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Chapter 29: Problem 65

A friend buys a used pickup truck that comes with a CB radio. However, the antenna is broken off, and your friend asks you to help make one out of a steel rod that he will affix to the rear bumper. You know that the CB channel frequency is \(27.3 \mathrm{MHz}\) and that the antenna must be a quarter- wavelength long. How long should you make the rod?

Short Answer

Expert verified
The length of the rod should be approximately 2.75 meters.

Step by step solution

01

Understand the Problem

First understand what the problem is asking for. It requires a determination of the rod's length which in turn is determined by the CB channel's frequency and the concept of a quarter wavelength long antenna.
02

Use the speed, frequency, and wavelength equation

Next, utilize the formula \(c = λν\), where c is the speed of light (approximately \(3.00 x 10^8 ms^-1\)), λ is the wavelength, and ν (nu) is the frequency. We need to find λ, so rearrange the formula as \(λ = c /ν\) and substitute known values into the equation. The given frequency must be converted to Hz as the SI unit for frequency is seconds^-1 (or Hz), and 1MHz equals \(1 x 10^6 Hz\). Thus, the frequency is \(27.3 x 10^6 Hz\).
03

Calculate the Wavelength

Calculate the necessary wavelength by dividing c by the frequency (ν). This will yield the length of a full wave antenna.
04

Calculate the Quarter Wavelength

To get the length of the quarter wavelength long (the length of the rod), divide the full wavelength by 4 because the antenna must be a quarter-wavelength long.

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