The microwave generator in FIGURE P21.47 can produce...

The microwave generator in FIGURE P21.47 can produce microwaves at any frequency between $10 \mathrm{GHz}$ and $20 \mathrm{GHz}$ The microwaves are aimed, through a small hole, into a "microwave cavity" that consists of a $10-\mathrm{cm}-$ long cylinder with reflective ends. a. Which frequencies will create standing waves in the microwave cavity? b. For which of these frequencies is the cavity midpoint an antinode?

The microwave generator in FIGURE P21.47 can produce microwaves at any frequency between $10 \mathrm{GHz}$ and $20 \mathrm{GHz}$ The microwaves are aimed, through a small hole, into a "microwave cavity" that consists of a $10-\mathrm{cm}-$ long cylinder with reflective ends.
a. Which frequencies will create standing waves in the microwave cavity?
b. For which of these frequencies is the cavity midpoint an antinode?

Key Concepts

Cavity Modes

Cavity modes refer to the distinct patterns or configurations of standing waves present within the resonator. Each mode corresponds to a particular resonant frequency and is characterized by a specific distribution of nodes and antinodes along the cavity. The mode number, which indicates the number of half-wavelength segments fitting within the cavity length, helps determine the spatial pattern and energy distribution of the standing wave.

Boundary Conditions

Boundary conditions describe the behavior of the electromagnetic wave at the endpoints of the cavity, typically characterized by reflective surfaces in microwave cavities. These conditions determine where nodes or antinodes must appear along the cavity. For example, in a cavity with perfectly reflecting ends, the field often has nodes at the boundaries, directly influencing the allowed resonant frequencies.

Resonance Frequency

Resonance frequency pertains to the specific frequencies at which a cavity will naturally support standing waves. For a cavity of fixed length, these frequencies occur when the length is an integer multiple of half the wavelength of the microwaves. This condition maximizes the energy stored in the cavity, as the wave reflects and reinforces itself after each round trip.

Standing Waves

Standing waves are the resultant formation when two waves of identical frequency and amplitude travel in opposite directions and interfere. In resonant cavities, such as those used for microwaves, these standing waves manifest as fixed patterns of nodes (points of no oscillation) and antinodes (points of maximum oscillation). The formation of standing waves depends on the wave frequency matching specific conditions dictated by the physical dimensions of the cavity.

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