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Ch 36: Diffraction
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 36: DiffractionProblem 39
Chapter 35, Problem 39

Two satellites at an altitude of 1200 km are separated by 28 km. If they broadcast 3.6 cm microwaves, what minimum receiving-dish diameter is needed to resolve (by Rayleigh’s criterion) the two transmissions?

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Understand the problem: The question involves resolving two sources of electromagnetic waves (microwaves) using Rayleigh's criterion. Rayleigh's criterion states that the minimum angular resolution θ is given by θ = 1.22 * (λ / D), where λ is the wavelength of the wave, and D is the diameter of the receiving dish. The goal is to find the minimum diameter D required to resolve the two sources.
Convert the given values into consistent units: The wavelength λ is given as 3.6 cm, which should be converted to meters (λ = 0.036 m). The separation between the satellites is 28 km (28,000 m), and the altitude of the satellites is 1200 km (1,200,000 m).
Calculate the angular separation θ between the two satellites: Using the small-angle approximation, θ ≈ s / d, where s is the separation between the satellites (28,000 m) and d is the distance from the observer to the satellites (1,200,000 m). Substitute these values to find θ.
Rearrange Rayleigh's criterion formula to solve for the diameter D: From θ = 1.22 * (λ / D), rearrange to get D = 1.22 * (λ / θ). Substitute the values of λ and θ into this formula.
Perform the calculation to determine the minimum diameter D of the receiving dish. Ensure that all units are consistent and the result is expressed in meters.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rayleigh's Criterion

Rayleigh's criterion is a formula used to determine the minimum angular resolution of an optical system, such as a telescope or antenna. It states that two point sources are resolvable when the central maximum of one diffraction pattern coincides with the first minimum of another. This criterion is crucial for understanding how closely spaced objects can be distinguished based on the wavelength of the signal and the diameter of the receiving dish.

Diffraction Limit

The diffraction limit refers to the fundamental limit on the resolution of an imaging system due to the wave nature of light or other electromagnetic waves. It is influenced by the wavelength of the signal and the aperture size of the receiving dish. A larger dish can collect more waves and reduce the effects of diffraction, allowing for better resolution of closely spaced signals.
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Diffraction

Antenna Gain

Antenna gain is a measure of how well an antenna converts input power into radio waves in a specified direction compared to a standard antenna. It is important for understanding how effectively a receiving dish can capture signals from satellites. Higher gain antennas can focus on narrower beams, improving the ability to resolve signals from closely spaced sources, which is essential in this scenario.
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Example 2
Related Practice
Textbook Question

If the planes of a crystal are 3.50 Å (1 Å = 10-10 m = 1 Ångstrom unit) apart, what wavelength of electromagnetic waves is needed so that the first strong interference maximum in the Bragg reflection occurs when the waves strike the planes at an angle of 22.0°, and in what part of the electromagnetic spectrum do these waves lie?

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Textbook Question

A wildlife photographer uses a moderate telephoto lens of focal length 135 mm and maximum aperture f/4.00 to photograph a bear that is 11.5 m away. Assume the wavelength is 550 nm. (a) What is the width of the smallest feature on the bear that this lens can resolve if it is opened to its maximum aperture? (b) If, to gain depth of field, the photographer stops the lens down to f/22.0, what would be the width of the smallest resolvable feature on the bear?

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Textbook Question

If you can read the bottom row of your doctor’s eye chart, your eye has a resolving power of 1 arcminute, equal to 1/60 degree. If this resolving power is diffraction-limited, to what effective diameter of your eye’s optical system does this correspond? Use Rayleigh’s criterion and assume λ = 550 nm.

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Textbook Question

The VLBA (Very Long Baseline Array) uses a number of individual radio telescopes to make one unit having an equivalent diameter of about 8000 km. When this radio telescope is focusing radio waves of wavelength 2.0 cm, what would have to be the diameter of the mirror of a visible-light telescope focusing light of wavelength 550 nm so that the visible-light telescope has the same resolution as the radio telescope?

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Textbook Question

(a) What is the wavelength of light that is deviated in the first order through an angle of 13.5° by a transmission grating having 5000 slits/cm? (b) What is the second-order deviation of this wavelength? Assume normal incidence.

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Textbook Question

A laser beam of wavelength λ = 632.8 nm shines at normal incidence on the reflective side of a compact disc. (a) The tracks of tiny pits in which information is coded onto the CD are 1.60 μm apart. For what angles of reflection (measured from the normal) will the intensity of light be maximum? (b) On a DVD, the tracks are only 0.740 μm apart. Repeat the calculation of part (a) for the DVD.

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