Maxwell’s electromagnetic theory of radiation predicts that a changing electric field induces a changing magnetic field, as described by Ampere’s law, which in turn induces a changing electric field in accordance with Faraday’s law. Heinrich Hertz experimentally confirmed Maxwell’s theory in 1881 by generating and detecting radio waves in the laboratory, and demonstrating that these waves behaved like visible light, exhibiting properties such as reflection, refraction, diffraction, and interference.[1] Maxwell’s theory and Hertz’s experiments led directly to the development of modern radio, radar, television, electromagnetic imaging, and wireless communications. For example, the two antennas shown were developed by radio engineers at Bell Labs in 1933 and 1961 respectively. They were used to discover the very first radio waves from space, and the cosmic microwave background radiation, which are two of the greatest astronomical discoveries of all time. [2]
We begin this chapter by discussing sources of light and the spherical wave equation. A straightforward analysis of a spherically symmetric light wave allows us to expand on the concept of retarded time. Later in the chapter we will apply the finite speed at which information travels to two different phenomena—radio antennas and moving charged particles. As it turns out, the key insight was to use the potential formulation of electrodynamics, while carefully accounting for the time of information travel between source and observer. This was accomplished, at the end of the nineteenth century, by the French engineer Alfred-Marie Liénard and the German physicist Emil Johann Wiechert, who independently derived the correct expressions for the potentials and the fields of a point charge in arbitrary motion. In fact, it is their expression, rather than those of Coulomb and Biot, that is the correct one for charges undergoing arbitrary motion.
We begin our discussion of antennas with the simplest antenna of them all, the Hertzian dipole radiator, or simply the Hertzian dipole, and derive the scalar and vector potentials, and electric and magnetic fields, of the oscillating dipole in the limit of large distances (i.e. the radiation zone or far field region) and long wavelengths (compared to the size of the dipole). From these we calculate the radiated power, Poynting vector, radiation resistance, and beam pattern of a Hertzian dipole antenna. We then show how the dipole can be used as both a receiver and transmitter, and discuss the weak and strong reciprocity theorems in the context of antennas. Hertz constructed his dipole antenna to test a key prediction of Maxwell’s theory—the existence of electromagnetic waves—while John William Strutt (Lord Rayleigh) modeled the molecules of the Earth’s atmosphere as tiny Hertzian dipoles that oscillate in response to an electromagnetic field. In this way, Rayleigh derived his famous law of scattering and explained the blue sky during the day and the red sun at dawn and dusk.
In the case of antennas, we focus on the change in currents as the source of radiation in the far field, and we examine key results of antenna theory and single dish radio telescopes. From there, we continue with antenna theory in earnest. It is here that we come to understand why an engineer like Alfred-Marie Liénard would follow such a painstaking line of theoretical research. After calculating beam patterns of both whip and dish antennas, we will discuss how to use arrays of dish antennas to make detailed maps of faint radio sources, such as those extragalactic jets that exhibit superluminal motion.
While it is frequently necessary to solve problems where the radiation is produced by time changing currents, it is sometimes the case that the radiation is being emitted by a single charge, or small number of charges, undergoing acceleration. In that case, we want to solve for the potentials and fields of such a charge (or charges). The result for point charges is the Liénard-Wiechert potentials, which are named after the two scientists who independently derived the correct expressions for the potentials and fields of an arbitrarily moving point charge. Although Liénard and Wiechert derived these potentials prior to Einstein’s discovery of special relativity, the Liénard-Wiechert potentials and the fields derived from them are perfectly consistent with relativity.
Wiechert, unlike Liénard, was what we would now call a particle physicist. He, along with Hendrik Lorentz, Joseph Larmor, Henri Poincaré, Max Abraham, Paul Langevin, and others, worked on the relationship between the mass of the newly discovered electron and theories of the aether. Despite working under the false premise that the aether exists, these scientists developed many of the important relations we now know to be a consequence of special relativity. These included, among others, the relativistically correct potentials of a point charge.
[1] F.K. Vreeland and Henri Poincare, Maxwell’s theory and wireless telegraphy, Part 1: Maxwell’s theory and Hertzian oscillations, (New York: McGraw Publishing Co., 1904), 31-45. See also Heinrich Hertz, Electric Waves, trans. D.E. Jones (London: MacMillan and Co., 1893).
[2] See Karl G. Jansky, “Electrical Phenomena that Apparently are of Interstellar Origin,” Popular Astronomy, 41 (1933), pp. 548-555, and A.A. Penzias and R.W. Wilson, “A Measurement of Excess Antenna Temperature at 4080 Mc/s.,” The Astrophsical Journal, 142 (1965), pp. 419-421. The Jansky photo is from the website of the National Radio Astronomy Observatory. The photo of the horn-antenna is figure 2 from: A.B. Crawford, D.C. Hogg, and L.E. Hunt, “Project Echo—Horn-Reflector for Space Communication,” NASA Technical Note, D-1131 (1961).
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