Chapter 7, Problem 7.1YT

Interpretation Introduction

Interpretation:

The wavelength of electromagnetic radiation used to produce a modulation in amplitude is to be calculated.

Concept Introduction:

The distance between the two consecutive crests of a wave is called its wavelength.

Electromagnetic radiation is the type of energy that surrounds us and may lie in the visible region.

Modification or variation in amplitude is called amplitude modulation.

The wavelength of a radiation is given as follows:

$\lambda =\frac{c}{\nu }$

Here, $\lambda$ is the wavelength of the radiation, $c$ is the speed of light and $\nu$ is the frequency of the radiation.

Answer to Problem 7.1YT

Solution: $3.5\times {10}^2$ m

Explanation of Solution

Given information: The frequency of the radio signal is 840KHz.

First, the relationship between kilohertz and hertz is as follows:

$\begin{array}{c}1\text{ KHz}=1\times {10}^3\text{Hz}\\ 840\text{ KHz}=\left(\frac{840\cancel{\text{KHz}}\times 1\times {10}^3\text{Hz}}{1\cancel{\text{KHz}}}\right)\\ =840\times {10}^3\text{Hz}\end{array}$

Also, the relationship between hertz and cycle per second is as follows:

$\begin{array}{c}1\text{Hz}=1\text{cycle}\text{per}\text{second}\\ \text{840}\times {\text{10}}^3\text{Hz}=\left(\frac{\text{840}\times {10}^3\cancel{\text{Hz}}\times 1\text{cycle}\text{per}\text{second}}{1\cancel{\text{Hz}}}\right)\\ =840\times {10}^3\text{cycles}\text{per}\text{second}\text{or}840\times {10}^3{\text{s}}^{-1}\end{array}$

The wavelength of the radiation is calculated as follows:

$\lambda =\frac{c}{\nu }$

Substitute $3\times {10}^8{\text{ ms}}^{-1}$ for $c$ and $840\times {10}^3{\text{ s}}^{-1}$ for the value of $\nu$ in the above expression as follows:

$\begin{array}{l}\lambda =\left(\frac{3\times {10}^8\text{m}\cancel{{\text{s}}^{-1}}}{840\times {10}^3\cancel{{\text{s}}^{-1}}}\right)\\ \text{}=3.5\times {10}^2\text{m}\end{array}$

Conclusion

The wavelength of the electromagnetic radiation is $3.5\times {10}^2\text{m}$.