Introduction to Electrodynamics · Problem 11.23
Electrodynamics and Relativity — David J. Griffiths
Physics · radiation-applications
Problem
A radio tower rises to height above flat horizontal ground. At the top is a magnetic dipole antenna, of radius , with its axis vertical. FM station KRUD broadcasts from this antenna at (angular) frequency , with a total radiated power (that's averaged, of course, over a full cycle). Neighbors have complained about problems they attribute to excessive radiation from the tower—interference with their stereo systems, mechanical garage doors opening and closing mysteriously, and a variety of suspicious medical problems. But the city engineer who measured the radiation level at the base of the tower found it to be well below the accepted standard. You have been hired by the Neighborhood Association to assess the engineer's report. (a) In terms of the variables given (not all of which may be relevant), find the formula for the intensity of the radiation at ground level, a distance from the base of the tower. You may assume that . [Note: We are interested only in the magnitude of the radiation, not in its direction—when measurements are taken, the detector will be aimed directly at the antenna.] (b) How far from the base of the tower should the engineer have made the measurement? What is the formula for the intensity at this location? (c) KRUD's actual power output is 35 kilowatts, its frequency is 90 MHz, the antenna's radius is 6 cm, and the height of the tower is 200 m. The city's radio-emission limit is 200 microwatts/cm. Is KRUD in compliance?Intuitive explanation
A vertical dipole antenna radiates most strongly in the horizontal plane, but the ground reflects this radiation, creating interference patterns. The intensity at ground level depends on both the direct wave from the antenna and the reflected wave bouncing off the earth—these can reinforce or cancel depending on distance. The key is finding where constructive interference occurs and recognizing that the engineer's measurement location may have hit a destructive interference minimum, missing the true radiation hotspot.
Intuitive explanation
A vertical dipole antenna radiates most strongly in the horizontal plane, but the ground reflects this radiation, creating interference patterns. The intensity at ground level depends on both the direct wave from the antenna and the reflected wave bouncing off the earth—these can reinforce or cancel depending on distance. The key is finding where constructive interference occurs and recognizing that the engineer's measurement location may have hit a destructive interference minimum, missing the true radiation hotspot.