Final answer:
It is true that many broadcast antennas are optimally one-fourth the wavelength of the electromagnetic waves they transmit for improved power and efficiency. A 50.0 m high antenna broadcasts most efficiently at 1.5 x 10¶ Hz, which is in the AM band. The analogy between resonances in air columns and antennas lies in how both systems use standing wave patterns for maximum energy transfer.
Step-by-step explanation:
To address the question regarding broadcast antennas and their length relative to the wavelength of the electromagnetic waves they transmit, we must understand the concept of resonance in antennas. It is true that to improve power and efficiency, many broadcast antennas are designed with a length that is one-fourth (1/4 or λ/4) of the wavelength of the electromagnetic waves being transmitted. This creates a condition of resonance, which allows for efficient transmission and reception of radio waves.
To determine the frequency that a 50.0 m high antenna would broadcast most efficiently, which is one-fourth the wavelength, we use the formula λ = c/f, where λ is the wavelength, c is the speed of light (~3.00 x 108 m/s), and f is the frequency. Since the antenna is one-fourth the wavelength, we have 4 * 50.0 m = 200 m as the full wavelength. Then f = c/λ, giving us a frequency of f = 3.00 x 108 m/s / 200 m = 1.5 x 106 Hz or 1500 kHz, which falls within the AM band.
The analogy between the fundamental resonant mode of an air column closed at one end and the resonance of currents on an antenna that is one-fourth their wavelength is that both systems are set into resonance at a frequency that corresponds to a standing wave pattern where the length of the system is an integral number of quarter wavelengths.
In both cases, the resonance allows for maximum energy transfer—sound in the air column and electromagnetic waves in the antenna scenario.