Wave Trapped in a Cubic Microwave Cavity: Calculating Lowest Frequency and Modes

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SUMMARY

The discussion focuses on calculating the lowest frequency of a wave trapped in a cubic microwave cavity measuring one foot on each side. The problem involves understanding electromagnetic boundary conditions, specifically that the electric field's parallel components and the magnetic field's perpendicular components vanish at the cavity's surfaces. The solution requires applying concepts from electromagnetic theory rather than quantum mechanics, leading to the determination of the lowest frequency and the number of modes associated with it.

PREREQUISITES
  • Understanding of electromagnetic boundary conditions
  • Familiarity with wave functions in confined spaces
  • Knowledge of the relationship between frequency and wavelength
  • Basic principles of cavity resonators
NEXT STEPS
  • Study the principles of electromagnetic waves in cavities
  • Learn about the derivation of frequency modes in rectangular cavities
  • Explore the concept of photon occupancy in electromagnetic fields
  • Investigate the mathematical formulation of wave functions in infinite potential wells
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Students and professionals in physics, particularly those focusing on electromagnetism, microwave engineering, and quantum mechanics applications in cavity resonators.

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Homework Statement



Consider a cubic microwave cavity one foot on a side. Call this length a for purposes of your calculation. Look for the lowest fre- quency a wave trapped in the cavity can have. How many different modes are there which have this frequency? Assume the surfaces of the cavity all are perfect conductors, implying the conditions that parallel components of E⃗ and perpendicular components of B⃗ vanish at the surface. Estimate the maximum amplitude of E⃗ in volts per meter, such that exactly one photon is present in the cavity.


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The Attempt at a Solution


I looked into the particle in the infinite well and saw some similarities. However with this problem, it appears that there is a single wave function. However, I'm really stuck on how to determine the lowest frequency, and how many modes occupy it.
 
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You don't need quantum mechanics to solve this. This is just an E&M boundary condition problem.
 

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