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Radio waves transmitted through empty space at the speed of light \(\left(v=c=3.00 \times 10^{8} \mathrm{m} / \mathrm{s}\right)\) by the Voyager spacecraft have a wavelength of \(0.120 \mathrm{m}\). What is their frequency?

Short Answer

Expert verified
The frequency of radio waves transmitted by the Voyager spacecraft is approximately \(2.50 \times 10^9 Hz\).

Step by step solution

01

Write down the wave equation

The wave equation is given by: v = λf Where v is the speed, λ is the wavelength, and f is the frequency.
02

Insert the given values

We know that the speed of radio waves through empty space is the speed of light (c), which is \(3.00 \times 10^8 m/s\), and the wavelength (λ) is \(0.120 m\). Insert these values into the wave equation: \(3.00 \times 10^8 m/s = (0.120 m)f\)
03

Solve for frequency (f)

To solve for the frequency (f), divide both sides of the equation by the wavelength (0.120 m): \(\frac{3.00 \times 10^8 m/s}{0.120 m} = f\)
04

Calculate the frequency

Now, perform the calculation to find the frequency: \(f = \frac{3.00 \times 10^8 m/s}{0.120 m} \approx 2.50 \times 10^9 Hz\) Thus, the frequency of the radio waves transmitted by the Voyager spacecraft is approximately \(2.50 \times 10^9 Hz\).

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Key Concepts

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

Radio Waves
Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. They play a crucial role in communication technologies. You can think of them as giant dancing waves, carrying information through space. These waves are emitted by various sources, such as broadcasting antennas and even celestial bodies like stars. As one of the cornerstones of wireless communication, radio waves make it possible for devices to transmit and receive data across distances without the need for direct physical connections.

Some important key points about radio waves include:
  • They have the longest wavelengths in the electromagnetic spectrum.
  • They can travel long distances, which makes them ideal for radio and TV broadcasts.
  • The speed at which radio waves travel through empty space is equal to that of light.
Understanding radio waves is key when learning about wireless communication, radar, and other widespread technologies.
Wavelength
Wavelength is a fascinating aspect of waves that helps define their characteristics. It refers to the distance between consecutive peaks or valleys in a wave. Imagine waving a rope up and down—one full up-and-down motion is akin to one wavelength.

Wavelength is expressed typically in meters (m) and is a critical factor in determining the wave's properties and behavior. The shorter the wavelength, the higher the frequency and energy. In the case of radio waves, a common feature is their long wavelength, allowing them to penetrate through obstacles like buildings and the atmosphere. This property makes them especially useful for broadcasting signals to multiple locations.
  • Shorter wavelengths correspond to higher frequencies and more energy.
  • Longer wavelengths can travel further and through various mediums.
  • Measured in meters within the electromagnetic spectrum context.
Wavelength is the bridge that helps us understand how different waves interact with their environment and with each other.
Frequency
The frequency of a wave is the number of complete wave cycles that pass a given point in one second. Frequency is measured in hertz (Hz), and it essentially dictates how fast the wave oscillates. To visualize this, imagine standing in the ocean: the number of waves that hit you per second would be the wave's frequency.

A higher frequency indicates more wave cycles are happening in a shorter period, and this generally translates to more energy. When dealing with radio waves, higher frequencies allow for higher data transmission rates, which is why different electromagnetic waves are utilized for different types of communication and broadcasting.
  • Measured in hertz (Hz), which corresponds to one cycle per second.
  • Related to the energy of the wave—higher frequency means higher energy.
  • Used in different contexts, such as radio, to determine signal strength and clarity.
Grasping the concept of frequency is crucial to understanding why certain waves are chosen for specific technological applications.
Speed of Light
The speed of light is a fundamental constant in physics that represents how fast light travels in a vacuum. Denoted by the letter 'c', the speed of light is approximately \[ c = 3.00 \times 10^8 \, \text{m/s} \]This constant is crucial in understanding electromagnetic waves, including radio waves, because these waves travel at this speed in a vacuum.

The speed of light affects how quickly information or energy can be transmitted over distances. It also plays a vital role in the theory of relativity, where it acts as a speed limit for all massless particles and waves. This means that as long as we're dealing with electromagnetic waves, like radio waves, they will travel at the speed of light when in empty space.
  • The speed of light is the ultimate speed limit in the universe.
  • All electromagnetic waves travel at this speed in a vacuum.
  • Essential for understanding the behavior and interaction of light and other electromagnetic waves.
Comprehending the speed of light gives context to the limitations and capabilities of wave-based technologies and expands the understanding of our physical universe.

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Most popular questions from this chapter

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of \(500.00 \mathrm{N}\) applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the \(3.00 \mathrm{m}\) of the string?

A string is fixed at both ends to supports 3.50 m apart and has a linear mass density of \(\mu=0.005 \mathrm{kg} / \mathrm{m}\) The string is under a tension of 90.00 N. A standing wave is produced on the string with six nodes and five antinodes. What are the wave speed, wavelength, frequency, and period of the standing wave?

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(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S-and P-waves, which travel at different speeds. If S- and P-waves travel at 4.00 and \(7.20 \mathrm{km} / \mathrm{s}\), respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined? (b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)

Two sinusoidal waves are moving through a medium in the positive \(x\) -direction, both having amplitudes of 7.00 \(\mathrm{cm},\) a wave number of \(k=3.00 \mathrm{m}^{-1},\) an angular frequency of \(\omega=2.50 \mathrm{s}^{-1}, \quad\) and a period of \(6.00 \mathrm{s},\) but one has a phase shift of an angle \(\phi=\frac{\pi}{12}\) rad. What is the height of the resultant wave at a time \(t=2.00 \mathrm{s}\) and a position \(x=0.53 \mathrm{m} ?\)

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