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Chapter 36: Q97P (page 1115)

A spy satellite orbiting at 160 km above Earth’s surface has a lens with a focal length of 3.6 m and can resolve objects on the ground as small as 30 cm. For example, it can easily measure the size of an aircraft’s air intake port. What is the effective diameter of the lens as determined by diffraction consideration alone? Assume λ=550nm.

Short Answer

Expert verified

The effective diameter of the lens is 0.36 m.

Step by step solution

01

Describe the diffraction by a circular aperture or a lens

The expression to calculate the effective diameter of the lens is given by,

sinθ=1.22λd

Here, λ is the wavelength, d is diameter, and θ is angle.

Let D be the distance between two objects, and L be the distance between Earth and satellite. Then,

Lθ=Dθ=DL

From the above equation,

θ=1.22λdDL=1.22λdd=1.22λLD ….(1)

02

Determine the effective diameter of the lens

Substitute 550×10-9m for λ, 160×103m for L, and 0.30m for D in equation (1).

d=1.22550×10-9m160×103m0.30m=0.36m

Therefore, the effective diameter of the lens is 0.36 m.

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Most popular questions from this chapter

(a) Figure 36-34a shows the lines produced by diffraction gratingsA and B using light of the same wavelength; the lines are of the same order and appear at the same angles θ. Which grating has the greater number of rulings? (b) Figure 36-34b shows lines of two orders produced by a single diffraction grating using light of two wavelengths, both in the red region of the spectrum. Which lines, the left pair or right pair, are in order with greater m? Is the center of the diffraction pattern located to the left or to the right in(c) Fig. 36-34a andd) Fig. 36-34b?

The distance between the first and fifth minima of a single slit diffraction pattern is 0.35mmwith the screen 40cmaway from the slit, when light of wavelength role="math" localid="1663070418419" 550nmis used. (a) Find the slit width. (b) Calculate the angle role="math" localid="1663070538179" θ of the first diffraction minimum.

How far from grains of red sand must you be to position yourself just at the limit of resolving the grains if your pupil diameter is 1.5mm, the grains are spherical with radius50μm , and the light from the grains has wavelength 650nm? (a) If the grains were blue and the light from them had wavelength 400nm, would the answer to (b) be larger or smaller?

Light at wavelength 589 nm from a sodium lamp is incident perpendicularly on a grating with 40,000 rulings over width 76 mm. What are the first-order (a) dispersion Dand (b) resolving power R, the second-order (c) Dand (d) R,and the third-order (e) Dand (f) R?

(a) How many rulings must a 4.00-cm-wide diffraction grating have to resolve the wavelengths 415.496 and 415.487 nm in the second order? (b) At what angle are the second-order maxima found?
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