Question

Three FM radio stations covering the same geographical area broadcast at frequencies \(91.1,91.3,\) and \(91.5 \mathrm{MHz},\) respectively. What is the maximum allowable wavelength width of the band-pass filter in a radio receiver so that the FM station 91.3 can be played free of interference from FM 91.1 or FM 91.5? Use \(c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\), and calculate the wavelength to an uncertainty of \(1 \mathrm{~mm} .\)

Short answer

Solution: 1. Calculate the wavelengths of the three radio stations using the formula: Wavelength (λ) = speed of light (c) / frequency (f). 2. Compute the differences in wavelengths between FM 91.3 MHz and FM 91.1 MHz and between FM 91.3 MHz and FM 91.5 MHz. 3. Identify the minimum of these differences as the maximum allowable wavelength width of the band-pass filter. 4. Round the result to an uncertainty of 1 mm.

Step-by-step solution

Calculate Wavelength of Each Frequency

We are given three frequencies, and we want to find their corresponding wavelengths. We can do this using the formula: Wavelength (λ) = speed of light (c) / frequency (f) Use this formula to find the wavelengths of 91.1 MHz, 91.3 MHz, and 91.5 MHz stations.

Compute the Differences in Wavelength

After calculating the wavelengths, we will find the differences between the wavelengths of the FM stations (91.3 MHz compared to 91.1 MHz and 91.5 MHz). This will help us determine the maximum allowable wavelength width of the band-pass filter that can isolate FM 91.3 without interference.

Identify the Maximum Allowable Wavelength Width

Now that we have determined the differences in wavelengths, we will identify the maximum allowable wavelength width of the band-pass filter by finding the minimum of these differences.

Round the Wavelength with an Uncertainty of 1 mm

In this final step, round the maximum allowable wavelength width to an uncertainty of 1 mm. This will give us the answer for the maximum allowable wavelength width of the band-pass filter in a radio receiver to avoid interference.

Key concepts

FM Radio Frequencies

FM radio frequencies are used by radio stations to broadcast and transmit signals over the airwaves. Each station operates at a specific frequency so that their transmissions don't overlap or interfere with one another. FM frequencies are usually between 87.5 MHz and 108 MHz.

In this exercise, we're looking at three specific frequencies: 91.1 MHz, 91.3 MHz, and 91.5 MHz. These numbers are crucial because they determine the specific electromagnetic waves that carry the audio signals.

If two FM stations broadcast too closely in terms of frequency, you may experience interference, causing your radio to play a mix of both stations or static noise.
That's why understanding FM radio frequencies is important for setting up and using radio receivers effectively.

Wavelength Calculation

To find the wavelength of a radio wave, we use the formula:

\[ \text{Wavelength} (\lambda) = \frac{\text{speed of light} (c)}{\text{frequency} (f)} \]
The speed of light \(c\) is a constant, approximately \(3.00 \times 10^8 \text{ m/s}\). Using this formula, we can calculate the wavelength for any given frequency.

In our situation, we want to calculate wavelengths for frequencies of 91.1 MHz, 91.3 MHz, and 91.5 MHz. The term MHz means megahertz, and one MHz is equal to \(10^6\) Hz.

For example, to calculate the wavelength for the 91.3 MHz frequency, convert the frequency to Hertz:

• 91.3 MHz = 91.3 × \(10^6\) Hz

Then apply the formula:

• \( \lambda = \frac{3.00 \times 10^8 \text{ m/s}}{91.3 \times 10^6 \text{ Hz}} \)

Perform this calculation for each frequency to find their respective wavelengths. This provides us the necessary information about the bandwidth and helps in determining interference issues.

Wavelength Interference

Wavelength interference occurs when two or more radio waves overlap, which can lead to distorted signals. To avoid interference between radio stations, a radio receiver uses a band-pass filter. A band-pass filter allows a specific range of wavelengths to pass through, blocking other wavelengths from interfering.

After calculating the wavelengths of the FM stations, the next step is to calculate the difference between them. For instance, the difference between the wavelengths for 91.3 MHz and 91.1 MHz, and for 91.3 MHz and 91.5 MHz.

This difference tells us how close these signals are in terms of their wavelengths.
Knowing the differences helps determine the maximum allowable wavelength width for the 91.3 MHz station to play without interference from the other stations.

The band-pass filter must be configured to this maximum allowable wavelength width to effectively isolate the desired station while minimizing interference. By rounding the maximum allowable wavelength width to an uncertainty of 1 mm, we ensure the filter is precise enough to capture the correct signal and reject nearby ones.