E_0\): The maximum strength of the electric field. It represents the wave's height if you imagine it as a sine wave.
  • Angular frequency \(\omega\): This factor influences how quickly the wave oscillates over time.
  • Wave number \(k\): Related to the wavelength, helps understand the wave's spatial oscillations.
Understanding these components allows us to derive important properties like angular frequency, wavelength, and speed in a medium.
Angular Frequency
Angular frequency, denoted as \( \omega \), is a key concept in understanding wave behavior. It provides insight into how often a wave oscillates in radians per second. Angular frequency is crucial in determining the frequency of a wave.In the context of the exercise, the wave equation \( E_z = E_0 \cos \left(\pi \times 10^{15} \left(t - \frac{x}{0.65c}\right)\right) \) directly gives us \( \omega = \pi \times 10^{15} \) rad/s. This not only describes how rapidly the electric field oscillates but also aids in calculating the linear frequency.鈥�
Index of Refraction
The index of refraction \( n \) is a dimensionless number that measures how much a ray of light bends, or refracts, as it passes through a medium. It is calculated as the ratio of the speed of light in a vacuum \( c \) to the speed of light in the medium \( v \).From the exercise, the speed of light in glass is stated as \( 0.65c \). Thus, the index of refraction can be calculated as:\[ n = \frac{c}{0.65c} = \frac{1}{0.65} \approx 1.54 \]This tells us that light travels slower in glass than in a vacuum, causing refraction. A higher index indicates more bending.
Wavelength
Wavelength \( \lambda \) is the distance between successive crests of a wave, particularly in sound or electromagnetic waves. Understanding wavelength is crucial as it determines the wave鈥檚 energy and traits.In the exercise, we find the wavelength using the formula \( \lambda = \frac{v}{f} \). Given the speed in the medium \( v = 0.65c \) and frequency \( f = 5 \times 10^{14} \) Hz, we calculate the wavelength as:\[ \lambda = \frac{0.65 \times 3 \times 10^8}{5 \times 10^{14}} \approx 3.9 \times 10^{-7} \text{ m} \]This result shows the distance light travels in one cycle within the medium.
Frequency
Frequency \( f \) refers to the number of wave crests passing a point in one second. It is measured in hertz (Hz) and indicates how fast the wave oscillates.In this scenario, we find the frequency from the angular frequency using the relation \( \omega = 2\pi f \). With \( \omega = \pi \times 10^{15} \), we calculate:\[ f = \frac{\pi \times 10^{15}}{2\pi} = 5 \times 10^{14} \text{ Hz} \]This frequency tells us how many cycles the light wave completes per second, crucial for understanding light's interaction with materials.

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Most popular questions from this chapter

How many photons per second are emitted from a \(100-\mathrm{W}\) yellow lightbulb if we assume negligible thermal losses and a quasimonochromatic wavelength of \(550 \mathrm{nm}\) ? In actuality only about \(2.5 \%\) of the total dissipated power emerges as visible radiation in an ordinary 100 -W lamp.

A 550 -nm harmonic EM-wave whose electric field is in the direction is traveling in the \(y\) -direction in vacuum. (a) What is the frequency of the wave? (b) Determine both \(\omega\) and \(k\) for this wave. (c) If the electric field amplitude is \(600 \mathrm{V} / \mathrm{m},\) what is the amplitude of the magnetic field? (d) Write an expression for both \(E(t)\) and \(B(t)\) given that each is zero at \(x=0\) and \(t=0 .\) Put in all the appropriate units.

Determine the index of refraction of a medium if it is 10 reduce the speed of light by \(10 \%\) as compared to its speed in vacuum?

Consider a linearly polarized plane electromagnetic wave traveling in the \(+x\) -direction in free space having as its plane of vibration the \(x y\) -plane. Given that its frequency is \(10 \mathrm{MHz}\) and its amplitude is \(E_{0}=0.08 \mathrm{V} \mathrm{m}\) (a) Find the period and wavelength of the wave. (b) Write an expression for \(E(t)\) and \(B(t)\) (c) Find the flux density. \(\langle S\rangle\), of the wave.

A light beam with an irradiance of \(2.00 \times 10^{6} \mathrm{W} / \mathrm{m}^{2}\) impinges normally on a surface that reflects \(70.0 \%\) and absorbs \(30.0 \% .\) Compute the resulting radiation pressure on the surface.
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