Chapter 3, Problem 3.19QA

Interpretation Introduction

To calculate:

the wavelengths (λ) corresponding to the broadcast frequencies (ν) of the following radio stations: a. KRNU (Lincoln, NE), 90.3 MHz b. WBEU (Providence, RI), 95.5 MHz c. WYLD (New Orleans, LA), 98.5 MHz d. WAAF (Boston, MA), 107.3 MHZ

Answer to Problem 3.19QA

Solution:

The wavelengths (λ) corresponding to the broadcast frequencies (ν) of the following radio stations:

a) KRNU (Lincoln, NE), 90.3 MHz is 3.32 m

b) WBEU (Providence, RI), 95.5 MHz is 3.14 m

c) WYLD (New Orleans, LA), 98.5 MHz is 3.04 m

d) WAAF (Boston, MA), 107.3 MHz is 2.79 m

Explanation of Solution

1) Concept: Wavelength and frequency are related each other by speed of light. Speed of light is the product of wavelength and frequency of electromagnetic wave. Thus got the above formula to calculate the wavelength.

2) Formula:

$\mathrm{\lambda }=\frac{\mathrm{C}}{\mathrm{\nu }}$

3) Given information: The broadcast frequencies (ν) of the following radio stations:

i. KRNU (Lincoln, NE), 90.3 MHz

ii. WBEU (Providence, RI), 95.5 MHz

iii. WYLD (New Orleans, LA), 98.5 MHz

iv. WAAF (Boston, MA), 107.3 MHZ.

Conversion factor: $1\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}=1\times {10}^6\mathrm{ }\mathrm{H}\mathrm{z}$, $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}=2.998\times {10}^8\mathrm{ }\mathrm{m}\mathrm{ }\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{s}$

4) Calculations:

Conversion of all given frequency from MHz to Hz. Hz is also same as per second.

$\mathrm{a})\mathrm{ }90.3\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{ }\times \frac{1\times {10}^6\mathrm{ }\mathrm{H}\mathrm{z}}{1\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}}=9.03\times \mathrm{ }{10}^7\mathrm{ }\mathrm{H}\mathrm{z}$

$\mathrm{b})\mathrm{ }95.5\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{ }\times \frac{1\times {10}^6\mathrm{ }\mathrm{H}\mathrm{z}}{1\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}}=9.55\times \mathrm{ }{10}^7\mathrm{ }\mathrm{H}\mathrm{z}$

$\mathrm{c}) 98.5\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{ }\times \frac{1\times {10}^6\mathrm{ }\mathrm{H}\mathrm{z}}{1\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}}=9.85\times \mathrm{ }{10}^7\mathrm{ }\mathrm{H}\mathrm{z}$

$\mathrm{d})\mathrm{ }107.3\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{ }\times \frac{1\times {10}^6\mathrm{ }\mathrm{H}\mathrm{z}}{1\mathrm{ }\mathrm{M}\mathrm{H}\mathrm{z}}=1.073\times \mathrm{ }{10}^8\mathrm{ }\mathrm{H}\mathrm{z}$

Wavelength calculation:

$\mathrm{a})\mathrm{ }\mathrm{\lambda }=\frac{2.998\times {10}^8\frac{\mathrm{m}}{\mathrm{s}}}{9.03\times \mathrm{ }{10}^7\mathrm{ }\mathrm{H}\mathrm{z}}=3.32\mathrm{ }\mathrm{m}$

$\mathrm{b})\mathrm{ }\mathrm{\lambda }=\frac{2.998\times {10}^8\frac{\mathrm{m}}{\mathrm{s}}}{9.55\times \mathrm{ }{10}^7\mathrm{ }\mathrm{H}\mathrm{z}}=3.14\mathrm{ }\mathrm{m}$

$\mathrm{c})\mathrm{ }\mathrm{\lambda }=\frac{2.998\times {10}^8\frac{\mathrm{m}}{\mathrm{s}}}{9.85\times \mathrm{ }{10}^7\mathrm{ }\mathrm{H}\mathrm{z}}=3.04\mathrm{ }\mathrm{m}$

$\mathrm{d})\mathrm{ }\mathrm{\lambda }=\frac{2.998\times {10}^8\frac{\mathrm{m}}{\mathrm{s}}}{1.073\times \mathrm{ }{10}^8\mathrm{ }\mathrm{H}\mathrm{z}}=2.79\mathrm{ }\mathrm{m}$

Conclusion: The wavelengths of electromagnetic radiation is related to its frequency trough the speed of light. The corresponding to the broadcast frequencies of the listed stations are:

a) KRNU (Lincoln, NE), 90.3 MHz is 3.32 m

b) WBEU (Providence, RI), 95.5 MHz is 3.14 m

c) WYLD (New Orleans, LA), 98.5 MHz is 3.04 m

d) WAAF (Boston, MA), 107.3 MHz is 2.79 m