Question

A microwave generator can produce microwaves at any frequency between 10.0 GHz and 20.0 GHz . As the figure shows, the microwaves are aimed, through a small hole, into a "microwave cavity" that consists of a 10.0 cm-long cylinder with reflective ends.


Part A
Which frequencies between 10 GHz and 20 GHz will create standing waves in the microwave cavity?
Enter your answers numerically in increasing order separated by commas.
f = _________________________GHz


Part B
For which of these frequencies is the cavity midpoint an antinode?
Enter your answers numerically in increasing order separated by commas.
f_a = ___________________________GHz

Answer 1

Concepts and reason

The concept used to solve this problem is standing waves on a string and microwaves.

Use the length of the cavity to find the expression for the wavelength that forms standing waves.

Use the relation between the length of the cavity and the speed of the wave to find the relation between the frequencies of the corresponding waves.

Fundamentals

Assume that the microwave cavity as a closed pipe, because the two ends are reflective.

Expression for the wavelength that forms standing waves is as follows:

Here, is the wavelength that forms standing waves, L is the length of the cavity, and n is the order.

Expression for the frequency of the wave is as follows:

Here, f is the frequency of the microwave and c is the speed of the light.

(A)

Expression for the frequency of the wave is as follows:

…… (1)

Expression for the wavelength that forms standing waves is as follows:

Replace for in Equation (1) to solve for .

Case 1:

The frequency for the corresponding wave of order is as follows:

Here, is the frequency for the corresponding wave of order

Substitute for c, for n, and for L.

Case 2:

The frequency for the corresponding wave of order is as follows:

Here, is the frequency for the corresponding wave of order .

Substitute for c, for n, and for L.

Case 3:

The frequency for the corresponding wave of order is as follows:

Here, is the frequency for the corresponding wave of order .

Substitute for c, for n, and for L.

Case 4:

The frequency for the corresponding wave of order is as follows:

Here, is the frequency for the corresponding wave of order .

Substitute for c, for n, and for L.

Case 5:

The frequency for the corresponding wave of order is as follows:

Here, is the frequency for the corresponding wave of order .

Substitute for c, for n, and for L.

Case 6:

The frequency for the corresponding wave of order is as follows:

Here, is the frequency for the corresponding wave of order .

Substitute for c, for n, and for L.

Case 7:

The frequency for the corresponding wave of order is as follows:

Here, is the frequency for the corresponding wave of order .

Substitute for c, for n, and for L.

Therefore, the frequencies are ranked on the basis of increasing order as follows:

.

(B)

Here, there are seven different frequencies of standing waves.

The even number of n creates a node point, while odd number of n creates antinodes.

The frequencies that have the midpoint called as antinodes that are as follows:

,

,

,

,

Ans: Part A

The increasing order of the frequencies between and in the microwave cavity that creates standing waves are , , , , , , .

Part B

The increasing order of the frequencies between and is the microwave cavity midpoint an antinodes are , , , .