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 such 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

Answer: The maximum allowable wavelength width is 4 mm ± 1 mm.

Step-by-step solution

Calculate the wavelength for each frequency station

Using the equation, λ = c/f, where λ is the wavelength, c is the speed of light, and f is the frequency, calculate the wavelengths for stations 91.1, 91.3, and 91.5 MHz.

Calculate the wavelength differences between stations

After calculating the wavelengths for each station, find the differences between them (e.g., Δλ1 = λ91.3 - λ91.1, Δλ2 = λ91.5 - λ91.3) to determine the required wavelength width necessary to avoid interference.

Find the maximum allowable wavelength width

Find the minimum of Δλ1 and Δλ2; this represents the maximum allowable wavelength width for the band-pass filter to avoid interference. Now, let's calculate the wavelengths and the allowable wavelength width:

Calculate the wavelength for each frequency station

Using the speed of light c = 3.00 * 10^8 m/s, we can calculate the wavelengths for the stations as follows: λ91.1 = c / f91.1 = 3.00*10^8 m/s / 91.1*10^6 Hz ≈ 3.291 m λ91.3 = c / f91.3 = 3.00*10^8 m/s / 91.3*10^6 Hz ≈ 3.287 m λ91.5 = c / f91.5 = 3.00*10^8 m/s / 91.5*10^6 Hz ≈ 3.279 m

Calculate the wavelength differences between stations

Calculate the differences in wavelengths between the stations: Δλ1 = λ91.3 - λ91.1 ≈ 3.287 m - 3.291 m ≈ -0.004 m Δλ2 = λ91.5 - λ91.3 ≈ 3.279 m - 3.287 m ≈ -0.008 m Since these values are negative, we need to multiply them by -1. Δλ1 ≈ 0.004 m Δλ2 ≈ 0.008 m

Find the maximum allowable wavelength width

Find the minimum of Δλ1 and Δλ2 to determine the maximum allowable wavelength width: Maximum allowable wavelength width = min(Δλ1, Δλ2) = 0.004 m However, we should provide the answer with an uncertainty of 1 mm. Thus, the final answer is: Maximum allowable wavelength width = 0.004 m ± 0.001 m = 4 mm ± 1 mm