Question

The range of wavelengths allotted to the radio broadcast band is from about \(190 \mathrm{m}\) to \(550 \mathrm{m} .\) If each station needs exclusive use of a frequency band \(10 \mathrm{kHz}\) wide, how many stations can operate in the broadcast band?

Short answer

Answer: Approximately 103 radio stations can operate within the given range of wavelengths for the radio broadcast band.

Step-by-step solution

Understand the relationship between frequency and wavelength

Wavelength (λ) and frequency (ν) are related by the speed of light (c). The equation is given by: c = λν where c = speed of light = \(3 \times 10^8\) m/s λ = wavelength in meters ν = frequency in Hertz (Hz)

Convert given wavelengths to frequencies

We are given the range of wavelengths allotted for the radio broadcast band as \(190 \mathrm{m}\) to \(550 \mathrm{m}\). We can now convert these wavelengths into frequencies using the equation from Step 1 by rearranging the formula: ν = c / λ For the lower limit of the wavelength: ν1 = c / λ1 ν1 = (3 x 10^8 m/s) / (550 m) ν1 ≈ 545454.55 Hz For the upper limit of the wavelength: ν2 = c / λ2 ν2 = (3 x 10^8 m/s) / (190 m) ν2 ≈ 1,578,947.37 Hz

Calculate the difference in frequencies

To find out how many stations can operate in the broadcast band, we need to know the difference in frequencies for the given range of wavelengths. To calculate the difference in frequencies, subtract the lower frequency from the higher frequency: Δν = ν2 - ν1 Δν ≈ 1,578,947.37 Hz - 545,454.55 Hz Δν ≈ 1,033,492.82 Hz

Determine the number of stations

Each station requires exclusive use of a frequency band that is 10 kHz wide. To find the number of stations that can fit in this range, we simply divide the difference in frequencies by the width of each band: Number of stations = Δν / Bandwidth Number of stations ≈ 1,033,492.82 Hz / 10,000 Hz Number of stations ≈ 103.35 Since we cannot have a fraction of a station, we round down to the nearest whole number.

Final answer

Based on our calculations, approximately 103 radio stations can operate within the given range of wavelengths for the radio broadcast band.