Problem 55 - College Physics

Question

\(\bullet\bullet\) The AM frequencies on a radio dial range from \(550 \mathrm{kHz}\) to \(1600 \mathrm{kHz},\) and the FM frequencies range from \(88.0 \mathrm{MHz}\) to \(108 \mathrm{MHz}\). All of these radio waves travel at a speed of \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\) (speed of light). (a) Compared with the FM frequencies, the AM frequencies have (1) longer, (2) the same, or (3) shorter wavelengths. Why? (b) What are the wavelength ranges of the AM band and the FM band?

Step-by-Step Solution

Answer

(a) AM has longer wavelengths than FM. (b) AM: 187.5 m to 545.45 m, FM: 2.78 m to 3.41 m.

Step 1: Formula for Wavelength

The wavelength \( \lambda \) of a wave is given by the formula \( \lambda = \frac{c}{f} \), where \( c \) is the speed of light (\( 3.00 \times 10^8 \ m/s \)) and \( f \) is the frequency of the wave.

Step 2: Relationship between Frequency and Wavelength

Given that the speed of light is constant, the wavelength is inversely proportional to the frequency. Therefore, a higher frequency \( f \) results in a shorter wavelength \( \lambda \).

Step 3: Compare AM and FM Frequencies

The AM frequency range is from \( 550 \ \text{kHz} \) to \( 1600 \ \text{kHz} \), which is lower than the FM frequency range of \( 88.0 \ \text{MHz} \) to \( 108 \ \text{MHz} \). Since AM frequencies are lower, their wavelengths are longer than those of FM frequencies.

Step 4: Convert Frequencies for AM Band

Convert the AM frequencies from kilohertz to hertz:- Lower AM frequency: \( 550 \ \text{kHz} = 550 \times 10^3 \ \text{Hz} \)- Upper AM frequency: \( 1600 \ \text{kHz} = 1600 \times 10^3 \ \text{Hz} \).

Step 5: Calculate AM Wavelength Range

Use \( \lambda = \frac{c}{f} \) to calculate:- Lower AM wavelength: \( \lambda = \frac{3.00 \times 10^8}{550 \times 10^3} \approx 545.45 \ \text{m} \).- Upper AM wavelength: \( \lambda = \frac{3.00 \times 10^8}{1600 \times 10^3} \approx 187.5 \ \text{m} \).Thus, the AM wavelength range is approximately from \( 187.5 \ \text{m} \) to \( 545.45 \ \text{m} \).

Step 6: Convert Frequencies for FM Band

Convert the FM frequencies from megahertz to hertz:- Lower FM frequency: \( 88.0 \ \text{MHz} = 88.0 \times 10^6 \ \text{Hz} \)- Upper FM frequency: \( 108 \ \text{MHz} = 108 \times 10^6 \ \text{Hz} \).

Step 7: Calculate FM Wavelength Range

Use \( \lambda = \frac{c}{f} \) to calculate:- Lower FM wavelength: \( \lambda = \frac{3.00 \times 10^8}{108 \times 10^6} \approx 2.78 \ \text{m} \).- Upper FM wavelength: \( \lambda = \frac{3.00 \times 10^8}{88 \times 10^6} \approx 3.41 \ \text{m} \).Thus, the FM wavelength range is approximately from \( 2.78 \ \text{m} \) to \( 3.41 \ \text{m} \).

Key Concepts

Radio Waves: The Basics

Radio waves are a type of electromagnetic radiation. They have the longest wavelengths in the electromagnetic spectrum, ranging from about one millimeter to more than 100 kilometers. Used primarily for communication, radio waves can be emitted by everyday household devices like radios, cell phones, and televisions, as well as the natural world like in lightning.

Radio waves have a wide range of applications:

• Television and radio broadcasting
• Cell phone communication
• Satellite communications
• Wireless internet signals

Compared to other types of electromagnetic radiation (like visible light or X-rays), radio waves have much lower frequencies and subsequently longer wavelengths. These lengthy wavelengths allow radio waves to travel long distances and penetrate through buildings and other obstacles with ease. Understanding the properties of radio waves helps us comprehend how communication technologies work and how various devices are interconnected in our world today.

Frequency-Wavelength Relationship

In the realm of waves, there is an important relationship between frequency and wavelength, especially for electromagnetic waves like radio waves. This relationship is defined by the equation:
\[\lambda = \frac{c}{f}\]where \( \lambda \) is the wavelength, \( c \) is the speed of light (approximately \( 3.00 \times 10^8 \ m/s \)), and \( f \) is the frequency.

Key aspects of this relationship include:

• Wavelength is inversely proportional to frequency.
• A higher frequency means a shorter wavelength.
• A lower frequency results in a longer wavelength.

When applying this equation to radio waves, understanding that the speed of light is constant helps us calculate one parameter if the other is known. For example, using AM and FM radio frequencies, we can find out the range of wavelengths that these waves span.

Understanding AM and FM Frequencies

AM and FM stand for Amplitude Modulation and Frequency Modulation, respectively. They represent two types of radio waves used for broadcasting.

**AM Frequencies:** - AM radio frequencies range from 550 kHz to 1600 kHz. - These frequencies are relatively low compared to FM frequencies, so they have longer wavelengths. - The wavelength range for AM frequencies is approximately 187.5 m to 545.45 m. This allows AM waves to cover longer distances, often clear reception in rural or remote areas.

**FM Frequencies:** - FM radio frequencies range from 88.0 MHz to 108 MHz. - These higher frequencies mean FM waves have shorter wavelengths, ranging approximately from 2.78 m to 3.41 m. - FM radio often offers better sound quality with less static for listeners within the local broadcast area. Overall, the difference between AM and FM reflects not just the type of modulation, but also the ways in which wave frequency and wavelength impact transmission quality and distance. Understanding these concepts is essential for appreciating everyday broadcasts and how they reach our radios.