Problem 27 - Chemistry The Science in Context

Question

FM radio stations broadcast at different frequencies. Calculate the wavelengths corresponding to the broadcast frequencies of the following college radio stations: (a) KCSU-FM (Fort Collins, CO), \(90.5 \mathrm{MHz} ;\) (b) WVUD (Newark, DE), \(91.3 \mathrm{MHz} ;\) (c) KUCR (Riverside, CA), \(88.3 \mathrm{MHz}\)

Step-by-Step Solution

Answer

Question: What are the wavelengths corresponding to the broadcast frequencies of the following college radio stations: (a) KCSU-FM (Fort Collins, CO): 90.5 MHz (b) WVUD (Newark, DE): 91.3 MHz (c) KUCR (Riverside, CA): 88.3 MHz Answer: The wavelengths corresponding to the broadcast frequencies of the college radio stations are: (a) KCSU-FM (Fort Collins, CO): λ ≈ 3.31 m (b) WVUD (Newark, DE): λ ≈ 3.29 m (c) KUCR (Riverside, CA): λ ≈ 3.40 m

Step 1: Write down given frequencies

The frequencies of the radio stations are given as: (a) KCSU-FM (Fort Collins, CO): \(90.5 \mathrm{MHz}\) (b) WVUD (Newark, DE): \(91.3 \mathrm{MHz}\) (c) KUCR (Riverside, CA): \(88.3 \mathrm{MHz}\)

Step 2: Convert MHz to Hz

Before we can calculate the wavelengths, we need to convert the frequencies from MHz to Hz by multiplying by \(10^6\): (a) \(90.5 \mathrm{MHz} = 90.5 \times 10^6 \mathrm{Hz}\) (b) \(91.3 \mathrm{MHz} = 91.3 \times 10^6 \mathrm{Hz}\) (c) \(88.3 \mathrm{MHz} = 88.3 \times 10^6 \mathrm{Hz}\)

Step 3: Calculate the wavelengths

Using the formula \(\lambda = \frac{c}{f}\) and the speed of light \(c = 3 \times 10^8 \mathrm{m/s}\), we can calculate the wavelengths of the radio stations: (a) \(\lambda_{\mathrm{KCSU-FM}} = \frac{3 \times 10^8 \mathrm{m/s}}{90.5 \times 10^6 \mathrm{Hz}} \approx 3.31 \mathrm{m}\) (b) \(\lambda_{\mathrm{WVUD}} = \frac{3 \times 10^8 \mathrm{m/s}}{91.3 \times 10^6 \mathrm{Hz}} \approx 3.29 \mathrm{m}\) (c) \(\lambda_{\mathrm{KUCR}} = \frac{3 \times 10^8 \mathrm{m/s}}{88.3 \times 10^6 \mathrm{Hz}} \approx 3.40 \mathrm{m}\)

Step 4: Present the final results

The wavelengths corresponding to the broadcast frequencies of the college radio stations are: (a) KCSU-FM (Fort Collins, CO): \(\lambda \approx 3.31 \mathrm{m}\) (b) WVUD (Newark, DE): \(\lambda \approx 3.29 \mathrm{m}\) (c) KUCR (Riverside, CA): \(\lambda \approx 3.40 \mathrm{m}\)

Key Concepts

Frequency Conversion

Frequency conversion is an essential step when dealing with radio waves, especially when the initial data is expressed in different units such as Megahertz (MHz).
To perform frequency conversion, particularly from MHz to Hertz (Hz), you need to remember that 1 MHz is equal to 1 million Hz (\(10^6\)).
Hence, the conversion from MHz to Hz involves multiplying the given frequency by \(10^6\).

• If the radio station broadcasts at 90.5 MHz, you calculate it in Hz as 90.5 \(\times 10^6\) Hz.
• Similarly, 91.3 MHz becomes 91.3 \(\times 10^6\) Hz, and 88.3 MHz turns into 88.3 \(\times 10^6\) Hz.

This conversion is mandatory because the subsequent calculations require frequencies in Hz in order to apply the wavelength formula accurately.

Speed of Light

The speed of light, denoted as \(c\), is a fundamental constant in physics integral for many calculations related to light and electromagnetic waves.
For calculations of radio wavelengths, we use the speed of light value of \(3 \times 10^8\) meters per second (\(\text{m/s}\)).
This constant helps connect the speed, frequency, and wavelength of waves. Because radio waves, like light, travel at the speed of light in a vacuum, this value is crucial.

• It ensures that we can calculate the wavelength by relating it to frequency.
• Additionally, understanding this value and its application solidifies our grasp of how electromagnetic waves propagate.

Using this constant in calculations isn't merely about memorization—it's about applying it correctly to solve for unknowns such as wavelengths.

Wavelength Formula

The wavelength formula illustrates the relationship between frequency, speed, and wavelength of a wave. The equation is \(\lambda = \frac{c}{f}\), where \(\lambda\) represents the wavelength in meters, \(c\) is the speed of light (\(3\times 10^8\) \(\text{m/s}\)), and \(f\) is the frequency in Hz.
To use this formula effectively:

• Ensure that the frequency is converted to Hz if not initially given in this unit.
• Substitute the values into the equation, dividing the speed of light by the frequency.
• Calculate to find the wavelength corresponding to the given frequency.

For example, using the University radio frequencies, after converting the frequency to Hz, we plug the values into the formula:
For KCSU-FM at 90.5 MHz (or \(90.5 \times 10^6\) Hz), the wavelength \(\lambda\) is \(\frac{3 \times 10^8}{90.5 \times 10^6} \approx 3.31\) meters.
This same method applies to the other frequencies, ensuring a consistent approach for calculating wavelengths.