Chapter 5, Problem 5.1P

Interpretation Introduction

To determine:

The wavelength of FM radio and medical X-ray in meters.

Answer to Problem 5.1P

Solution:

The frequency of FM radio and medical X-ray are $\text{2}\text{.93 m}$ and $\text{3}{\text{.14×10}}^{\text{-10}}\text{ m}$ respectively.

Explanation of Solution

Given:

Frequency of an FM radio wave = $\text{102}\text{.5 MHz}$

Frequency of medical X-ray = $\text{9}{\text{.55×10}}^{\text{17}}\text{ Hz}$

Wavelength:
Wavelength of any wave is the distance between consecutive peaks or troughs. It is denoted by $\text{λ}$. The unit of wavelength is meter, $\text{cm}$ , $\text{nm}$ , $\text{mm}$ etc.

Frequency:
Frequency of any wave is the number of oscillations per second which implies the number of oscillation in one second. It is denoted by $\nu$. The unit of frequency is per second. Its SI unit is $\text{Hz}$ , which is equal to ${\text{s}}^{-1}$.

Formula used:

  $\text{ν = }\frac{\text{c}}{\text{λ}}$

Here,
$\nu$ = frequency
$\lambda$ = wavelength
$\text{c}$ = speed of light ( $\text{2}\text{.998 }\times {\text{ 10}}^{\text{8}}\text{ m/s}$ )

Calculation:

First change the unit of frequency from $\text{MHz}$ to $\text{Hz}$

:

  $\text{102}\text{.5 MHz×}\frac{{\text{10}}^{\text{6}}\text{ Hz}}{\text{1 MHz}}\text{=102}{\text{.5×10}}^{\text{6}}\text{ Hz or/s}$

To calculate the wavelength uses the following expression:

  $\begin{array}{l}\text{ν = }\frac{\text{c}}{\text{λ}}\\ \text{λ=}\frac{\text{c}}{\text{v}}\\ \text{λ= }\frac{\text{2}{\text{.998 × 10}}^{\text{8}}\text{ m/s}}{\text{102}{\text{.5 ×10}}^{\text{6}}{\text{ s}}^{\text{-1}}}\\ \text{ =2}\text{.93 m}\end{array}$

and,
  $\begin{array}{l}\text{λ=}\frac{\text{c}}{\text{v}}\\ \text{λ= }\frac{\text{2}{\text{.998 × 10}}^{\text{8}}\text{ m/s}}{\text{9}{\text{.55×10}}^{\text{17}}{\text{ s}}^{\text{-1}}}\\ \text{ =3}{\text{.14×10}}^{\text{-10}}\text{ m}\end{array}$

Conclusion

Therefore,the frequency of FM radio and medical X-ray are $\text{2}\text{.93 m}$ and $\text{3}{\text{.14×10}}^{\text{-10}}\text{ m}$ respectively.