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Chapter 31: Problem 34

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

Expert verified
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

01

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.
02

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.
03

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.
04

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.

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