Questions: Cavity Resonators and Standing Wave Patterns

Question 1

A rectangular cavity resonator has dimensions a × b × d. All three dimensions are doubled to 2a × 2b × 2d. How do the resonant frequencies change?

• A. All resonant frequencies double, because the cavity is larger
• B. All resonant frequencies are halved, because larger dimensions allow longer wavelengths
• C. Only the lowest resonant frequency changes; higher modes are unaffected
• D. Resonant frequencies change unpredictably, depending on which mode is excited

Answer: (B)

The resonant frequencies of a rectangular cavity are f_mnp = (c/2)√[(m/a)² + (n/b)² + (p/d)²]. When all dimensions double (a→2a, b→2b, d→2d), each term under the square root is divided by 4, so the whole expression is divided by 2. Every resonant frequency is halved. This makes physical sense: larger cavities support longer wavelengths, which correspond to lower frequencies.

Question 2

Why does a metal cavity resonator typically have a much higher quality factor Q than a simple resistive LC circuit?

• A. Because the cavity stores no energy between driving cycles, reducing dissipation
• B. Because energy losses occur only in the thin skin-depth layer at the conducting walls, while the bulk of stored energy resides in the fields inside the cavity
• C. Because the resonant frequency is fixed by geometry, eliminating thermal noise
• D. Because standing waves in a cavity reflect energy rather than dissipating it

Answer: (B)

Q = (energy stored)/(energy lost per cycle). In a metal cavity, the electromagnetic field fills the entire interior volume, storing substantial energy. But resistive losses only occur in the thin skin-depth layer (order of micrometers) at the surface — a very small fraction of the volume. The ratio of total stored energy to surface-loss energy is enormous, giving Q ~ 10⁴–10⁵. In a lumped LC circuit, resistive elements pervade the circuit and the stored energy is much smaller relative to dissipation.

Question 3

In a cavity resonator, energy alternates between electric and magnetic field storage — analogous to a mass-spring oscillator exchanging potential and kinetic energy.

• T. True
• F. False

Answer: True

At resonance, when the electric field is maximum (charges maximally separated, analogous to maximum potential energy in a spring), the magnetic field is zero. A quarter cycle later, the electric field is zero and the magnetic field is maximum (currents flowing at maximum, analogous to maximum kinetic energy). This oscillation is the electromagnetic analogue of the LC resonator circuit, where energy alternates between capacitor (electric) and inductor (magnetic) storage.

Question 4

A cavity resonator supports a continuous range of resonant frequencies, like a transmission line, but reflects power back at both ends instead of transmitting it.

• T. True
• F. False

Answer: False

A cavity supports only discrete resonant frequencies, not a continuous spectrum. Each resonant mode is labeled by three integers (m, n, p), and only these specific combinations of standing-wave patterns fit the boundary conditions on all six walls simultaneously. This discreteness is the key consequence of closing the waveguide at both ends — it is analogous to the discrete energy levels of a quantum particle in a box and is what makes cavities useful as precise frequency-selective filters.

Question 5

Why does enclosing a waveguide at both ends with conducting walls produce discrete resonant modes rather than a continuous range of propagating frequencies?

Model answer: A waveguide confines waves in the transverse directions, producing discrete modes with a minimum cutoff frequency. Along the propagation direction z, fields are free to travel continuously. When you add conducting end walls, the fields must also vanish (or satisfy appropriate conditions) at z = 0 and z = d — introducing a standing-wave condition in the z-direction. Only wavelengths that fit an integer number of half-waves into the length d satisfy this condition. The result is that all three spatial directions impose quantization: only specific triplets (m, n, p) of integers satisfy all boundary conditions, and each triplet corresponds to a discrete resonant frequency.

This is analogous to a 1D particle in a box having discrete energy levels, or a string fixed at both ends supporting only harmonics. Confining the field in all three directions simultaneously quantizes all three components of the wave vector, producing a discrete spectrum of allowed modes.