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Free oscillation in a circuit is generally short-lived because of the presence of a resistance. NormallyACcurrent flows continuously, as the generator keeps supplying energy, but it is not simple to obtain a source of persisting electrical oscillations at a desired frequency. Today, vacuum tubes and semi-conducting devices, oscillators, signal amplifiers, and other electronically controlled circuits are available at almost any frequency, thereby, basic electromagnetic experiments can be performed with precision in modern electronics. A self-sustaining oscillator with a sizable output power is a particularly important device to study high-frequency phenomena.

Maxwell formulated the laws of electromagnetism by hydrodynamic analogy, basing them on Faraday’s concept of fields. The Maxwell equations constitute fundamental laws of physics, as recognized in Einstein’s special theory of relativity.We have, so far, followed these pioneers’ footsteps in arriving at Maxwell’s equations, the physical implications of which are formally discussed in this chapter.

Hertz demonstrated that electromagnetic radiation could be emitted from a spark gap in an induction current, thereby giving evidence supporting the Maxwell theory. Then, Marconi (1895) invented wireless communication by means of electromagnetic waves at radio frequencies and opened the modern era of telecommunication technology. Using an antenna , electromagnetic waves can be transmitted through open space, and detected by another antenna at a distant location. Although not clearly established in Hertz’s experiment, the spark gap played the role of an antenna for emitting a high-frequency radiation. Further, he demonstrated a persistent oscillation is required for continuous radiation from the antenna.

In Newton’s mechanics, a particle of mass m in motion under a force F is described by the equation

Although a static conductor is characterized by E = 0, the electric field E can penetrate into a conducting medium if it is varying as a function of time. In a conductor, charge carriers are mobilized by an applied electric field so that currents flow through a conductor under time-dependent conditions. Distributed currents, called eddy currents , flow at high densities in the close vicinity of surfaces to certain depth, called the spin-depth . In free space, electromagnetic waves are reflected by dielectric boundaries as well as conducting surfaces where surface currents play a significant role.

Wave equations have a simple plane-wave solution that can be obtained in free space and used for a field bounded by rectangular boundaries. In such cases, as a consequence of the linear differential equations involved, solutions can be expressed by a Fourier series of sinusoidal functions of the phase variable Φ = k.r - ω t . Normally, a light beam propagates as a free wave but is coupled with a reflected wave at a boundary point, as discussed in the previous section. Nevertheless, such a reflection in optics is a significant concept for general electromagnetic waves in open and bound space. For electromagnetic applications the radiation needs to be guided in a desired direction, where it is necessary to specify directions of the propagating and reflecting power of the wave from a target object. In this chapter we discuss the basic principle for such guided waves in modern communication and guidance.

In the previous chapter, we discussed TE and TM modes of electromagnetic propagations in a rectangular waveguide, and a TEM mode in a coaxial cable. In a waveguide, axial components of E or H vectors behave like scalar variables of the phase of propagation, and in a coaxial cable the uniaxial vector potential A plays such a role as a scalar variable. Thus, transversal fields depend on the type of waveguide; on the other hand, the axial component plays essential role for propagation.

In Section 17.3, we discussed the property of a resonant cavity in a rectangular TE mode. Primarily, it is a matter of analytical convenience, but in principle resonance can be found at a given frequency for a cavity of arbitrary shape. In this chapter, following Slater, we discuss resonant properties of a cavity of general shape, for which the Maxwell equations can provide oscillating modes with intense amplitudes (J. C. Slater, Microwave Electronics , Van Nostrand, 1950.)

A resonator is an essential device for sustained oscillations, such as a triode oscillator, in which the oscillating current is controlled by a resonant LC load. In microwave oscillators a cavity resonator controls the electronic current to sustain oscillation in the feedback mechanism. The electronic current density j can interact with a cavity when passing through an area dominated by the E vector. Either from electrons traveling to the cavity or in the opposite direction, the transferred energy operates the device like an oscillator or an electron accelerator. Sharing a similar principle, a betatron accelerator is operated at low frequencies for accelerating electrons by an alternating dynamic E field that arises from a sinusoidal magnetic flux.

Regarding conducting properties, materials were traditionally classified into categories of “conductor” or “insulator.” Today, many types of semi-conductors as well as magnetic materials are used for a variety of applications. Modern materials can be discussed properly with respect to structure, but such a discussion is beyond the scope of this book. Nevertheless, modern empirical knowledge of electronic charge and spin, among other properties, must be taken for granted in order to understand classical electromagnetic theory. In addition, the presence of electric and magnetic moments in constituent species is empirical.