Problem 10 - Applied Physics

Question

Find the wavelength of a radio wave from an FM station broadcasting at a frequency of \(10 \overline{0}\) MHz.

Step-by-Step Solution

Answer

The wavelength of the radio wave is 3 meters.

Step 1: Understanding the Relationship

The wavelength (\(\lambda\)) of a wave is related to its speed (\(c\), which for electromagnetic waves is the speed of light) and its frequency (\(f\)) by the formula:\[\lambda = \frac{c}{f}\]where \(c = 3 \times 10^8 \text{ m/s}\) (speed of light). In this exercise, we want to find \(\lambda\).

Step 2: Convert Frequency to Hertz

The given frequency is 100 MHz. To use it in our formula, we need to convert it to Hertz:\[100 \text{ MHz} = 100 \times 10^6 \text{ Hz} = 1 \times 10^8 \text{ Hz}\]Now, we have \(f = 1 \times 10^8 \text{ Hz}\).

Step 3: Calculate the Wavelength

Substitute the values of \(c\) and \(f\) into the wavelength formula:\[\lambda = \frac{3 \times 10^8 \text{ m/s}}{1 \times 10^8 \text{ Hz}}\]This simplifies to:\[\lambda = 3 \text{ meters}\]Thus, the wavelength of the radio wave is 3 meters.

Key Concepts

Frequency to Wavelength Conversion

To discover the wavelength of a wave given its frequency, we must understand the direct relationship between these two properties. Electromagnetic waves, like radio waves, travel at the speed of light, approximately \(3 \times 10^8 \text{ m/s}\). The formula used for converting frequency to wavelength is:
\[\lambda = \frac{c}{f}\]where \(\lambda\) is the wavelength, \(c\) is the speed of light, and \(f\) is the frequency.
Let's break down the steps:

• **Identify the frequency**: In our exercise, it's given as \(100 \text{ MHz}\).
• **Convert Frequency to Hertz**: Remember that \(1 \text{ MHz} = 1 \times 10^6 \text{ Hz}\). This means \(100 \text{ MHz} = 100 \times 10^6 \text{ Hz} = 1 \times 10^8 \text{ Hz}\).
• **Substitute in the Formula**: Using the frequency in hertz and the constant speed of light.

Understanding this process helps visualize how a wave's frequency inversely affects its wavelength. Higher frequencies result in shorter wavelengths and vice versa.

Speed of Light

The speed of light is a fundamental constant in physics, denoted typically by the letter \(c\). Its value is close to \(3 \times 10^8 \text{ m/s}\). Let's dive a little deeper into why this constant is so vital:

• **Universal Speed Limit**: Light travels at this maximum speed in a vacuum. It's the ultimate speed at which energy, information, and matter can travel.
• **Role in Formulas**: In wave calculations, the speed of light allows us to connect frequency and wavelength seamlessly.
• **Electromagnetic Spectrum**: Light's speed ensures that electromagnetic waves, including radio waves, move steadily through space.

Whenever working with electromagnetic wave properties—whether calculating wavelengths or frequencies—understanding \(c\) is imperative because it assures consistency and accuracy.

Radio Waves

Radio waves are a part of the electromagnetic spectrum with some unique and fascinating properties. They are used extensively for communication purposes:

• **Wavelength Range**: Radio waves typically have wavelengths that range from about one millimeter to 100 kilometers, categorizing them as some of the longest waves in the spectrum.
• **Applications**: From broadcasting music (like our FM station example) to television and cellular communications, everyday technology relies heavily on radio waves.
• **Frequency Range**: Radio waves usually operate at frequencies from 30 Hz to 300 GHz.

Their ability to travel over long distances and penetrate through objects like walls makes radio waves indispensable in technology. That's why knowing how to calculate a radio wave's wavelength from its frequency is essential for designing effective communication systems.