Two radio antennas separated by 300 m as shown in Figure P 37.3 simultancously broadcast identic

Two radio antennas separated by 300 m as shown in Figure P...

Two radio antennas separated by 300 $\mathrm{m}$ as shown in Figure $\mathrm{P} 37.3$ simultancously broadcast identical signals at the same wavelength. A radio in a car traveling due north receives the signals. (a) If the car is at the position of the second maximum, what is the wavelength of the signals? (b) How much farther must the car travel to encounter the next minimum in reception? (Note: Do not use the small-angle approximation in this problem.)

Two radio antennas separated by 300 $\mathrm{m}$ as shown in Figure $\mathrm{P} 37.3$ simultancously broadcast identical signals at the same wavelength. A radio in a car traveling due north
receives the signals. (a) If the car is at the position of the second maximum, what is the wavelength of the signals? (b) How much farther must the car travel to encounter the next minimum in reception? (Note: Do not use the small-angle approximation in this problem.)

Key Concepts

Wave Interference

This concept describes how two or more waves overlapping in space combine to form a new wave pattern. The interference can result in regions of higher amplitude (constructive interference) or lower amplitude (destructive interference). Understanding this concept is crucial to analyzing scenarios where multiple sources, such as radio antennas, emit waves that meet and overlap.

Path Difference

Path difference refers to the difference in distances that waves travel from their respective sources to a common point. This difference causes a phase shift between the waves, which directly determines whether they will interfere constructively (adding up) or destructively (canceling out). It is a key parameter in determining the locations of maxima and minima in an interference pattern.

Constructive and Destructive Interference Conditions

These conditions provide the criteria for when the superposed waves will be in phase (constructive interference) or out of phase (destructive interference). Constructive interference occurs when the path difference equals an integer multiple of the wavelength, while destructive interference occurs when the path difference equals an odd multiple of half wavelengths. These conditions form the foundation for analyzing and predicting the interference patterns from coherent sources.

Geometric Analysis in Wave Interference

In problems involving wave interference where approximations (like the small-angle approximation) cannot be used, careful geometric analysis using precise trigonometric relationships is required. This involves calculating exact distances and angles to determine accurate path differences between waves from different sources, ensuring the correct identification of interference maxima and minima.

Transcript

00:01 Question number three asks, if two radio antennas broadcast signals at the same wavelength, what wavelength do the signals have when there is a car at the second maximum? i'll illustrate this problem to do as best i can to replicate the diagram from the book.

00:20 Here are the two radio sources, the radio antennas.

00:26 Then out here we have the car.

00:30 At the point of the second maximum.

00:33 Now the car is traveling northwards along the y -axis.

00:39 The x -axis is the line drawn between the two antennas, and the y -axis aligns with the car's direction of travel.

00:50 Now the two radio antennas are at a distance d apart, and they're both broadcasting radio signals at the same wavelength.

00:59 The thing is, the signal from the antenna closer to the car will have less distance to travel than the one further away.

01:08 This path difference is important to us, and it's denoted by the greek letter delta.

01:15 Now, delta is equal to the difference of the two path lengths of the radio waves, r2 minus r1, but it can also be equated to the distance d times the sign of the angle theta.

01:31 Theta is the angular difference between the car's position on the y -axis and the x -axis.

01:39 Now, it's said that the car is at the second maximum, so that means it's at a place where there is constructive interference.

01:47 In that case, our equation d -sign theta will be equal to m times, lambda.

01:55 M denotes the maxima at which the car is located, which we found is equal to 2.

02:01 The car is at the second maximum.

02:04 Our distance d was given to us as 300 meters.

02:09 The car is said to be at a position where x is equal to a thousand meters, and its distance along the y -axis is 400 meters.

02:20 Now, we want to find the sign of our angle theta.

02:23 If we draw a line from the x -axis to the position of our car, the angle theta will be right here.

02:32 Important to our calculation for the sine of theta is the hypotenuse of this right triangle we just made.

02:40 I'm going to denote the hypotenuse by the letter h.

02:43 Sign theta will be equal to y over h.

02:48 The question is, how do we find the value for h? i said that we were working with a right triangle here, so we can use the pythagorean theorem to find h squared.

03:01 H squared will be equal to x squared plus y squared, and then if we take the square root of both sides, we can find h.

03:12 That means h will be equal to the square root of x squared plus y squared.

03:20 We already know our values for both x and y.

03:25 We're given them right up there.

03:27 So let's go ahead and plug our values in to find h.

03:32 H is going to equal the square root of 1 ,000 meters squared plus 400 meters squared.

03:44 When we do our calculation, we find that h is at a value of of about 1 ,077 meters.

03:55 So the sine of theta is going to be equal to 400 meters, our value for y, divided by our value for h of 1 ,077 meters.

04:09 When we divide the two, we find that sine theta is equal to .3714...

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