Difficulty: Easy
Correct Answer: 75 m
Explanation:
Introduction / Context:
In radio communication, the size of the transmitting antenna is related to the wavelength of the carrier wave that it radiates efficiently. A common design uses a quarter-wavelength (λ/4) vertical antenna. This question asks you to use the carrier frequency of 1000 kHz to determine the approximate length of such a quarter-wave antenna, reinforcing your understanding of the relation between frequency, wavelength and antenna size.
Given Data / Assumptions:
Concept / Approach:
The wavelength of a radio wave is found using λ = c / f. For f = 1 MHz, λ ≈ 3 × 10^8 / 1 × 10^6 = 300 m. A quarter-wave antenna has a physical length equal to λ/4, so we divide the wavelength by 4 to get the desired antenna length. This gives 300 m / 4 = 75 m. Among the options, 75 m is therefore the correct approximate length. Shorter lengths such as 3 m or 0.3 m would correspond to much higher frequencies, while 300 m corresponds to a full wavelength rather than a quarter-wavelength radiator.
Step-by-Step Solution:
Step 1: Convert the frequency into hertz: 1000 kHz = 1000 × 10^3 Hz = 1 × 10^6 Hz.Step 2: Use the wave relation λ = c / f with c ≈ 3 × 10^8 m/s.Step 3: Compute the wavelength: λ ≈ (3 × 10^8 m/s) / (1 × 10^6 Hz) = 3 × 10^2 m = 300 m.Step 4: For a quarter-wave antenna, length L = λ/4.Step 5: Calculate L = 300 m / 4 = 75 m.Step 6: Match 75 m with the correct option.
Verification / Alternative check:
You can perform a quick mental check: 1 MHz signals are known to have wavelengths of a few hundred metres, so a fraction of that wavelength on the order of tens of metres for an antenna is reasonable. Practical radio engineering often uses quarter-wave vertical antennas of similar sizes for medium wave broadcasts. This rule of thumb supports the calculation and confirms that 75 m is a realistic answer.
Why Other Options Are Wrong:
300 m corresponds to a full wavelength (λ) and not to a quarter-wave antenna; while antennas of that length can be used, the question specifically asks for a quarter-wave radiator. Lengths such as 3 m or 0.3 m would match carrier frequencies far higher than 1 MHz (in the VHF or UHF range), not the given medium frequency. Therefore, these lengths are not appropriate for a 1000 kHz quarter-wave transmitting antenna.
Common Pitfalls:
A common error is to forget to convert kilohertz to hertz correctly or to confuse the wavelength with the antenna length directly. Some students also mistakenly use c ≈ 3 × 10^6 m/s, leading to wrong wavelengths. To avoid such mistakes, always write down λ = c / f with c = 3 × 10^8 m/s and carefully handle the powers of ten before dividing by 4 for a quarter-wave antenna.
Final Answer:
For a carrier wave of 1000 kHz, the length of a quarter-wave transmitting antenna is approximately 75 m.
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