Wave Guides and Resonant Cavities

Abstract

We investigate the propagation of electromagnetic waves in wave guides and resonant cavities, that is, in hollow, cylinder-like metal bodies. The metal body has the same cross-sectional area everywhere, and its surface is assumed to be an ideal conductor. See Figure18.1.If the body is closed, one talks of a resonant cavity, if its ends are open it is denoted a wave guide. Let a uniform material of permittivity ɛ and permeability µ be inside the cylinder. For the time dependence e ^—iωt of the electromagnetic field in the interior of the cylinder, Maxwell’s equations take the following form:

$$ \begin{array}{*{20}{c}} {\nabla \times E = i\frac{\omega }{c}B,{\text{ }}\nabla \cdot E = 0} \\ {\nabla \times B = - i\mu \varepsilon \frac{\omega }{c}E,{\text{ }}\nabla \cdot B = 0{\text{ }}} \end{array} $$

(18.1)

In these equations, E and B are still coupled. Again, we take the curl, and after rewriting we obtain the wave equations known from Chapter 16 (see equations (16.6a) and (16.6b))

$$ {\nabla ^2}E + \mu \varepsilon \frac{{{\omega ^2}}}{{{c^2}}}E = 0,{\text{ }}{\nabla ^2}B + \mu \varepsilon \frac{{{\omega ^2}}}{{{c^2}}}B = 0 $$

(18.2)