Double-slit experiment with radio towers

[gigli]

Homework Statement

Two shortwave radio antennas broadcast identical, in-phase signals at the same frequency. The transmitters are 176.0 m north, and 176.0 m south of Western Ave, respectively, as shown (that is, they are separated by 352.0 m). Western Ave is 452.0 m long. Starting at the end of that avenue, a car drives north along Negundo Street, which lies parallel to the line joining the two radio antennas. The car first encounters a minimum in reception after it travels 124.0 m. What is the wavelength of the radio waves? Assume that the car and the transmitters are all at the same altitude.

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Homework Equations

2-slit interference for dark fringes (minimums)
sin[tex]\theta[/tex]=[(m+.5)[tex]\lambda[/tex]]/d

The Attempt at a Solution

First encounters a minimum of reception means the first dark spot from the midpoint. Therefore m, the fringe number=0.
Solving for the angle of separation between the first dark spot at 124m and the midpoint. We have two sides of a triangle so arctan(124/452)= 15.3408908 degrees

sin(15.3408908)=[(.5)[tex]\lambda[/tex]]/352
2*sin(15.3408908)*352=[tex]\lambda[/tex]=186.251202 m

However this is incorrect! Any help you can give me would be excellent!




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[alphysicist]

Homework Statement

Two shortwave radio antennas broadcast identical, in-phase signals at the same frequency. The transmitters are 176.0 m north, and 176.0 m south of Western Ave, respectively, as shown (that is, they are separated by 352.0 m). Western Ave is 452.0 m long. Starting at the end of that avenue, a car drives north along Negundo Street, which lies parallel to the line joining the two radio antennas. The car first encounters a minimum in reception after it travels 124.0 m. What is the wavelength of the radio waves? Assume that the car and the transmitters are all at the same altitude.

http://www.instantimagehosting.com/storage/Untitled_6.jpg

Homework Equations

2-slit interference for dark fringes (minimums)
sin[tex]\theta[/tex]=[(m+.5)[tex]\lambda[/tex]]/d

The Attempt at a Solution

First encounters a minimum of reception means the first dark spot from the midpoint. Therefore m, the fringe number=0.
Solving for the angle of separation between the first dark spot at 124m and the midpoint. We have two sides of a triangle so arctan(124/452)= 15.3408908 degrees

sin(15.3408908)=[(.5)[tex]\lambda[/tex]]/352
2*sin(15.3408908)*352=[tex]\lambda[/tex]=186.251202 m

However this is incorrect! Any help you can give me would be excellent!




http://www.instantimagehosting.com/storage/Untitled_6.jpg

The condition for destructive interference here is that the path length difference be related to the wavelength by:

[tex]
\Delta L= \left(m+\frac{1}{2}\right)\lambda
[/tex]
where [itex]\Delta L=L_1-L_2[/itex]. So on the left hand side you find the distance from one source to the receiver, find the distance from the other source to the receiver, and then find the difference between those two paths.

In cases where the distance between the sources is much smaller than the distance to the receiver (like in Young's experiment), there is an approximation that can be used:

[tex]
\Delta L = d \sin\theta
[/tex]

This is what you were using; however, for this to be valid, you would need the distance between the transmitters to be much smaller than the length of Western Avenue. Since this is not the case, I believe you would need to use the original formulation.

 

[baba89]

The double-slit experiment with radio towers is a fascinating application of the principles of wave interference. In this experiment, two shortwave radio antennas are broadcasting identical, in-phase signals at the same frequency. The antennas are separated by a distance of 352.0 m, and a car is driving along a road parallel to the line connecting the two antennas. The car first encounters a minimum in reception after traveling 124.0 m from the end of Western Ave.

To find the wavelength of the radio waves, we can use the equation for 2-slit interference for dark fringes (minimums): sinθ = [(m+0.5)λ]/d. In this case, m = 0 since we are looking at the first dark spot from the midpoint. We can also calculate the angle of separation between the first dark spot and the midpoint using the given values: arctan(124/452) = 15.3408908 degrees.

Substituting these values into the equation, we get: sin(15.3408908) = [(0.5)λ]/352. Rearranging, we can solve for the wavelength: λ = (2*sin(15.3408908)*352) = 186.251202 m.

However, this answer does not match the given wavelength of 176.0 m in the problem. This could be due to several factors, such as the altitude of the car and transmitters not being exactly the same, or the presence of other interfering signals. It is important to consider these factors when conducting experiments and analyzing results.