Problem 7 - Physics

Question

A truck driver is broadcasting at a frequency of \(26.965 \mathrm{MHz}\) with a CB (citizen's band) radio. Determine the wavelength of the electromagnetic wave being used.

Step-by-Step Solution

Answer

The wavelength is approximately 11.12 meters.

Step 1: Understand the Relationship

The relationship between the speed of light, frequency, and wavelength is given by the formula \( c = f \lambda \), where \( c \) is the speed of light in vacuum (approximately \( 3 \times 10^8 \text{ m/s} \)), \( f \) is the frequency, and \( \lambda \) is the wavelength.

Step 2: Rearrange the Formula

To find the wavelength \( \lambda \), rearrange the formula: \( \lambda = \frac{c}{f} \).

Step 3: Plug in Known Values

Substitute the known values into the equation: \( c = 3 \times 10^8 \text{ m/s} \) and \( f = 26.965 \times 10^6 \text{ Hz} \).

Step 4: Calculate the Wavelength

Calculate: \[ \lambda = \frac{3 \times 10^8}{26.965 \times 10^6} = 11.12 \text{ meters} \].

Key Concepts

Electromagnetic Waves

Electromagnetic waves are fascinating forms of energy transfer that travel through space at the speed of light. They are made up of oscillating electric and magnetic fields, which makes them unique. These waves do not require a medium to travel through, meaning they can move through the vacuum of space.

Different types of electromagnetic waves make up the electromagnetic spectrum, each differing in wavelength and frequency. They include radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Among these, radio waves have the longest wavelengths and lowest frequencies.

Characteristics of electromagnetic waves:

• Travel at the speed of light (approximately \(3 \times 10^8\) m/s in vacuum).
• Can travel through a vacuum without requiring a medium.
• Include a wide range of wavelengths and frequencies, determining their type and application.

Understanding these properties helps us see how electromagnetic waves play a crucial role in technology, such as in citizen's band radios.

Frequency and Wavelength Relationship

The relationship between frequency and wavelength is central to understanding electromagnetic waves. They are inversely related, meaning as the frequency of a wave increases, its wavelength decreases, and vice versa. This relationship is described by the formula:
\[ c = f \lambda \]
Where:

• \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8\) m/s.
• \(f\) represents the frequency of the wave, measured in hertz (Hz).
• \(\lambda\) is the wavelength, measured in meters.

When solving problems like calculating the wavelength of a CB radio broadcast, this formula becomes essential. Rearranging it allows us to find the wavelength if we know the frequency:
\[ \lambda = \frac{c}{f} \]
This simple rearrangement provides a direct method to calculate the wavelength, highlighting the elegance of physics in describing everyday phenomena.

Citizen's Band Radio

Citizen's band (CB) radio is a popular form of short-distance communication. It operates in a specific range of frequencies assigned by governmental authorities for public use. In the United States, CB radios typically operate around 27 MHz.

CB radios are widely used among truckers, hobbyists, and others needing reliable communication over short distances. The typical range depends on various factors, including terrain and atmospheric conditions, but it usually spans around 1 to 5 miles on flat, open ground.

Key features of CB radio:

• Provides a convenient way for individuals to communicate without a license (in most countries).
• Operates on 40 channels within the 27 MHz band.
• Used in mobile settings, like trucks, or can be set up as base stations.

CB radios rely on the characteristics of electromagnetic waves, particularly radio waves, allowing users to send signals wirelessly. Understanding the frequency and wavelength of these signals is crucial for optimizing their use and improving communication effectiveness.