Problem: 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 = _________________________ GHzPart BFor which of these frequencies is the cavity midpoint an antinode?Enter your answers numerically in increasing order separated by commas.fa = ___________________________ GHz

FREE Expert Solution

Since two ends of the microwave cavity are reflective, we'll assume that it is a closed pipe. 

The wavelength that forms standing waves is expressed as:

$\boxed{\mathbf{\lambda }\mathbf{=}\frac{\mathbf{2}\mathbf{L}}{\mathbf{n}}}$, where λ is the wavelength forming standing waves, L is the length of the cavity, and n is the order.

The frequency of the wave is:

$\boxed{\mathbf{f}\mathbf{=}\frac{\mathbf{c}}{\mathbf{\lambda }}}$, where c is the speed of light 

Part A

We get the frequencies that will create standing waves in the microwave cavity by substituting λ = 2L/n in f = c/λ.

So, we have:

$\boxed{\begin{array}{rcl}\mathbf{f}& \mathbf{=}& \frac{\mathbf{c}}{\mathbf{\left(}{\displaystyle \frac{\mathbf{2}\mathbf{L}}{\mathbf{n}}}\mathbf{\right)}}\\ & \mathbf{=}& \frac{\mathbf{cn}}{\mathbf{2}\mathbf{L}}\end{array}}$

In the first case, the standing wave will be formed when n = 7.

83% (102 ratings)