Question

Microwaves pass through a small hole into the “microwave cavity” of FIGURE EX17.13. What frequencies between 10 GHz and 20 GHz will create standing waves in the cavity?

Short answer

$m$	$f_m\left(GHz\right)$
7	$10.5$
8	$12.0$
9	$13.5$
10	$15.0$
11	$16.5$
12	$18.0$
13	$19.5$

Step-by-step solution

Given information

The length of the microwave cavity is $L=10cm=0.10m$

Frequencies of the standing waves

There are reflectors at both ends, so the electromagnetic standing wave acts like the standing wave on a string that is tied at both ends.

The electromagnetic waves of all frequencies travel at the speed of light c.

The frequencies of the standing waves are given by

$$\begin{gathered} f_m=m\frac{v_{light}}{2L} \\ =m\frac{c}{2L} \end{gathered}$$

Substitute the given values

$$\begin{gathered} f_m=m\left(\frac{3\times {10}^8}{2\times 0.10}\right) \\ =m\left(1.5\times {10}^9\right) \\ =1.5mGHz \end{gathered}$$

Frequencies between 10 GHz and 20 GHz

The generator can produce standing waves at any frequency between $10GHz$and $20GHz$.

The table below shows the frequencies produce by the generator between $10GHz$and $20GHz$.

$m$	$f_m\left(GHz\right)$
7	$10.5$
8	12
9	$13.5$
10	15
11	$16.5$
12	18
13	$19.5$