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Chapter 15: Problem 2121

Astronomers have found that electromagnetic waves of wavelength $21 \mathrm{~cm}$ are continuously reaching the Earth's surface. Calculate the frequency of this radiation. $\left(\mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$ (A) \(14.28 \mathrm{GHz}\) (B) \(1.428 \mathrm{kHz}\) (C) \(1.428 \mathrm{MHz}\) (D) \(1.428 \mathrm{GHz}\)

Short Answer

Expert verified
The short answer to the given question is: The frequency of the electromagnetic radiation with a wavelength of \(21\) cm is approximately \(1.428 \text{ GHz}\) (Option (D)).

Step by step solution

01

Write down the given information

We are given: - Wavelength (\(λ\)) = \(21\) cm - Speed of light (\(c\)) = \(3 \times 10^{8}\) m/s
02

Convert the wavelength to meters

The given wavelength is in centimeters, so we need to convert it to meters before we can use it in the formula with the speed of light. 1 cm = 0.01 m So, \(λ = 21\) cm = \(0.21\) m
03

Use the formula to find the frequency

The formula relating the speed of light, wavelength, and frequency is: \(c = λν\) We can solve for frequency (\(ν\)) by dividing both sides by \(λ\): \(ν = \cfrac{c}{λ}\) Now we plug in the given values: \(ν = \cfrac{3 \times 10^{8} \text{ m/s}}{0.21 \text{ m}}\)
04

Calculate the frequency

Calculate the frequency: \(ν = \cfrac{3 \times 10^{8}}{0.21}\) \(ν ≈ 1.42857 \times 10^9 \text{ Hz}\)
05

Convert the frequency to an appropriate unit

The given answer choices are in GHz, kHz, and MHz. Let's convert the frequency to GHz: \(1 \text{ GHz} = 10^9 \text{ Hz}\) \(ν ≈ 1.42857 \times 10^9 \text{ Hz} = 1.42857 \text{ GHz}\)
06

Choose the correct answer

The calculated frequency is approximately 1.42857 GHz, which is closest to option (D): (D) 1.428 GHz

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