A microwave generator can produce microwaves at any...

A microwave generator can produce microwaves at any frequency between $10 \mathrm{GHz}$ and $\begin{array}{llll}20 & \text { GHz. As Figure } & \text { P16.51 }\end{array}$ shows, 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 between $10 \mathrm{GHz}$ and $20 \mathrm{GHz}$ will create standing waves in the microwave cavity? b. For which of these frequencies is the cavity midpoint an antinode?

A microwave generator can produce microwaves at any frequency between $10 \mathrm{GHz}$ and $\begin{array}{llll}20 & \text { GHz. As Figure } & \text { P16.51 }\end{array}$ shows, 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 between $10 \mathrm{GHz}$ and $20 \mathrm{GHz}$ will create standing waves in the microwave cavity?
b. For which of these frequencies is the cavity midpoint an antinode?

Key Concepts

Standing Waves

Standing waves are the result of interference between two waves traveling in opposite directions within a bounded medium. In a cavity, these waves reflect off the walls and interfere to produce fixed points of zero amplitude (nodes) and maximum amplitude (antinodes), resulting in a stable pattern that is characteristic of resonant systems.

Resonant Frequencies

The resonant frequencies in a cavity are determined by the geometry and boundary conditions of the cavity. For a cavity with reflective ends, the condition for resonance is that an integer number of half-wavelengths must fit exactly into the length of the cavity. This relation allows one to calculate the specific frequencies at which standing waves can exist.

Mode Numbers

Mode numbers represent the integer count of half-wavelengths that can span the length of the resonant cavity. Each mode corresponds to a different standing wave pattern. The fundamental mode corresponds to one half-wavelength, while higher modes include additional half-wavelength segments, each introducing additional nodes and antinodes within the cavity.

Nodes and Antinodes

Nodes are points along a standing wave where the amplitude is always zero, while antinodes are points where the amplitude reaches its maximum value. The position of nodes and antinodes within a cavity is determined by the mode number and the boundary conditions. In many practical problems, such as microwave cavities, determining which mode results in an antinode at a particular location (for example, the midpoint) is crucial for optimizing energy coupling or field intensity.

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