A radio station, broadcasting at 89.1 MHz on the FM dial, has two transmission towers d = 40.0 m away from each other, producing signals that are completely in phase. An avid fan of this station is listening to it on her portable radio while jogging directly toward one of the towers from an initially large distance away, as shown below. Let x be her distance from the closest tower.
A)At what distance x will the jogger first experience no radio signal, due to destructive interference?
B)The jogger stops when she gets to the bottom tower. During her jog, how many times total did she experience a loss of signal? (A whole number is required here.)
first. IT is FM. If the frequency is not varying, I wonder how she is "listening" to it, as it will make no sound on her radio.
FM does not use towers as antennas, the wavelengths are too short, they use short antenna arrays, which may be installed on a tower, or not. Figure the wavelength of an FM (100Mhz) or AM (1100kh) and you see why the antennnas are different.
Not the question. Interference. So if the change in position is one quarter wavelength from tower one, it must be now also a quarter wavelength change in the distance from tower two, or be 180 deg out of phase.
b. if the jogger goes another 1/4 wave, the other wavy is now 1/4 shift, so the total shift now is 1 wavelength, or in phase. Constructive nterference. Repeat that going to tower B. So you get one destructive waveldngth per 1/2 wave length of x, so how many half wavelength are between A and B?
To answer these questions, we need to understand the concept of destructive interference and how it affects the reception of radio signals.
Destructive interference occurs when waves with the same frequency and opposite phases superpose, resulting in cancellation or a significant decrease in amplitude. In the context of radio waves, this would result in a loss of signal.
A) At what distance x will the jogger first experience no radio signal, due to destructive interference?
To determine this distance, we need to consider the conditions for destructive interference between the two transmission towers.
The key concept here is the path difference between the waves reaching the jogger from the two towers. To experience destructive interference, the path difference should be equal to an integer multiple of the wavelength.
The wavelength, λ, of a radio wave can be calculated using the formula:
λ = c / f
Where:
λ is the wavelength
c is the speed of light (approximately 3.0 x 10^8 m/s)
f is the frequency of the radio wave (89.1 MHz or 89.1 x 10^6 Hz)
Plugging in these values, we find:
λ = (3.0 x 10^8 m/s) / (89.1 x 10^6 Hz)
≈ 3.37 meters
Since the jogger is moving towards one of the transmission towers, the distance traveled by the waves from the closer tower to the jogger is shorter than the distance traveled by the waves from the farther tower.
Let's denote the distance from the jogger to the closer tower as x. The path difference can be calculated as:
Path Difference = (distance from farther tower) - (distance from closer tower)
= (40 m + x) - x
= 40 m
To experience destructive interference, the path difference should be equal to an integer multiple of the wavelength (nλ). Setting the path difference equal to nλ, we have:
40 m = n * 3.37 m
Dividing both sides by 3.37 m, we get:
n ≈ 11.88
Since n should be an integer, the smallest integer greater than 11.88 is 12.
So, the jogger will first experience no radio signal, due to destructive interference, when the distance x is equal to the path difference divided by 40 m (which is the spacing between the towers):
x = 40 m / 12 ≈ 3.33 m
B) The jogger stops when she gets to the bottom tower. During her jog, how many times total did she experience a loss of signal?
To determine the number of times the jogger experiences a loss of signal, we need to consider the intervals at which destructive interference occurs.
From the previous calculation, we found that the distance x should be equal to approximately 3.33 m for destructive interference to occur.
As the jogger moves from a large distance towards the bottom tower, she passes through intervals where the path difference (40 m) is equal to nλ, where n is an integer.
Since the jogger starts from a large distance away, the first interval occurs when n = 12 (as we calculated earlier).
The jogger will then continue through each subsequent interval where the path difference is equal to an integer multiple of the wavelength until she reaches the bottom tower.
Therefore, the number of times the jogger experiences a loss of signal corresponds to the number of intervals she passes through.
Since n = 12 corresponds to the first interval, and she stops at the bottom tower (where there is no interference), the total number of intervals can be found by subtracting 1 from n.
So, the jogger experiences a loss of signal a total of:
Total = n - 1
= 12 - 1
= 11 times