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

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In summary, a cubic microwave cavity with a side length of one foot is being considered. The lowest frequency of a wave trapped in the cavity is being sought after, assuming all surfaces are perfect conductors. It is also assumed that the parallel components of E⃗ and perpendicular components of B⃗ vanish at the surface. The maximum amplitude of E⃗ in volts per meter for exactly one photon to be present in the cavity is estimated. The problem does not require the use of quantum mechanics, as it is simply an electromagnetic boundary condition problem.
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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.
 

1. What is a cubic microwave cavity?

A cubic microwave cavity is a type of resonant chamber used in microwave engineering. It is a box-shaped structure made of conductive materials, such as metal, that is designed to trap and amplify microwave energy within its walls.

2. How does a wave get trapped in a cubic microwave cavity?

When a microwave signal is introduced into the cavity, it bounces back and forth between the walls, creating a standing wave pattern. As the waves continue to reflect off the walls, they become amplified and trapped within the cavity.

3. What is the lowest frequency that can be trapped in a cubic microwave cavity?

The lowest frequency that can be trapped in a cubic microwave cavity is determined by the dimensions of the cavity itself. The wavelength of the lowest frequency mode must be able to fit within the dimensions of the cavity without creating any nodes or null points.

4. How can the lowest frequency and modes of a wave trapped in a cubic microwave cavity be calculated?

The lowest frequency and modes can be calculated using the dimensions of the cavity and the speed of light. The lowest frequency is determined by dividing the speed of light by twice the length of the cavity, while the modes can be calculated using the formula m = 2n + p + q, where n, p, and q are integers representing the number of nodes along each side of the cavity.

5. What are the applications of studying wave trapped in a cubic microwave cavity?

Studying waves trapped in a cubic microwave cavity has practical applications in microwave engineering, such as designing and optimizing microwave circuits and devices. It is also used in fields like spectroscopy, where it can be used to accurately measure the properties of molecules and materials.

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