Problem — Rectangular microwave cavity

Problem

A metal cavity has dimensions $a=5$ cm, $b=3$ cm, $c=2$ cm. Calculate the frequency of the fundamental mode $\text{TE}_{110}$ using $f_{110}=\frac{c}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}}$ where $c$ is the speed of light.

Solution

Data in metres. $a=0{,}05$ m, $b=0{,}03$ m, $c=3{,}00\cdot 10^8$ m/s.

Terms under the square root.

$\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{0{,}05^2}+\frac{1}{0{,}03^2}=400+1111=1511\ \text{m}^{-2}$

Frequency.

$f_{110}=\frac{3{,}00\cdot 10^8}{2}\sqrt{1511}=1{,}5\cdot 10^8\cdot 38{,}9\approx 5{,}8\cdot 10^9\ \text{Hz}$

$\ev{f_{110}\approx 5{,}8\ \text{GHz}}$

This falls within the microwave band (domestic oven $\approx 2{,}45$ GHz).

Collegamenti

Argomenti: Onde elettromagnetiche Concetti: Onda elettromagnetica