Chapter 32: Problem 9
Consider electromagnetic waves propagating in air. (a) Determine the frequency of a wave with a wavelength of (i) 5.0 km, (ii) 5.0 mm, (iii) 5.0 nm. (b) What is the wavelength (in meters and nanometers) of (i) gamma rays of frequency 6.50 \(\times\) 10\(^{21}\) Hz and (ii) an AM station radio wave of frequency 590 kHz?
Short Answer
Step by step solution
Understanding the formula
Solving Problem (a) - Finding Frequency
Calculating Frequency for 5.0 km Wavelength
Calculating Frequency for 5.0 mm Wavelength
Calculating Frequency for 5.0 nm Wavelength
Solving Problem (b) - Finding Wavelength
Calculating Wavelength for Gamma Rays
Calculating Wavelength for AM Radio Wave
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wavelength
In an electromagnetic wave, the wavelength can vary widely depending on the type of wave. For example:
- Radio waves have long wavelengths, often measured in kilometers.
- Visible light has wavelengths in the nanometer range.
- Gamma rays have extremely short wavelengths, even smaller than visible light.
To calculate the wavelength when you know the frequency and speed of light, you can use: \(\lambda = \frac{c}{f}\), where \(c\) is the speed of light.
Frequency
A wave’s frequency is inversely related to its wavelength: as frequency increases, wavelength decreases, and vice versa. This relationship is captured in the equation: \(f = \frac{c}{\lambda}\).
Frequency can be very different across the electromagnetic spectrum:
- Radio waves have frequencies ranging from about 3 kHz to 300 GHz.
- Visible light frequencies are in the range of hundreds of THz.
- Gamma rays exhibit extremely high frequencies, in the range of billions of GHz.
Speed of Light
In a vacuum, light travels at this speed, and it remains remarkably constant except in different media where it may slow down. However, for most calculations in air, especially in introductory physics, the vacuum speed of light is used.
This speed is crucial for calculations involving electromagnetic waves, as demonstrated in the core equation: \(c = \lambda f\). This relationship allows us to determine either the frequency or wavelength when the other is known. The constancy of the speed of light also provides a basis for measuring astronomical distances and exploring the vast universe.
Gamma Rays
Gamma rays are emitted in nuclear reactions, such as those inside stars, and during radioactive decay processes. Their high energy makes them capable of penetrating most materials, which is why they are used in medical treatment to target cancer cells in radiotherapy.
Despite their utility, gamma rays can be harmful to living organisms as they can damage cellular structures and DNA.
Dealing with gamma rays often involves understanding their wavelength to assess their energy level: the shorter the wavelength, the more energy the gamma ray possesses. This information is crucial in both protecting against and utilizing gamma rays safely.
Radio Waves
These waves are primarily used in communication systems. They form the backbone of broadcasting technologies such as radio, television, and mobile phone signals.
One significant feature of radio waves is their ability to travel long distances and penetrate through the atmosphere, making them ideal for wireless communication.
Understanding the frequency of radio waves is essential for tuning broadcast devices to the correct channel or signal. Moreover, radio waves are crucial in many modern technologies like radar, satellite communications, and even astronomy observations.