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Ch. 35 - Diffraction
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 35 - DiffractionProblem 68
Chapter 34, Problem 68

You want to design a spy satellite to photograph license plate numbers. Assuming it is necessary to resolve points separated by 2 cm with 550-nm light, and that the satellite orbits at a height of 130 km, what minimum lens aperture (diameter) is required?

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Understand the problem: The goal is to determine the minimum lens aperture (diameter) required for a spy satellite to resolve two points separated by 2 cm using light of wavelength 550 nm. This involves the concept of diffraction and the Rayleigh criterion for resolution.
Apply the Rayleigh criterion for resolution: The angular resolution θ is given by the formula θ = 1.22 * (λ / D), where λ is the wavelength of light, and D is the aperture diameter. Rearrange this formula to solve for D: D = 1.22 * (λ / θ).
Relate angular resolution θ to the physical separation of the points and the distance to the satellite: θ = s / h, where s is the separation between the points (2 cm = 0.02 m), and h is the height of the satellite (130 km = 130,000 m). Substitute these values into the formula for θ.
Substitute the values for θ and λ into the rearranged Rayleigh criterion formula: Use λ = 550 nm = 550 × 10⁻⁹ m and the calculated θ from the previous step to find D. Ensure all units are consistent (meters).
Simplify the expression to calculate the minimum aperture diameter D. This will give the required lens diameter for the satellite to resolve the points separated by 2 cm at the given height.

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Key Concepts

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

Diffraction Limit

The diffraction limit is a fundamental concept in optics that describes the smallest detail that can be resolved by an optical system due to the wave nature of light. It is determined by the wavelength of light used and the aperture size of the lens. For a given wavelength, a larger aperture allows for better resolution, enabling the system to distinguish between closely spaced points.
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Diffraction

Rayleigh Criterion

The Rayleigh criterion provides a formula to calculate the minimum resolvable detail in an optical system. It states that two point sources are considered resolvable when the central maximum of one diffraction pattern coincides with the first minimum of another. This criterion is essential for determining the required aperture size to resolve specific distances, such as the 2 cm separation in the question.

Aperture Diameter

The aperture diameter of a lens is the effective opening through which light enters the optical system. A larger aperture allows more light to enter and improves resolution, while a smaller aperture can lead to diffraction effects that limit resolution. In satellite design, calculating the minimum aperture diameter is crucial for ensuring that the satellite can capture clear images of objects, such as license plates, from a significant distance.
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Related Practice
Textbook Question

A diffraction grating has 15,000 rulings in its 1.9 cm width. Determine (a) its resolving power in first and second orders, and (b) the minimum wavelength resolution (∆λ) it can yield for λ = 410 nm.

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Textbook Question

What is the highest spectral order that can be seen if a grating with 6800 slits per cm is illuminated with 633-nm laser light? Assume normal incidence.

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Textbook Question

(II) X-rays of wavelength 0.138 nm fall on a crystal whose atoms, lying in planes, are spaced 0.315 nm apart. At what angle Φ (relative to the surface, Fig. 35–28) must the X-rays be directed if the first diffraction maximum is to be observed?

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Textbook Question

A slit of width D = 22 μm is cut through a thin aluminum plate. Light with wavelength λ = 620nm passes through this slit and forms a single-slit diffraction pattern on a screen a distance ℓ = 2.0 m away. Defining x to be the distance between the two first minima on either side of the center in this diffraction pattern ( m = +1 and m = -1), find the change ∆x in this distance when the temperature T of the metal plate is changed by an amount ∆T = 55 C°. [Hint: Since λ ≪ D, the first minima occur at a small angle.]

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Textbook Question

X-rays of wavelength 0.10 nm fall on a microcrystalline powder sample. The sample is located 15 cm from a photographic sensor. The crystal structure of the sample has an atomic spacing of 0.22 nm. Calculate the radii of the diffraction rings corresponding to first- and second-order scattering. Note in Fig. 35–28 that the X-ray beam is deflected through an angle 2Φ.

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Textbook Question

(II) (a) Suppose for a conventional X-ray image that the X-ray beam consists of parallel rays. What would be the magnification of the image? (b) Suppose, instead, that the X-rays come from a point source (as in Fig. 35–31) that is 15 cm in front of a human body which is 25 cm thick, and the film is pressed against the person’s back. Determine and discuss the range of magnifications that result.


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