Problem: You are listening to the FM radio in your car. As you come to a stop at a traffic light, you notice that the radio signal is fuzzy. By pulling up a short distance, you can make the reception clear again. In this problem, we work through a simple model of what is happening.Our model is that the radio waves are taking two paths to your radio antenna:the direct route from the transmitteran indirect route via reflection off a buildingBecause the two paths have different lengths, they can constructively or destructively interfere. Assume that the transmitter is very far away, and that the building is at an a 45° angle from the path to the transmitter.Point A in the figure is where you originally stopped, and point B is where the station is completely clear again. Finally, assume that the signal is at its worst at point A, and at its clearest at point B.Part AWhat is the distance d between points A and B?Express your answer in wavelengths, as a fraction.Part BYour FM station has a frequency of 100 megahertz. The speed of light is about 3.00x108 meters per second. What is the distance d between points A and B?Express your answer in meters, to two significant figures.

FREE Expert Solution

Part A

At point A, there is destructive interference. 

Destructive interference is expressed as:

 xA=(n+12)λ

At point B, there is constructive interference, expressed as:

xB=nλ

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Problem Details

You are listening to the FM radio in your car. As you come to a stop at a traffic light, you notice that the radio signal is fuzzy. By pulling up a short distance, you can make the reception clear again. In this problem, we work through a simple model of what is happening.

Our model is that the radio waves are taking two paths to your radio antenna:

  • the direct route from the transmitter
  • an indirect route via reflection off a building

Because the two paths have different lengths, they can constructively or destructively interfere. Assume that the transmitter is very far away, and that the building is at an a 45° angle from the path to the transmitter.

Point A in the figure is where you originally stopped, and point B is where the station is completely clear again. Finally, assume that the signal is at its worst at point A, and at its clearest at point B.

Part A

What is the distance d between points A and B?

Express your answer in wavelengths, as a fraction.


Part B

Your FM station has a frequency of 100 megahertz. The speed of light is about 3.00x108 meters per second. What is the distance d between points A and B?

Express your answer in meters, to two significant figures.