How Thick Should a Plastic Plate Be for Maximum Microwave Reflection?

[V0ODO0CH1LD]

Homework Statement

A plastic plate with refraction index of 1.5 is placed in the interior of a micro-wave oven which operates with a frequency of 2.5 x 10^9 Hz. If the micro-waves are perpendicular to the surface of the plate, what is the minimum thickness of the plate so that the maximum reflection of micro-waves occurs?

Homework Equations

The Attempt at a Solution

I honestly don't even understand the premises of the problem.. I feel like the maximum reflection of micro-waves will occur when some destructive interference is minimized. Right? But what are the interfering waves in this case? Are some waves reflecting out of the top surface of the plate and some go all the way through and reflect at the bottom surface? In that case, why is that happening?

 

[ehild]

Homework Statement

A plastic plate with refraction index of 1.5 is placed in the interior of a micro-wave oven which operates with a frequency of 2.5 x 10^9 Hz. If the micro-waves are perpendicular to the surface of the plate, what is the minimum thickness of the plate so that the maximum reflection of micro-waves occurs?

Homework Equations

The Attempt at a Solution

I honestly don't even understand the premises of the problem.. I feel like the maximum reflection of micro-waves will occur when some destructive interference is minimized. Right? But what are the interfering waves in this case? Are some waves reflecting out of the top surface of the plate and some go all the way through and reflect at the bottom surface? In that case, why is that happening?

Yes. The waves reflected from the top of the plate interfere with those, reflected from the bottom surface of the plate. Microwaves are electromagnetic waves, like light waves. Incident upon a boundary between two different media, part of the beam is reflected, some refracted and travel in the other medium, and reaching the second interface, partly reflected again. It is the same as with light waves. What is the condition for maximum reflection?

ehild

 

[V0ODO0CH1LD]

Okay so..

The waves that go all the way through and reflect at the bottom of the plate have to travel a distance $2d$ inside the plate, where $d$ is the thickness of the plate. The minimum thickness so that those waves won't cause destructive interference is $\frac{1}{2}\lambda_{plate}$. Because if $2d=\frac{1}{2}\lambda_{plate}$ the waves that bounce back will have inverted phase.

And if $\frac{\lambda_{plate}}{\lambda_{air}}=\frac{n_{air}}{n_{plate}}$ from snell's law, then $\frac{\lambda_{plate}}{\lambda_{air}}=\frac{1}{1.5}$. And $v=f\lambda$, so $\lambda_{air}=\frac{c}{f}=\frac{3x10^8}{2.5x10^9}$.

Therefore $\lambda_{plate}=\frac{3x10^8}{1.5*2.5^10^9}=0.08m=8cm$

So the minimum thickness $d=\frac{1}{4}\lambda_{plate}$ is $\frac{1}{4}8cm=2cm$

But I still don't get why some waves bounce back at the top and some bounce at the bottom.. Is it probabilistic??

 

[ehild]

But I still don't get why some waves bounce back at the top and some bounce at the bottom.. Is it probabilistic??

You have to imagine the incident wave like water waves coming into the shore. They never stop.
Part of the wave reflects, the other part enters into the plate, reaches the bottom, and part of it reflects again. The intensity of the wave decreases at every reflection. That back-reflected wave reaches the first surface and part of it goes into the air again. That wave would interfere with the wave, just arrived and reflected back directly from the first surface. But this, directly reflected wave changes phase at reflection: Its phase changes by pi. If the incident wave had maximum, the reflected one has minimum.
When the wave reflected from the back surface has the same phase as the first one, there is constructive interference, the reflection is high.
In order to get phase opposite to that of the incident wave, the refracted beam has to travel λ/2 when traversing the plate forward and back. Your solution is correct.

ehild

 

[Nickie]

I would approach this problem by first understanding the properties of microwaves and their interaction with materials. Microwaves are electromagnetic waves with a wavelength of about 12 cm and are commonly used for cooking in microwave ovens. When these waves encounter a material, they can be absorbed, transmitted, or reflected based on the material's properties.

In this problem, we are dealing with a plastic plate with a refractive index of 1.5. Refractive index is a measure of how much a material slows down the speed of light passing through it. In general, materials with a higher refractive index tend to reflect more light. Therefore, we can assume that the plastic plate will also reflect a significant amount of microwaves.

The frequency of the microwaves is given as 2.5 x 10^9 Hz. This means that the microwaves have a wavelength of about 12 cm, which is similar to the size of the plate. When the microwaves encounter the plate, some of them will be reflected from the surface of the plate, while others will pass through and be reflected from the bottom surface. This creates a phenomenon known as interference, where the waves interfere with each other and can either enhance or cancel out the reflected waves.

To find the minimum thickness of the plate for maximum reflection, we need to consider the concept of phase difference. When two waves interfere, their phase difference determines whether they will enhance or cancel each other out. In this case, the phase difference between the reflected waves from the top and bottom surfaces of the plate needs to be 180 degrees for maximum reflection. This means that the distance the waves travel from the top surface to the bottom surface and back needs to be half the wavelength of the microwaves.

Using this information, we can calculate the minimum thickness of the plate as 6 cm (half the wavelength of the microwaves). This is the thickness at which the reflected waves will have a phase difference of 180 degrees and interfere constructively, resulting in maximum reflection.

In conclusion, the maximum reflection of microwaves from the plastic plate will occur when the plate has a minimum thickness of 6 cm. This problem highlights the importance of understanding the properties of waves and their interactions with materials in order to solve complex problems.