A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding wavelength band?
The corresponding wavelength band is 25 m to 40 m.
The relationship between wavelength (λ) and frequency (ν) for electromagnetic waves is given by λ = c/ν, where c = ⁸ m/s is the speed of light in vacuum. All electromagnetic waves travel through vacuum with the same speed.
Step 1: Calculate wavelength for lower frequency (7.5 MHz) ν₁ = 7.5 MHz = ⁶ Hz λ₁ = c/ν₁ = (⁸)/(⁶) = 40 m
Step 2: Calculate wavelength for higher frequency (12 MHz) ν₂ = 12 MHz = ⁶ Hz λ₂ = c/ν₂ = (⁸)/(⁶) = 25 m
Since higher frequency corresponds to shorter wavelength, the wavelength band extends from 25 m to 40 m. This falls within the radio wave range (wavelength > 0.1 m).
Explanation
This question tests the student's understanding of the inverse relationship between wavelength and frequency for electromagnetic waves. The context establishes that all electromagnetic waves travel through vacuum with the same speed (c = ⁸ m/s), and the basic difference between various types lies in their wavelengths or frequencies. The formula λ = c/ν is fundamental. Students must recognize that lower frequency (7.5 MHz) gives longer wavelength (40 m), and higher frequency (12 MHz) gives shorter wavelength (25 m). The calculated wavelengths fall within the radio wave category mentioned in the context (wavelength > 0.1 m).
Solution Steps
Step 1: Convert frequencies from MHz to Hz: 7.5 MHz = ⁶ Hz and 12 MHz = ⁶ Hz
Step 2: Apply the formula λ = c/ν where c = ⁸ m/s
Step 3: Calculate λ₁ = (⁸)/(⁶) = 40 m for the lower frequency
Step 4: Calculate λ₂ = (⁸)/(⁶) = 25 m for the higher frequency
Step 5: State the wavelength band as 25 m to 40 m (shorter wavelength corresponds to higher frequency)