Two radio antennas are 100 m apart along a north-south line....

Two radio antennas are 100 m apart along a north-south line. They broadcast identical radio waves at a frequency of $3.0 \mathrm{MHz}$ Your job is to monitor the signal strength with a handheld receiver. To get to your first measuring point, you walk $800 \mathrm{m}$ east from the midpoint between the antennas, then 600 m north. a. What is the phase difference between the waves at this point? b. Is the interference at this point maximum constructive, perfect destructive, or somewhere in between? Explain. c. If you now begin to walk farther north, does the signal strength increase, decrease, or stay the same? Explain.

Two radio antennas are 100 m apart along a north-south line. They broadcast identical radio waves at a frequency of $3.0 \mathrm{MHz}$ Your job is to monitor the signal strength with a handheld receiver. To get to your first measuring point, you walk $800 \mathrm{m}$ east from the midpoint between the antennas, then 600 m north.
a. What is the phase difference between the waves at this point?
b. Is the interference at this point maximum constructive, perfect destructive, or somewhere in between? Explain.
c. If you now begin to walk farther north, does the signal strength increase, decrease, or stay the same? Explain.

Key Concepts

Path Length Difference

The difference in the distances that waves travel from their sources to a given point determines the phase difference between them. This path length difference, when expressed in terms of wavelength, indicates how the waves combine at that location, thereby affecting the observed signal strength and the interference pattern.

Interference

Interference occurs when two or more waves overlap, leading to a resultant amplitude that is the sum of the individual waves. Depending on the phase difference, the waves can interfere constructively (amplitudes add to enhance the signal) or destructively (amplitudes subtract, reducing the signal), with many real situations falling somewhere in between these extremes.

Radio Wave Properties

Radio waves are electromagnetic waves that propagate through space and are characterized by their frequency and wavelength. They are used for various forms of wireless communication, and their behavior—such as interference, diffraction, and reflection—is fundamental in understanding how signals are transmitted and received.

Wavelength and Frequency Relationship

The wavelength of a radio wave is determined by its frequency and the speed of light, following the equation ? = c/f. This relationship is crucial because it sets the scale for interference effects, as differences in path length are compared to the wavelength to determine phase differences.

Phase Difference

Phase difference refers to the difference in the phase angles of two waves arriving at a point, which is related to their path length difference relative to the wavelength. A phase difference of 2? (or multiples thereof) corresponds to the waves being in phase, while a phase difference of ? corresponds to them being out of phase, leading to maximal cancellation.

Transcript

00:01 All right, i've drawn a diagram of the situation here.

00:05 And the first thing that i want to do in order to answer part a is i want to calculate delta r.

00:13 So, delta r is going to be this distance, which i'm going to call r2, minus this distance, which i'm going to call r1.

00:36 Okay, r2 is going to be the square root.

00:40 Using the pythagorean theorem of 800 squared plus we've got 50 here plus the 600.

01:02 So that would be 650 squared minus r2, which would be 8, i mean r1, which would be 800 squared.

01:15 And then we've got 600 minus 50.

01:19 So that would be 550.

01:22 All right.

01:27 So now that we know delta r, and since these begin at in phase, delta phi equals 2 pi delta r over lambda plus delta phi, zero, but like i said, these are in phase, so that's zero.

01:56 Speed of light equals wavelength times frequency.

02:02 So wavelength is speed of light over frequency.

02:08 So the phase difference is 2 pi f delta r over c, where c is 3 times 10 to the 8th power meters per second.

02:30 So i'm going to put this in a calculator.

02:36 2 times pi times the frequency.

02:42 And the frequency was 3 megahertz.

02:49 3 times 10 to the 6th power hertz.

02:55 5 times 3 times 10 to the 6th power.

02:58 And then i'm going to go over c, 3 times 10.

03:05 To the eighth power times delta r, which is the square root of 800 squared plus 650 squared minus square root 800 squared plus 550 squared.

03:36 And that gives me, let me see how many significant digits i have, probably 2, yep, 3 .8 b.

04:17 Okay, so we want to consider what type of interference this is.

04:23 So, when delta phi equals some number times 2 pi, well then it's constructive.

04:46 So let's go back to my calculator.

04:53 2 times pi is approximately 6 .28.

05:04 So it is definitely not constructive interference.

05:15 It's not maximum constructive interference because delta phi would have to be some multiple of 2 pi, which it is not.

05:40 Now, to be destructive, it would have to be some multiple, that's one -half of two -pie.

05:58 So if m is zero, that would be half of two -pie, which is pi, 3 .14.

06:06 But this number is not 3 .14.

06:18 And so the answer is somewhere in between.

06:32 And i guess to summarize the explanation, it is not a multiple of pie.

06:49 Looking at what we have here for constructive and destructive interference, if it's an even multiple of pie, it'll be constructive, and if it's an odd multiple of pie, it would be destructive.

07:02 But this is not a multiple of pie at all...

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