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Chapter 33: Problem 2

Project Seafarer was an ambitious program to construct an enormous antenna, buried underground on a site about 10000 \(\mathrm{km}^{2}\) in area. Its purpose was to transmit signals to submarines while they were deeply submerged. If the effective wavelength were \(1.0 \times 10^{4}\) Earth radii, what would be the (a) frequency and (b) period of the radiations emitted? Ordinarily, electromagnetic radiations do not penetrate very far into conductors such as seawater, and so normal signals cannot reach the submarines.

Short Answer

Expert verified
The frequency is 4.71 Hz and the period is 0.212 seconds.

Step by step solution

01

Understanding the Wavelength

The problem gives the effective wavelength of the radiation emitted by Project Seafarer as \(1.0 \times 10^{4}\) Earth radii. The radius of Earth is approximately \(6.371 \times 10^{6}\) meters. Therefore, first, we calculate the effective wavelength in meters:\[ \lambda = 1.0 \times 10^{4} \times 6.371 \times 10^{6} \text{ m} = 6.371 \times 10^{10} \text{ m} \]
02

Calculating the Frequency

The frequency \( f \) of a wave is related to its speed \( v \) and wavelength \( \lambda \) by the formula:\[ f = \frac{v}{\lambda} \]For electromagnetic waves, the speed \( v \) is the speed of light, \( c = 3.0 \times 10^{8} \text{ m/s}\). Therefore,\[ f = \frac{3.0 \times 10^{8} \text{ m/s}}{6.371 \times 10^{10} \text{ m}} \approx 4.71 \text{ Hz} \]
03

Calculating the Period

The period \( T \) of the wave is the reciprocal of the frequency:\[ T = \frac{1}{f} \]Using the frequency from Step 2:\[ T = \frac{1}{4.71 \text{ Hz}} \approx 0.212 \text{ s} \]

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Key Concepts

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

Wavelength calculation
To calculate the wavelength, it's essential to understand the relationship between physical dimensions and wave properties. In the exercise, the effective wavelength is given as a multiple of Earth's radius. The Earth's radius is approximately \(6.371 \times 10^{6}\) meters. By multiplying this value by \(1.0 \times 10^{4}\), we obtain the effective wavelength of the electromagnetic wave.
  • Effective wavelength: \(\lambda = 1.0 \times 10^{4} \times 6.371 \times 10^{6} \text{ m} = 6.371 \times 10^{10} \text{ m} \)
This calculation shows the enormous scale of the wavelength used in Project Seafarer, emphasizing the challenge of reaching deeply submerged submarines. Remember, wavelength is a measure of the distance over which the wave's shape repeats, and electromagnetic waves in this case have been adapted to optimize communication underwater.
Frequency of a wave
Frequency is a fundamental concept in wave mechanics, describing how often the wave's crests pass a fixed point per unit of time. It's calculated through its relationship with the speed of the wave and its wavelength.For electromagnetic waves like those used in Project Seafarer, the speed is the speed of light, approximately \(3.0 \times 10^{8} \text{ m/s}\). The formula for frequency \( f \) is:\[ f = \frac{v}{\lambda} \]Plugging in the values:
  • Speed of light \( v = 3.0 \times 10^{8} \text{ m/s}\)
  • Wavelength \( \lambda = 6.371 \times 10^{10} \text{ m} \)
  • Frequency \( f \approx 4.71 \text{ Hz} \)
This low frequency is typical for long wavelengths, which are necessary for penetrating materials like seawater. Understanding frequency helps in comprehending how often signals can be sent and received at these long wavelengths.
Period of a wave
The period of a wave is the duration for one complete cycle of the wave to pass a given point. It is the inverse of the wave's frequency. A simple formula to remember is:\[ T = \frac{1}{f} \]Where:
  • \( T \) is the period
  • \( f \) is the frequency
Using the calculated frequency of approximately \(4.71 \text{ Hz}\), we find the period:
  • \( T \approx \frac{1}{4.71} \approx 0.212 \text{ seconds} \)
Wave period is crucial in timing the transmissions correctly and synchronizing communication. Understanding the period is vital, especially in complex systems like submarine communications, where timing and synchronization can be pivotal to the successful transmission of information.

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Most popular questions from this chapter

What is the intensity of a traveling plane electromagnetic wave if \(B_{m}\) is \(1.0 \times 10^{-4} \mathrm{~T} ?\)

A beam of partially polarized light can be considered to be a mixture of polarized and unpolarized light. Suppose we send such a beam through a polarizing filter and then rotate the filter through \(360^{\circ}\) while keeping it perpendicular to the beam. If the transmitted intensity varies by a factor of \(5.0\) during the rotation, what fraction of the intensity of the original beam is associated with the beam's polarized light?

An electromagnetic wave is traveling in the negative direction of a \(y\) axis. At a particular position and time, the electric field is directed along the positive direction of the \(z\) axis and has a magnitude of \(100 \mathrm{~V} / \mathrm{m}\). What are the (a) magnitude and (b) direction of the corresponding magnetic field?

Light that is traveling in water (with an index of refraction of \(1.33\) ) is incident on a plate of glass (with index of refraction 1.53). At what angle of incidence does the reflected light end up fully polarized?

In about A.D. 150, Claudius Ptolemy gave the following measured values for the angle of incidence \(\theta_{1}\) and the angle of refraction \(\theta_{2}\) for a light beam passing from air to water: $$\begin{array}{cccl} \theta_{1} & \theta_{2} & \theta_{1} & \theta_{2} \\ \hline 10^{\circ} & 8^{\circ} & 50^{\circ} & 35^{\circ} \\ 20^{\circ} & 15^{\circ} 30^{\prime} & 60^{\circ} & 40^{\circ} 30^{\prime} \\ 30^{\circ} & 22^{\circ} 30^{\prime} & 70^{\circ} & 45^{\circ} 30^{\prime} \\ 40^{\circ} & 29^{\circ} & 80^{\circ} & 50^{\circ} \\ \hline \end{array}$$ Assuming these data are consistent with the law of refraction, use them to find the index of refraction of water. These data are interesting as perhaps the oldest recorded physical measurements.
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