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Chapter 24: Problem 9

In a certain UHF radio wave, the shortest distance between positions at which the electric and magnetic fields are zero is \(0.34 \mathrm{m} .\) Determine the frequency of this UHF radio wave.

Short Answer

Expert verified
The frequency is approximately \(4.41 \times 10^8 \text{ Hz}\."

Step by step solution

01

Determine the wavelength

The distance between the points where the electric and magnetic fields are zero corresponds to half a wavelength of the wave. Therefore, the wavelength \( \lambda \) can be determined by doubling the given distance: \( \lambda = 2 \times 0.34 \text{ m} = 0.68 \text{ m} \).
02

Use the speed of light formula

Use the formula for the speed of light: \( c = \lambda f \), where \( c \) is the speed of light (approximately \( 3 \times 10^8 \text{ m/s} \)), \( \lambda \) is the wavelength, and \( f \) is the frequency. Rearrange this formula to solve for frequency: \( f = \frac{c}{\lambda} \).
03

Calculate the frequency

Using the formula from Step 2, substitute the values: \( f = \frac{3 \times 10^8 \text{ m/s}}{0.68 \text{ m}} \). Performing the division gives \( f \approx 4.41 \times 10^8 \text{ Hz} \).

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Key Concepts

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

Wavelength Determination
To find the wavelength of any wave, it is crucial to understand the concept of how waves move. Wavelength is defined as the distance between two consecutive points in phase on a wave. For instance, from crest to crest or trough to trough.
In the problem you've been given, the distance between the points where the electric and magnetic fields are zero corresponds to half a wavelength. This is a key concept in wave mechanics, as it tells us that if you observe two zero points on a radio wave, they are actually half a wavelength apart. Thus, if we're given that distance as 0.34 meters, to find the full wavelength, we simply double it. Therefore, the wavelength for this UHF radio wave is:
  • Full Wavelength: \[ \lambda = 2 \times 0.34\text{ m} = 0.68\text{ m} \]
Understanding this principle helps in deciphering the wave's physical properties, crucial for tasks like frequency determination.
Speed of Light Formula
The speed of light is a fundamental constant essential in electromagnetic wave calculations. Its value is approximately \(3 \times 10^8 \text{ m/s}\). This value is used in combination with the wavelength to determine the frequency of a wave.
The relationship is governed by the formula \(c = \lambda f\), where:
  • \(c\) is the speed of light
  • \(\lambda\) is the wavelength
  • \(f\) is the frequency
To isolate the frequency, the formula can be rearranged to:
  • \( f = \frac{c}{\lambda} \)
This formula is pivotal in transforming a known light speed and measured wavelength into a frequency, allowing scientists and engineers to understand and utilize electromagnetic waves effectively in technology and communication.
Electromagnetic Wave Analysis
Electromagnetic waves are waves of electric and magnetic fields that travel through space carrying energy. They are essential in many areas, including communication technologies, such as UHF radio waves. Understanding these waves involves grasping both their electric and magnetic properties, as well as how they interact.
UHF (Ultra High Frequency) radio waves span from 300 MHz to 3 GHz in frequency. These waves are vital for modern communication, transmitting data over long distances swiftly. In analyzing an electromagnetic wave, the frequency鈥攄etermined by dividing the speed of light by the wavelength鈥攑lays a critical role in knowing the wave鈥檚 energy and potential applications.
  • This detailed analysis involves:
    • Identifying wavelength and corresponding frequency
    • Relating wave properties to their technological uses鈥攍ike broadcast signals
Therefore, by analyzing UHF radio waves, we better understand how they convey information, allowing seamless global communication and advanced electronics operations.

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Most popular questions from this chapter

An electromagnetic wave strikes a \(1.30-\mathrm{cm}^{2}\) section of wall perpendicularly. The rms value of the wave's magnetic field is determined to be \(6.80 \times 10^{-4}\) T. How long does it take for the wave to deliver \(1850 \mathrm{J}\) of energy to the wall?

A Magneto-optic Device. The Faraday effect is a magneto-optic phenomenon where the polarization of linearly polarized light rotates when the light passes through a magnetized material. You and your team are tasked with analyzing the results of an experiment designed to measure how much a certain thin magnetic film rotates the polarization of the light that passes through it. As the drawing shows, an unpolarized laser beam of average intensity \(\bar{S}_{0}=1000 \mathrm{W} / \mathrm{m}^{2}\) passes through a vertical polarizer. It then passes through a thin magnetic film that can be in one of three states: (i) unmagnetized, (ii) magnetized to the right (when looking from the laser), and (iii) magnetized to the left. It is assumed that the film does not absorb any of the light. The light then passes through another polarizer (called the analyzer) whose transmission axis is rotated \(7.50^{\circ}\) clockwise from the horizontal (when looking from the laser). (a) What is the intensity of the beam after it passes through the first polarizer? (b) In the case that the film is unmagnetized, the polarization will not rotate when it passes through the film. What should be the light intensity measured at the detector? (c) When the film is magnetized to the left, the intensity measured by the detector is \(I_{\mathrm{L}}=12.2 \mathrm{W} / \mathrm{m}^{2}\) and when magnetized to the right the intensity is \(I_{\mathrm{R}}=5.46 \mathrm{W} / \mathrm{m}^{2} .\) In each case, determine the direction (clockwise or counterclockwise when looking from the laser) and the magnitude of the angle through which the magnetic film rotates the polarization of the light from the vertical.

A politician holds a press conference that is televised live. The sound picked up by the microphone of a TV news network is broadcast via electromagnetic waves and heard by a television viewer. This vicwer is seated \(2.3 \mathrm{m}\) from his television set. A reporter at the press conference is located \(4.1 \mathrm{m}\) from the politician, and the sound of the words travels directly from the celebrity's mouth, through the air, and into the reporter's ears. The reporter hears the words exactly at the same instant that the television viewer hears them. Using a value of \(343 \mathrm{m} / \mathrm{s}\) for the speed of sound, determine the maximum distance between the television set and the politician. Ignore the small distance between the politician and the microphone. In addition, assume that the only delay between what the microphone picks up and the sound being emitted by the television set is that due to the travel time of the electromagnetic waves used by the network.

The average intensity of light emerging from a polarizing sheet is \(0.764 \mathrm{W} / \mathrm{m}^{2},\) and the average intensity of the horizontally polarized light incident on the sheet is \(0.883 \mathrm{W} / \mathrm{m}^{2}\). Determine the angle that the transmission axis of the polarizing sheet makes with the horizontal.

A laser emits a narrow beam of light. The radius of the beam is \(1.0 \times\) \(10^{-3} \mathrm{m},\) and the power is \(1.2 \times 10^{-3} \mathrm{W} .\) What is the intensity of the laser beam?
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