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Chapter 35: Problem 2

Two radio antennas \(A\) and \(B\) radiate in phase. Antenna \(B\) is \(120 \mathrm{~m}\) to the right of antenna \(A .\) Consider point \(Q\) along the extension of the line connecting the antennas, a horizontal distance of \(40 \mathrm{~m}\) to the right of antenna \(B .\) The frequency, and hence the wavelength, of the emitted waves can be varied. (a) What is the longest wavelength for which there will be destructive interference at point \(Q ?\) (b) What is the longest wavelength for which there will be constructive interference at point \(Q ?\)

Short Answer

Expert verified
The longest wavelength for destructive interference at point Q is 80 m and the longest wavelength for constructive interference at point Q is 40m.

Step by step solution

01

Understand the Scenario

There are two radio antennae A and B with a distance of 120 m between them. Point Q is further 40 m from antenna B along the line connecting the two antennae. This means the total distance from A to Q is 160 m. We need to determine the longest wavelengths that cause destructive and constructive interferences at point Q. For destructive interference, the additional distance travelled by the wave from antenna B compared to antenna A must be an odd multiple of half the wavelength. For constructive interference, the additional distance travelled by the wave from antenna B compared to antenna A must be an integer multiple of the wavelength.
02

Determine Longest Wavelength for Destructive Interference

For destructive interference, the path difference must be an odd multiple of half the wavelength i.e. (2n - 1) x (\(位/2\)) where n is a positive integer. Given the path difference is 40 m, this gives us the equation: (2n - 1) x (\(位/2\)) = 40. For the longest wavelength, n should be minimum, which is 1. Solving the equation with n = 1, we obtain \(位 = (40/0.5)= 80 m\)
03

Determine Longest Wavelength for Constructive Interference

For constructive interference, the path difference must be a multiple of the wavelength, i.e. n x \(位\), where n is a positive integer. Given the path difference is 40 m, this gives us the equation: n x \(位\) = 40. For the longest wavelength, n should be minimum, which is 1. Solving the equation with n = 1, we obtain \(位\) = 40m.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radio Waves
Radio waves are a form of electromagnetic radiation used for communication purposes, such as radio and television broadcasting.
They travel at the speed of light and can propagate over long distances. Their wavelength can range from one millimeter to several kilometers.
  • Properties: Radio waves are low-frequency and can penetrate through various materials.
  • Usage: Antennas transmit and receive radio waves after converting them into electrical signals.
  • Wavelength: The wavelength determines the wave's frequency, meaning how many wave peaks pass a given point in one second.
In the context of the exercise, two antennas are emitting radio waves. The phase (timing) of these waves is important when considering interference, where waves can either add together or cancel each other out based on their relative phase and wavelength.
Constructive Interference
Constructive interference occurs when two waves meet and combine to form a wave with higher amplitude.
This happens when the crest of one wave aligns perfectly with the crest of another, leading to their energies adding up to amplify the wave.
  • Condition: The path difference between the waves must be an integer multiple of the wavelength, i.e., n x \(位\), where \(位\) is the wavelength and n is any positive integer.
  • Resulting Amplitude: The resulting wave is stronger due to the superimposed peaks.
In the exercise's scenario, at point Q, constructive interference occurs when the path difference between the waves from both antennas is a whole multiple of the wavelength. The challenge was to find the longest wavelength for which this happens, and it turned out to be 40 m when the waves constructively interfere at point Q.
Destructive Interference
Destructive interference takes place when waves meet in such a manner that their amplitudes cancel each other out.
This occurs when the crest of one wave aligns with the trough of another.
  • Condition: The path difference must be an odd multiple of half-wavelength, i.e., \((2n - 1) \times (\frac{位}{2})\).
  • Canceling Amplitude: The waves diminish or completely nullify each other, resulting in minimal or zero amplitude.
In the context of the problem, destructive interference at point Q is achieved with a path difference of 40 m when the wave from antenna B travels 40 m further than from antenna A. The longest wavelength for destructive interference, when \(n = 1\), is found to be 80 m, indicating that the waves interfere destructively at point Q.

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Most popular questions from this chapter

A plastic film with index of refraction 1.70 is applied to the surface of a car window to increase the reflectivity and thus to keep the car's interior cooler. The window glass has index of refraction \(1.52 .\) (a) What minimum thickness is required if light of wavelength \(550 \mathrm{nm}\) in air reflected from the two sides of the film is to interfere constructively? (b) Coatings as thin as that calculated in part (a) are difficult to manufacture and install. What is the next greater thickness for which constructive interference will also occur?

Coherent light of wavelength \(500 \mathrm{nm}\) is incident on two very narrow and closely spaced slits. The interference pattern is observed on a very tall screen that is \(2.00 \mathrm{~m}\) from the slits. Near the center of the screen the separation between two adjacent interference maxima is \(3.53 \mathrm{~cm}\). What is the distance on the screen between the \(m=49\) and \(m=50\) maxima?

Jan first uses a Michelson interferometer with the \(606 \mathrm{nm}\) light from a krypton-86 lamp. He displaces the movable mirror away from him, counting 818 fringes moving across a line in his field of view. Then Linda replaces the krypton lamp with filtered \(502 \mathrm{nm}\) light from a helium lamp and displaces the movable mirror toward her. She also counts 818 fringes, but they move across the line in her field of view opposite to the direction they moved for Jan. Assume that both Jan and Linda counted to 818 correctly. (a) What distance did each person move the mirror? (b) What is the resultant displacement of the mirror?

Coherent light with wavelength \(450 \mathrm{nm}\) falls on a pair of slits. On a screen \(1.80 \mathrm{~m}\) away, the distance between dark fringes is \(3.90 \mathrm{~mm} .\) What is the slit separation?

The index of refraction of a glass rod is 1.48 at \(T=20.0^{\circ} \mathrm{C}\) and varies linearly with temperature, with a coefficient of \(2.50 \times 10^{-5} / \mathrm{C}^{\circ} .\) The coefficient of linear expansion of the glass is \(5.00 \times 10^{-6} / \mathrm{C}^{\circ} .\) At \(20.0^{\circ} \mathrm{C}\) the length of the rod is \(3.00 \mathrm{~cm}\) A Michelson interferometer has this glass rod in one arm, and the rod is being heated so that its temperature increases at a rate of \(5.00 \mathrm{C}^{\circ} / \mathrm{min} .\) The light source has wavelength \(\lambda=589 \mathrm{nm},\) and the rod initially is at \(T=20.0^{\circ} \mathrm{C}\). How many fringes cross the field of view each minute?
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