Question

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

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

Write down the given information

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

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

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: \(ν = \cfrac{3 \times 10^{8} \text{ m/s}}{0.21 \text{ m}}\)

Calculate the frequency

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

Convert the frequency to an appropriate unit

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

Choose the correct answer

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

Key concepts

Wavelength

When we talk about electromagnetic waves, the term "wavelength" comes up often. In simple terms, wavelength is the distance between two consecutive peaks of a wave.
It is usually denoted by the Greek letter lambda (\(\lambda\)). This quantity is an essential parameter for describing waves, as it determines properties such as the energy and type of electromagnetic radiation.
Wavelength is commonly measured in meters (m), centimeters (cm), or nanometers (nm).
In this particular problem, we're dealing with a wavelength of 21 cm. Since we often use meters in scientific equations, we'll need to convert 21 cm into meters, resulting in 0.21 m. This conversion is crucial because calculations involving the speed of light (\(c\)) usually require the wavelength and speed to be in compatible units, such as meters and meters per second.

Frequency Calculation

Within the realm of electromagnetic waves, frequency is equally as important as wavelength. Frequency, which is represented by the Greek letter nu (\(u\)), tells us how often a wave peak passes a particular point per second.
The unit for frequency is Hertz (Hz), where 1 Hz equals one cycle per second.
The frequency of a wave is inversely proportional to its wavelength according to the relationship \(c = \lambda u\), where \(c\) represents the speed of light.
In simple words, as the wavelength gets longer, the frequency gets lower and vice versa.To find the frequency when the wavelength is known, you rearrange the formula to \(u = \cfrac{c}{\lambda}\).
Here, we have the speed of light (\(3 \times 10^8\) m/s) and the wavelength (0.21 m) plugged into this rearranged equation to calculate the frequency. Performing this division gives us the frequency in Hertz, allowing us to identify the nature of the wave.

Speed of Light

The speed of light is a fundamental physical constant that is denoted by the letter \(c\). It represents the speed at which electromagnetic waves, such as light, travel in a vacuum.
The accepted value for the speed of light is \(3 \times 10^8\) meters per second (m/s).This constant value is essential in physics, linking many equations and concepts, such as the equation \(c = \lambda u\), which connects the speed of light with the wavelength and frequency of a wave.
It is because of this constant speed that we can express how incredibly fast light and other electromagnetic waves traverse space, allowing us to communicate this way.When calculating such problems, ensure that the units for the speed of light, wavelength, and frequency are compatible to yield accurate results. In our calculation, the converted wavelength into meters allowed us to seamlessly utilize the speed of light for frequency determination.

Unit Conversion

Unit conversion is an integral part of scientific calculations, particularly when dealing with different measurement systems for the same property.
Ensuring the correct units is key to accurate calculations.In our exercise, two unit conversions are specifically important:

• Converting centimeters into meters: Since 1 meter equals 100 centimeters, we convert 21 cm to meters by multiplying by the conversion factor (0.01), which gives us 0.21 meters.
• Converting Hertz into gigahertz: For our final frequency calculation, which comes out in Hertz, it is vital to express this in gigahertz. We use the conversion factor \(1 \text{ GHz} = 10^9 \text{ Hz}\). This conversion transforms \(1.42857 \times 10^9\) Hz into 1.42857 GHz.

Managing these conversions correctly ensures that all values align with the choices provided and make sense within the context of typical measurements for electromagnetic radiation.